Parallelism and Adaptivity in Student-Teacher Witnessing

P arallelism and Adapti vity in Student-T eacher W itnessing Ond ˇ rej Je ˇ zil * Faculty of Mathematics and Physics Charles Univ ersity Dimitrios Tsintsilidas † Department of Computer Science Univ ersity of W arwick February 2026 Abstract Student-T eacher games are a model of computation where a computationally restricted student tries to ﬁnd a string satisfying some refutable property , and e very time the student outputs a candidate answer an all-kno wing teacher tries to refute it if possible. These games are a classical computational model for the witnessing of ∀∃∀ -formulas in bounded arithmetic by the well-kno wn result of Kraj ´ ı ˇ cek, Pudl ´ ak and T akeuti [ KPT91 ]. W e introduce subclasses of total search problems in the polynomial hierarchy which are characterized by the number of rounds and the number of candidate answers per round a student from a related lev el of the polynomial hierarchy would need to solve the giv en problem. Our main results are as follows: (a) W e ﬁnd theories of bounded arithmetic whose ∀∃ Π b i -consequences are witnessed by Student- T eacher games with a giv en amount of adaptivity and parallelism. As a consequence, we sho w that assuming NP ⊆ P / p oly the theories PV 1 , PV 1 + BB (Σ b 1 ) , PV 1 + LLIND ( s Σ b 1 ) , PV 1 + LLIND (Σ b 1 ) and S 1 2 , are all distinct. This extends the results of Zambella [ Zam96 ], Cook and Thapen [ CT06 ] and Garl ´ ık [ Gar15 ] who separated each of the pairs in this sequence under stronger or seemingly in- comparable assumptions. (b) Generalizing the pre vious to all le vels of polynomial hierarchy and assuming Σ p i +1 ⊆ ∆ p i +1 / p oly , we obtain an analogous result for T i 2 . Thus, under plausible complexity theoretic assumptions, we giv e a solution to an open problem of Buss and Ressayre [ Bus85 , CK93 ] regarding the strength of the BB (Σ b i +1 ) scheme, and another problem of Pollett [ Pol97 ] regarding the strength of the schemes LLIND (Σ b i +1 ) and LLIND ( s Σ b i +1 ) . (c) W e revisit two major unconditional unprovability results for PV 1 . Namely the unprov ability of circuit upper bounds of Kraj ´ ı ˇ cek and Oliv eira [ K O17 ], which we sho w holds for the theory PV 1 + BB (Σ b 1 ) and the unprov ability of strong co-nondeterministic circuit lower -bounds of Pich and Santhanam [ PS21 ], which we show holds for the theory PV 1 + LLIND ( s Σ b 1 ) . It follows that both of these unprov ability results hold simultaneously for a speciﬁc theory which extends PV 1 , and this theory is a proper extension of PV 1 assuming NP ⊆ P / p oly . (d) As a technical tool, we b uild upon the result of Kraj ´ ı ˇ cek, Pudl ´ ak and Sgall [ KPS90 ] and assuming Σ p i +1 ⊆ ∆ p i +1 / p oly we separate different classes of TF Σ p i +2 which can be solved in a Student- T eacher game with r -many rounds (adaptivity) and q -many candidate answers per round (paral- lelism). Notably , the addition of a single round cannot be substituted by polynomial blow-up in the number of parallel answers. W e also provide a general witnessing theorem for ∀∃∀ -consequences of ﬁrst-order theories, which we use as a base result for our other witnessing theorems. * Email: ondrej.jezil@email.cz † Email: dimitrios.tsintsilidas@warwick.ac.uk 1 Contents 1 Introduction 3 1.1 Context and Moti v ation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Our results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.1 Student–T eacher Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.2 W itnessing with Student-T eacher Games . . . . . . . . . . . . . . . . . . . . . . . 8 1.2.3 Separations of Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2.4 Conditional Solutions to Open Problems in Bounded Arithmetic . . . . . . . . . . . 11 1.2.5 Unprov ability Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.3 Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.4 Org anisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2 Preliminaries 14 3 Student-T eacher Games 17 3.1 Adapti vity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.2 Parallelism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4 Witnessing Theorems 21 4.1 General W itnessing Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.2 Sharply Bounded Replacement Scheme BB (Σ b j , b ) . . . . . . . . . . . . . . . . . . . . . . 22 4.3 Length b -induction LIND (Σ b j , b ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 5 Separations of Theories 27 6 Unprov ability Results 29 6.1 Unprov ability of circuit upper bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 6.2 Unprov ability of a v erage-case circuit lo wer bounds . . . . . . . . . . . . . . . . . . . . . . 31 6.3 Unprov ability of both in a single theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 A Function Calculations 37 A.1 Transformation of Input Sizes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 A.2 Growth of Sublinear Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2 1 Intr oduction 1.1 Context and Motivation Bounded Arithmetic refers to weak theories of arithmetic that were introduced to study the relation of computational complexity with mathematical logic. The theories in Buss’ hierarchy are the most used theories for this goal, since they closely relate and formalise reasoning with functions in the corresponding class of the polynomial hierarchy [ Bus85 ]. In fact, the theories S i 2 , T i 2 of the i th lev el of the hierarchy are axiomatisable by dif ferent axiom schemes, length induction ( LIND ) and usual induction ( IND ), respecti v ely , on predicates from the class Σ p i (in logic as bounded formulas Σ b i ). At the same time, by Buss’ s theorem all the functions with Σ b i graphs which are prov ably total in the theory S i 2 are exactly the functions in the class FP Σ p i − 1 . By their deﬁnition, it is easy to prov e that S i 2 ⊆ T i 2 , and as expected by the polynomial hierarchy , we also hav e T i 2 ⊆ S i +1 2 , thus getting an increasing sequence of theories. Another widely used Bounded Arithmetic theory is PV 1 , which is placed at the bottom of Buss’ s hierar - chy . This theory was introduced by Stephen Cook [ Coo75 ] as the equational theory PV , based on Cobham’ s axioms for deﬁning symbols for all FP functions [ Cob65 ], and later extended to ﬁrst-order as a fragment of Peano Arithmetic [ KPT91 ] with open formula ( P -time predicate) induction. PV 1 is prov en to be equi v alent to T 0 2 , if we consider an extended language for the latter [ Je ˇ r06 ], but they are also equiv alent theories if we get T 0 2 on the language of PV . The question if Buss’ s hierarchy is proper was addressed very early in the literature. The separations between these theories rely on a fundamental tool of Bounded Arithmetic, the witnessing theorems. These theorems show that from a proof of a suitable existential statement in a gi ven theory , one can extract a computational procedure that “witnesses” the existential quantiﬁer . In [ KPT91 ], a ne w witnessing theorem is de v eloped, which was later called KPT theorem, after the authors, in order to sho w that the inclusion T i 2 ⊆ S i +1 2 (or PV 1 ⊆ S 1 2 ) is proper , under the assumption that Σ p i +1 ⊆ ∆ p i +1 / p oly . In general, if the polynomial hierarchy does not collapse, then Buss’ s hierarchy does not collapse, as well. The case of S i 2 versus T i 2 was addressed in [ Kra93 ], where another witnessing theorem is used, and the separation is achie ved under the assumption that P Σ p i [ O (log n )] (logarithmic number of Σ p i -oracle queries) is not equal to ∆ p i +1 (which is P Σ p i with unlimited number of queries). The KPT theorem is particularly interesting, because the computational procedure extracted by a proof is represented by an interacti ve game, called Student-T eacher Game. In this game, the Student tries to witness the existential quantiﬁer of a ∃∀ -formula and sends their attempts to the T eacher , who either accepts if the Student guessed the correct answer or otherwise, pro vides a countere xample, which corresponds to the uni versal quantiﬁer . The Student then can use the countere xamples adapti vely to guess a better answer . The KPT theorem asserts that if the statement is prov able in the theory PV 1 or some theory T i 2 , there is a Student represented by uniform functions in the corresponding level of the polynomial hierarchy that succeeds in a constant number of rounds. This framework of Student-T eacher Games has been prov ed useful for other applications, too, such as unprov ability results. For example, two important unprovability results are that for any k ∈ N , PV 1 ⊬ P ⊆ SIZE [ n k ] due to Kraj ´ ı ˇ cek-Oli veira [ KO17 ], and also that PV 1 ⊬ NSubExp ⊆ Avg - coNSIZE [2 n δ ] 1 due to Pich–Santhanam [ PS21 ]. These results belong to a wider family of results that are concerned with the meta- mathematics of complexity theory [ Oli25 ], which has as its main goal to prov e kno wn results of algorithms and complexity in the weakest theories possible (e.g. PV 1 prov es the Cook-Le vin theorem and PCP theorem 1 This unprov ability statement can also be e xtended to a fragment of the theory APC 1 , b ut in this paper we only consider theories in Buss’ s hierarchy . 3 [ Pic15b ], T 2 2 prov es the circuit size hierarchy [ CKK + 25 ], etc.), or to establish the unprovability of open problem statements in these theories. It is worth noting that, by [ KPT91 ], the separation of the theories in Buss’ s hierarchy is itself equi v alent to the unpro v ability of certain open problem statements about the polynomial-time hierarchy . Another axiom considered in [ Bus85 ] is the bounded replacement axiom, which is also called collection or choice axiom, since it can be considered as a feasible analogue to the axiom of choice. It is useful because it provides a theory with a quantiﬁer e xchange property , where you can mov e all the bounded quantiﬁers of the formula outside the sharply bounded quantiﬁers (quantiﬁers with smaller scope). This is in analogy with the case of unbounded fragments of Peano Arithmetic, where the replacement axiom can change the order of unbounded and bounded quantiﬁers. In fact in this case, the theory axiomatised by the collection axiom on Σ i +1 formulas (with i + 1 unbounded quantiﬁers) is strictly between the theories axiomatised by induction on Σ i and Σ i +1 formulas [ PK78 ]. W e write that as I Σ i ⊊ B Σ i +1 ⊊ I Σ i +1 . W e e xpect the same to be true in the bounded case, too. As a matter of fact, Buss proved the analogous inclusions S i 2 ⊆ S 1 2 + BB (Σ b i +1 ) ⊆ S i +1 2 , where S 1 2 is used as a base theory and BB (Σ b i +1 ) denotes the bounded replacement axiom. Howe ver , it was left open whether these inclusions are proper . It was later proposed as a major open problem in Bounded Arithmetic by Buss and Ressayre (Problem 12 in Section 1.2 of the survey on open problems in Bounded Arithemetic, Proof Complexity and Fragments of Peano Arithmetic [ CK93 ]). The answer for the second part was gi v en by Zambella [ Zam96 ], who sho ws by a model theoretic argu- ment that S i +1 2 ⊆ T i 2 + BB (Σ b i +1 ) is equi v alent with S i +1 2 ⊆ T i 2 . Thus, he separated the bounded replacement axiom from the length induction axiom for the same class of formulas, under the same assumptions [ KPT91 ] uses to separate T i 2 and S i +1 2 . Nonetheless, the ﬁrst part, whether S i 2 ⊢ BB (Σ b i +1 ) was not considered, or the e ven stronger , whether T i 2 ⊢ BB (Σ b i +1 ) . This question has been open since then. The ﬁrst answer to wards that direction was gi v en by Cook and Thapen [ CT06 ]. Here, they prov e that under the hardness of factoring against probabilistic polynomial time, PV 1 ⊢ BB (Σ b 1 ) . Also, using similar model theoretic ideas with [ Zam96 ], they sho w a new witnessing theorem for the theory PV 1 + BB (Σ b 0 ) , where we get a version of the Student-T eacher Game with constant rounds and polynomially many parallel queries per round. Provided this witnessing theorem, Je ˇ zil prov ed the separation of PV 1 + BB (Σ b 0 ) from S 1 2 under the new cryptographic assumption that non-uniform polynomial-time algorithms cannot factor a constant fraction of all semiprimes [ Je ˇ z25 ]. Ho we ver , the separation from [ CT06 ] is only kno wn for the ﬁrst level of the hierarchy and cannot gen- eralise to all lev els, due to the nature of the factoring problem. Hence, more general hardness assumptions would be preferable to completely solv e the problem. Another problem, concerning theories between PV 1 and S 1 2 comes from the v ariant of length induction axiom, where length is replaced by double length ( LLIND ). It is not kno wn whether LLIND accepted for the bounded existential formulas in the language of PV prov es the suitable quantiﬁer e xchange property . The generalised version about the separation of theories axiomatised by LLIND (Σ b i +1 ) and, on the other hand, LLIND ( strict Σ b i +1 ) has been a long-standing open problem mentioned in the works of Pollett [ Pol97 , Pol99 , Pol18 ]. Only a partial answer to the problem was kno wn by Garl ´ ık, who sho ws by a model theoretic argument that on the ﬁrst le vel of the hierarchy the corresponding theories are actually separated, assuming the ex- istence of one-way permutations secure against polynomial-size circuits [ Gar16 ]. Nonetheless, it was still open to prov e this, under worst-case assumptions, or to generalise this to all le vels of Buss’ s hierarchy . 4 Our Contributions. In this paper , motiv ated by the aforementioned open problems on separations of theo- ries and the extensi v e use of the Student-T eacher witnessing in Bounded Arithmetic, we sho w the following: 1. Solutions to open problems about separations of theories: W e giv e conditional solutions for all le vels of Buss’ s hierarchy to the two aforementioned open problems about fundamental separations of theories in Bounded Arithmetic [ CK93 , Pol97 ]. The condition we use is the same as in [ KPT91 ] about the separation of theories T i 2 and S i 2 . Our results e xtend the results of [ CT06 , Gar15 ], who solved the two problems only on the ﬁrst le v el of the hierarchy using stronger or seemingly incomparable assumptions. 2. Generalised separation of theories: W e provide a bigger family of different theories between PV 1 and S 1 2 , which form hierarchies ov er variations of the length induction axioms and over the bounded replacement axioms. All these theories are separated under the assumption NP ⊆ P / poly (see Fig- ure 1 ). Again, these separations are generalised to all le v els of Buss’ s hierarchy , where the assumption becomes Σ p i +1 ⊆ ∆ p i +1 / p oly . 3. Lifting unprovability results: As an attempt towards Open Problems 5.1 and 5.8(d) in [ Oli25 ], we lift the unprov ability results PV 1 ⊬ P ⊆ SIZE [ n k ] of Kraj ´ ı ˇ cek-Oli veira [ K O17 ], and PV 1 ⊬ NSubExp ⊆ Avg − coNSIZE [2 n δ ] of Pich-Santhanam [ PS21 ] to the theories PV 1 + BB (Σ b 1 ) and PV 1 + LLIND ( s Σ b 1 ) , respecti vely . These theories, and also a theory in the intersection where both results hold, are strictly stronger than PV 1 , under NP ⊆ P / p oly , from the previous results. 4. Separations of Student-T eacher Games classes: Our main technical tool is the study of Student- T eacher Games. F or that, we classify total search problems in the polynomial hierarchy into new complexity classes we introduce, which are characterised by the number of rounds and the number of candidate solutions per round that a Student needs to solve the problem in the Student-T eacher Game manner . Then, building on the result of [ KPS90 ], under the assumption that Σ p i ⊆ ∆ p i / p oly , we separate these classes for both the direction of rounds and the direction of parallel queries. These separations will be the core for the separation of different theories, while the ﬁne-grained version of these classes will be the one that will improv e the unprov ability results. 5. Unifying witnessing theorems: In order to connect these ﬁne-grained Student-T eacher classes with the corresponding theories, we pro vide a model-theoretic proof that uniﬁes all the witnessing theo- rems of this kind. T o use this general result, we also pro vide the description of these classes in an appropriate theory . 1.2 Our results In what follo ws, we gi ve an o v ervie w of the main results from the paper . 1.2.1 Student–T eacher Games Our main technical tool is the computational frame work provided by Student-T eacher Games, where we establish lower bounds against different v ariants of them. W e start by considering them as a computational model for search problems in TFPH . W e adopt the deﬁnition of TFΣ p i + 2 for i ≥ 0 , from [ KKMP21 ]. 5 S 1 2 ≡ PV 1 + LIND (Σ b 1 ) polynomial rounds, constant queries PV 1 + BB (Σ b 1 ) constant rounds, polynomial queries PV 1 + LLIND ( s Σ b 1 ) polylog rounds, constant queries PV 1 + LIND ( s Σ b 1 , log log) double-log rounds, constant queries PV 1 + BB ( s Σ b 1 , log) constant rounds, polylog queries PV 1 + BB ( s Σ b 1 , log log) constant rounds, double-log queries PV 1 constant rounds, constant queries Figure 1: Partial order of theories between PV 1 and S 1 2 . There are two hierarchies, one with e xtensions axiomatised by length induction axioms (right), and one with extensions axiomatised by bounded replace- ment axioms (left). The le v els are for the bounds p oly , log , log log , etc. Under the hardness assumption NP ⊆ P / p oly , all these theories are distinct. The arro ws symbolise the only possible inclusions. Deﬁnition 1.1. A relation R ( x, y ) belongs to TFΣ p i + 2 if it is polynomially bounded and total (that is, for e very x there exists y such that ( x, y ) is in the relation), and there exist a predicate φ ∈ ∆ p i +1 and a polynomial p ( n ) such that R ( x, y ) ⇐ ⇒ ∀ z ∈ { 0 , 1 } p ( | x | ) φ ( x, y , z ) . W e now deﬁne the notion of a Student–T eacher Game and introduce the corresponding complexity classes. Deﬁnition 1.2 (Student–T eacher Game) . A TFΣ p i + 2 relation R ( x, y ) , speciﬁed by a polynomial-time pred- icate φ ∈ ∆ p i +1 and a polynomial p ( n ) , can be solved by a Student-T eacher Game with r ( n ) rounds and q ( n ) parallel queries per round if, for n = | x | , there exists a uniform function f ∈ FP Σ p i , representing a Student, that takes as input the number x of size n , the index of the round i ∈ [ r ( n )] , and an additional input depending on the round, which has size ( i − 1) · p ( n ) · q ( n ) , and produces outputs y i 1 , y i 2 , . . . , y i q ( n ) , such that: ∀ z 1 = ( z 1 1 , . . . , z 1 q ) , . . . , z r = ( z r 1 , . . . , z r q )  ∃ j ∈ [ q ] φ ( x, f ( x, 1) j , z 1 j )  ∨  ∃ j ∈ [ q ] φ ( x, f ( x, 2 , z 1 ) j , z 2 j )  ∨ · · · ∨  ∃ j ∈ [ q ] φ ( x, f ( x, r , z 1 , . . . , z r − 1 ) j , z r j )  , where we abbreviate r := r ( n ) and q := q ( n ) , and we consider these as functions computable in polynomial time for the sake of well-deﬁnedness. The class S T Σ p i [ r ( n ) , q ( n )] consists of all TFΣ p i + 2 relations R ( x, y ) that can be solved by a Student- T eacher Game with r ( n ) rounds and q ( n ) parallel queries per round. The interpretation of this computational model is as an interaction between a Student trying to witness the existential quantiﬁer of the TFΣ p i + 2 problem and a T eacher that provides counterexamples for the ne xt uni versal quantiﬁer . Thus, the Student represented by the FP Σ p i function f , at ev ery round receives the input, 6 the number of the round and the corresponding counterexamples pro vided by the T eacher for their previous attempts, and computes q ( n ) new candidate witnesses. If the round is more than r ( n ) , then the Student must reject. Also, the T eacher’ s counterexamples can be checked by the Student using the Σ b i -oracle. For Student-T eacher Games with one query per round, we have the following foundational theorem from [ KPS90 ], where the frame work was ﬁrst introduced. Theorem 1.3 ([ KPS90 ]) . If NP ⊆ P / p oly , then for any sublinear , unbounded, increasing , polynomial-time function r ( m ) , S T [ r ( m ) , 1] ⊊ S T [ r ( m ) + 1 , 1] . The theorem can be easily generalised to S T Σ p i [ r ( m ) − 1 , 1] ⊊ S T Σ p i [ r ( m ) , 1] under the assumption Σ p i +1 ⊆ ∆ p i +1 / p oly . Howe ver , from this theorem, it is not clear where this po wer of the extra round is coming from; the adaptivity through the interaction between the Student and the T eacher or the number of counterexamples that the Student is able to use. Moti v ated by this question, we study the Student-T eacher Games with multiple (up to polynomially many) parallel queries per round and we e xtend the aforementioned result by establishing separations in both the r ound and query dimensions. That is both adapti vity through interaction and number of parallel queries (along the respecti ve countere xamples) can gi v e more po wer to the Student. Our ﬁrst result generalizes Theorem 1.3 by sho wing that the separation between r ( m ) and r ( m ) + 1 rounds persists ev en when the Student may issue an arbitrary polynomial number of parallel queries. This demonstrates that adaptivity acr oss r ounds is a crucial source of po wer in Student-T eacher interactions. Theorem 1.4. If Σ p i +1 ⊆ ∆ p i +1 / p oly , then for any sublinear , unbounded, increasing , polynomial-time func- tion r ( m ) and any unbounded, increasing , polynomial-time function 1 ≤ q ( m ) ≤ p oly ( m ) , S T Σ p i [ r ( m ) + 1 , 1] ⊆ S T Σ p i [ r ( m ) , q ( m )] . The idea for the proof is that there exists a hard search problems, which can be solved in r ( m ) + 1 rounds, b ut if the Student has one less round, ev en with many queries per round, they do not get enough information for the teacher . If the opposite is true, then we use this powerful Student to decide any Σ p i +1 problem with a ∆ p i +1 -algorithm that uses polynomial-size advice. The proof is quite technical and it is resumed for Section 3.1 . Our second result highlights the power gained from parallelism ; that is, from allo wing the Student to make multiple simultaneous queries to the T eacher . Essentially , each additional query , which results to a counterexample, combined with one extra adaptiv e round, allo ws the Student to solve strictly more problems. Theorem 1.5. If Σ p i +1 ⊆ ∆ p i +1 / p oly , then for any sublinear , unbounded, increasing , polynomial-time func- tion 2 ≤ q ( m ) ≤ m , S T Σ p i [2 , q ( m ) − 1] ⊊ S T Σ b i [2 , q ( m )] , and, mor e g enerally , for any sublinear , unbounded, incr easing functions 1 ≤ r 1 ( m ) , q 1 ( m ) , r 2 ( m ) , q 2 ( m ) , S T Σ p i [1 + r 1 ( m ) , q 1 ( m )] ⊆ S T Σ p i [1 + r 2 ( m ) , q 2 ( m )] whenever r 1 ( m ) q 1 ( m ) > r 2 ( m ) q 2 ( m ) . Note that in this theorem, we only consider Student-T eacher Games with more than one round, since otherwise there is no interaction with the T eacher . The idea of the proof is similar with Theorem 1.5 , b ut we use another kind of search problem. The full proof is in Section 3.2 . 7 1.2.2 Witnessing with Student-T eacher Games The original moti v ation for deﬁning Student–T eacher Games arises from the witnessing theorems of Bounded Arithmetic. The KPT Theorem, due to Kraj ´ ı ˇ cek, Pudl ´ ak, and T akeuti [ KPT91 ], is a Herbrand- type result for uni v ersal theories, which in our case applies to the theories PV 1 and univ ersal conserv ati ve extensions of T i 2 . W e ﬁrst recall the general version. Theorem 1.6 (General KPT Theorem) . Let T be a universal theory in a L , and let φ be a Σ 1 - L -formula. Suppose that T ⊢ ∀ x ∃ y ∀ z φ ( x, y , z ) . Then ther e e xists a ﬁnite sequence s 1 , . . . , s k of L -terms such that T ⊢ ∀ x, z 1 , . . . , z k  φ ( x, s 1 ( x ) , z 1 ) ∨ φ ( x, s 2 ( x, z 1 ) , z 2 ) ∨ · · · ∨ φ ( x, s k ( x, z 1 , . . . , z k − 1 ) , z k )  . The theory PV 1 deﬁned in [ KPT91 ] is a univ ersal theory whose terms denote polynomial-time functions. In a similar manner , the theories PV i +1 are universal theories whose terms are the FP Σ b i -functions, but they are conservati ve over T i 2 (under the same language they prove the same theorems), thus we do not distinguish them here. Combining the KPT theorem with the deﬁnition of Student-T eacher Games allows us to characterize the Σ b i +2 -consequences of T i 2 , or for PV 1 if i = 0 . Theorem 1.7. Let i ≥ 0 and T be PV 1 for i = 0 and T i 2 otherwise. Let φ be a Σ b i -formula 2 , and suppose that T ⊢ ∀ x ∃ y ∀ b z φ ( x, y , z ) . Then there exists a polynomial p ( n ) , given by the bound on ∀ b z , such that the relation R ( x, y ) 3 char acterized by φ and p ( n ) is total, and ther e exists a constant k ∈ N such that R ( x, y ) ∈ S T Σ p i [ k , 1] . The theories S i 2 are not uni versal, b ut b uilding on the KPT theorem we can obtain an analogous result. Theorem 1.8 ( S i 2 -analogue to KPT [ Kra92 , Pud92 ]) . Let φ be a Σ b i -formula 4 , and suppose that S i +1 2 ⊢ ∀ x ∃ y ∀ b z φ ( x, y , z ) . Then there exists a polynomial p ( n ) , given by the bound on ∀ b z , suc h that the r elation R ( x, y ) characterized by φ and p ( n ) is total, and ther e exists a polynomial r ( n ) such that R ( x, y ) ∈ S T Σ p i [ r ( n ) , 1] . As we see, the Σ b i +2 -consequences of the theories T i 2 correspond to total search problems in the class S T Σ p i [ O (1) , 1] , whereas the Σ b i +2 -consequences of S i +1 2 correspond to total search problems in the class S T Σ p i [ p oly ( n ) , 1] . Therefore, there is a fundamental dif ference between these theories. Follo wing the intermediate classes of Student-T eacher Games deﬁned abov e, we can also consider theo- ries between T i 2 and S i +1 2 whose Σ b i +2 -consequences are captured by the corresponding classes of Student- T eacher Games, thus having more ﬁne-grained separations of theories. 2 One can also apply the general KPT theorem to a ∃ Π b i -formula φ and with the quantiﬁer ∀ b z unbounded, obtaining a char- acterisation on a greater variety of consequences for T i 2 ; howe v er , in this work we restrict attention to Σ b i +2 -consequences only , which correspond to total problems in TFΣ p i + 2 , which enables us to classify them in the classes S T Σ b i [ r ( m ) , q ( m )] . It is noted that we allow the Student to be able to check the v alidity of the counterexamples in this deﬁnition of the Games. 3 The relation R ( x, y ) is guaranteed to be polynomially bounded by Parikh’ s Theorem [ Par71 ], which states that if there is a proof with an unbounded e xistential quantiﬁer, then a Bounded Arithmetic theory can also prov e the same theorem with the quantiﬁer being bounded. 4 Again, this theorem can be sho wn for φ ∈ Σ b i +1 and unbounded ∀ z , too. 8 It is known that T i 2 ⊆ S i +1 2 , but if we weaken the axioms of S i +1 2 to get weaker theories corresponding to weaker classes of Student-T eacher Games, it may be the case that these theories do not include T i 2 any more. That is why we work with T i 2 as the base theory , and deﬁne all other theories as its extensions. T o begin with, S i +1 2 can be regarded as an extension of T i 2 by the Length Induction Axiom Scheme for Σ b i +1 formulas, denoted LIND (Σ b i +1 ) . The formal deﬁnition of the axiom for some φ ∈ Σ b i +1 is LIND ( φ ) := ∀ x h ( φ (0) ∧ ∀ n < | x | ( φ ( n ) → φ ( n + 1))) → φ ( | x | ) i . This scheme is analogous to standard induction, but the induction is carried out on the length of the v ariable rather than on the v ariable itself. Since φ is Σ b i +1 , the axiom LIND ( φ ) is a ∀ Σ b i +2 sentence, which can be trivially witnessed in S T Σ b i [ p oly ( | x | ) , 1] . W e can easily see that if the Student starting from 0 tries to ﬁnd a contradiction in φ ( n ) → φ ( n + 1) step by step by acquiring the witnesses for the existential quantiﬁer for φ from the T eacher . W e can also deﬁne a variant where induction proceeds up to the double length of the number: LLIND ( φ ) := ∀ x h ( φ (0) ∧ ∀ n < || x || ( φ ( n ) → φ ( n + 1))) → φ ( || x || ) i . There are two cases to consider here. If φ is a strict Σ b i +1 formula (denoted s Σ b i +1 ), this means that all the sharply bounded quantiﬁers are after the bounded quantiﬁers (which will make the last part veriﬁable in polynomial time). In this case, this ∀ Σ b i +2 axiom can be witnessed in S T Σ p i [log( | x | ) , 1] , in the same w ay as in length induction. Ho we v er , if φ is not strict — i.e., it contains sharply bounded quantiﬁers preceding the outer bounded e xistential quantiﬁer — then polynomial number of parallel v ersions of the game are required to eliminate all the cases indicated by the sharply bounded quantiﬁers. This implies an upper bound for the witnessing of φ in the class S T Σ p i [ p olylog ( | x | ) , p oly ( | x | )] . W e see that these p oly ( | x | ) parallel queries are unnecessary in the case of LIND , as they can be simulated by the polynomially many rounds av ailable in that setting. This distinction between strictly Σ b i +1 and general Σ b i +1 corresponds to the exchange property for sharply bounded quantiﬁers guaranteed by Bounded Replacement Axiom Scheme ( BB (Σ b i +1 ) ). The BB axiom for a formula φ ∈ Σ b i +1 , is deﬁned as follo ws: BB ( φ ) := ∀ x, t h ( ∀ j ≤ | x | ∃ y ≤ t φ ( j, y )) → ( ∃ w ∀ j ≤ | x | φ ( j, w j )) i , where w encodes a sequence of length | x | with numbers less than t . When a theory includes the axiom BB (Σ b i +1 ) , one can sho w that for an y Σ b i +1 formula ψ , there exists an equiv alent strict Σ b i +1 formula ψ ′ , which is prov able in the theory . The axioms BB (Σ b i +1 ) are ∀ Σ b i +2 sentences that can be witnessed in S T Σ p i [ O (1) , p oly ( n )] . Moreov er , they are prov able in S i +1 2 and T i 2 + LLIND (Σ b i +1 ) [ All91 ], which aligns with the intuition we ha ve from the Student-T eacher Games. It is prov en in [ CT06 ] that the Σ b 2 -consequences of PV 1 + BB (Σ b 1 ) correspond to total search problems in S T [ O (1) , p oly ( n )] . The proof can be generalised to cov er the Σ b i +2 -consequences of T i 2 + BB (Σ b i +1 ) , which gi ve us total search problems in S T Σ p i [ O (1) , p oly ( n )] . Cook and Thapen also introduced ﬁne-grained v ariants of BB (Σ b 0 ) parametrised by dif ferent growth rates of lengths. For that we can use any sublinear, increasing function b ( x ) prov ably total in PV 1 , and we get the axiom: BB ( φ, b ) := ∀ x, t h ( ∀ j ≤ b ( | x | ) ∃ y ≤ t φ ( j, y )) → ( ∃ w ∀ j ≤ b ( | x | ) φ ( j, w j )) i . 9 They show that witnessing in PV 1 + BB (Σ b 0 , b ) corresponds to total search problems in S T [ O (1) , p oly ( b ( n ))] , and they separate all these different theories assuming the hardness of factoring against probabilistic algo- rithms. In this work, we also consider a ﬁne-grained version of the length induction axioms, denoted LIND (Σ b i +1 , b ) , and deﬁned as: LIND ( φ, b ) := ∀ x h  φ (0) ∧ ∀ n < b ( | x | ) ( φ ( n ) → φ ( n + 1))  → φ ( b ( | x | )) i . In this way , LLIND (Σ b i +1 ) is the same as LIND (Σ b i +1 , log ) and LIND (Σ b i +1 ) is the same as LIND (Σ b i +1 , p oly ) . W e aim to separate all the theories which extend T i 2 with these dif ferent axioms, by showing that their Σ b i +2 -consequences correspond to the natural Student-T eacher Game class. W e already kno w these for T i 2 , S i +1 2 and T i 2 + BB (Σ b i +1 ) . Ho we v er , in this work, using ideas from [ A vi02 , Tha02 , Kra92 ], we provide a generalised model-theoretic argument that uniﬁes all these witnessing theorems, and enables us to sho w the witnessing for all these extensions. This framework has the potential to be used in various situations to facilitate the proof of other witnessing theorems, too. Informally , we take a theory T which is an extension of T i 2 by an axiomatic scheme with formula complexity in Σ b i +2 . W e also suppose that the axiomatic scheme can be witnessed by an FP Σ b i -Student in a speciﬁc class of Student-T eacher Games that must be closed under deﬁnition by cases and ﬁnite composition. Then the Σ b i +2 -consequences of T can also be witnessed within the same class of Student-T eacher Games. The detailed exposition is in Section 4 . 1.2.3 Separations of Theories W e may thus consider all e xtensions of T i 2 by the different axiomatic schemes LIND ( s Σ b i +1 , b ) and BB ( s Σ b i +1 , b ) . Each one admits a witnessing theorem within the naturally corresponding S T Σ p i class. Ap- plying the ideas underlying Theorems 1.4 and 1.5 , we obtain the follo wing separations. Theorem 1.9. If Σ p i +1 ⊆ ∆ p i +1 / p oly , then for all incr easing PV functions b 1 ( x ) , b 2 ( x ) ≤ x , such that for any k ∈ N and for lar ge enough x , b 1 ( x ) ≥ b 2 ( x ) k , T i 2 + BB ( s Σ b i +1 , b 2 ) ⊢ BB ( s Σ b i +1 , b 1 ) , or equivalently , T i 2 + BB ( s Σ b i +1 , b 1 ) ⊆ T i 2 + BB ( s Σ b i +1 , b 2 ) . The above theorem is also prov ed in [ CT06 ], but only for the ﬁrst lev el, and under a cryptographic hardness assumption instead of a worst-case one. Theorem 1.10. If Σ p i +1 ⊆ ∆ p i +1 / p oly , then for all incr easing PV functions b 1 ( x ) , b 2 ( x ) ≤ x , such that for any k ∈ N and for lar ge enough x , b 1 ( x ) ≥ b 2 ( x ) k , T i 2 + LIND ( s Σ b i +1 , b 2 ) ⊢ LIND ( s Σ b i +1 , b 1 ) , or equivalently , T i 2 + LIND ( s Σ b i +1 , b 1 ) ⊆ T i 2 + LIND ( s Σ b i +1 , b 2 ) . Therefore, we have two hierarchies for the extensions of LIND and BB , and the lev els can be index ed by p oly , log , log log , . . . etc. W e also kno w that T i 2 + LIND ( s Σ b i +1 , b ) ⊢ BB ( s Σ b i +1 , b ) . by parametrising the proof of S i +1 2 ⊢ BB (Σ b i +1 ) (Theorem 2.9 ), which means that ev ery le vel of the LIND hierarchy pro v es the corresponding theory in the BB hierarchy . Ho we v er , this is the best we can ha v e by the two ﬁnal theorems: Theorem 1.11. If Σ p i +1 ⊆ ∆ p i +1 / p oly , then for all incr easing PV functions b ( x ) ≤ x , T i 2 + BB ( s Σ b i +1 ) ⊢ LIND ( s Σ b i +1 , b ) , or equivalently and T i 2 + LIND ( s Σ b i +1 , b ) ⊆ T i 2 + BB ( s Σ b i +1 ) . 10 Theorem 1.12. If Σ p i +1 ⊆ ∆ p i +1 / p oly , then for all incr easing PV functions b 1 ( x ) , b 2 ( x ) ≤ x , such that for any k ∈ N and for lar ge enough x , b 1 ( x ) ≥ b 2 ( x ) k , T i 2 + LIND ( s Σ b i +1 , b 2 ) ⊢ BB ( s Σ b i +1 , b 1 ) , or equivalently , T i 2 + BB ( s Σ b i +1 , b 1 ) ⊆ T i 2 + LIND ( s Σ b i +1 , b 2 ) The ﬁrst theorem shows that no theory in the BB hierarchy includes an y theory from the LIND hierarchy , and the second shows that the inclusions per le v el are the only ones possible. Therefore, an interesting consequence of that is that the theory T i 2 + LIND ( s Σ b i +1 , log ) is independent from T i 2 + BB (Σ b i +1 , p oly ) . The proofs for all the abov e theorems are applications of Theorems 1.4 and 1.5 , where we sho w that the sentence used for the separation is provable in the corresponding class. W e can see all these relations in Figure 1 , if we transfer them to the le vel i of Buss’ s hierarchy . 1.2.4 Conditional Solutions to Open Problems in Bounded Arithmetic The separations described in the previous section giv e some natural solutions to two open problems in Bounded Arithmetic regarding the po wer of the bounded collection axiom and the difference of double length induction for strict and non-strict formulas. 5 The difference that these problems hav e with the abov e separations is that the corresponding theories are deﬁned ov er a different base theory . Howe v er , the same results still hold. The assumption we use is, as always in the paper , Σ p i +1 ⊆ ∆ p i +1 / p oly , which is the best kno wn assumption for separations of theories in Buss’ s Hierarchy . Buss and Rassayre’ s Open Problem [ CK93 ]. As mentioned abov e, it was prov en in [ Bus85 ] that over the base theory S 1 2 , the axiomatic scheme BB (Σ b i +1 ) can be proven by the scheme LIND (Σ b i +1 ) , while it can prove LIND (Σ b i ) (see in the Preliminaries Theorems 2.8 to 2.10 ), which means we have S i 2 ⊆ S 1 2 + BB (Σ b i +1 ) ⊆ S i +1 2 . The problem is whether these theories can be separated. It is easy to see that under the usual assumption Σ p i +1 ⊆ ∆ p i +1 / p oly , which is the same as the one used to separate T i 2 from S i +1 2 , we can also sho w the desired separations. By Theorem 1.9 , for b 2 = 0 , we get that T i 2 ⊢ BB ( s Σ b i +1 , b ) for any b which is super-constant, which also implies that S i 2 ⊢ BB ( s Σ b i +1 , b ) . For b ( x ) = x , this directly sho ws that BB (Σ b i +1 ) ⊆ S i 2 , thus achieving the ﬁrst separation. For the second part, we can use Theorem 1.11 with b ( x ) = x , which gi ves us that T i 2 + BB (Σ b i +1 ) ⊬ LIND ( s Σ b i +1 ) , which also means, by weakening the base theory , that S 1 2 + BB (Σ b i +1 ) ⊬ LIND ( s Σ b i +1 ) . This separates the second part. Double-length induction on strict vs. non-strict Σ b 1 -f ormulas [ P ol97 ]. The theories concerned in this problem hav e the basic set of axioms which is used in Buss’ s theories, denoted by BASIC (see Preliminaries), but they extend it by two versions of double length induction. As we see in Theorem 2.9 , the length induction for strict formulas can prove the bounded collection axiom, which provides the sharply bounded quantiﬁer exchange property Theorem 2.7 . This means that length induction on strict formulas is equiv alent with length induction with formulas from the corresponding non-strict class. Ho we ver , this w as not known for double length induction; the problem is if double length induction on strict formulas can prove the version for non-strict ones. W e show that this is not true ( BASIC + LLIND ( s Σ b i ) ⊬ LLIND (Σ b i ) ). This is the case, since we already kno w that BASIC + LLIND (Σ b i ) ⊬ BB (Σ b i ) [ All91 ], and it cannot happen that BASIC + LLIND ( s Σ b i ) ⊬ 5 Amusingly in both problems the concerned theories are both denoted by the letter R . R i = S 1 2 + BB (Σ b i ) which was deﬁned already in Buss’ s thesis [ Bus85 ], and R i 2 = BASIC + LLIND ( Σ b 1 ) , which is deﬁned in the same manner as S i 2 [ Pol97 ]. W e do not use these names to av oid confusion. 11 BB (Σ b i ) , under the assumption that Σ p i +1 ⊆ ∆ p i +1 / p oly . If the latter was true, then we would also hav e PV 1 + LLIND ( s Σ b i ) ⊢ BB (Σ b i ) , which is contradictory with Theorem 1.12 . 1.2.5 Unprov ability Results In the ﬁnal section, we revisit two major unprov ability results in PV 1 — the unprov ability of circuit upper bounds by Kraj ´ ı ˇ cek–Oli veira [ K O17 ] and the unprov ability of average-case circuit lower bounds by Pich–Santhanam [ PS21 ] — and in vestigate whether their conclusions can be strengthened to prov ably stronger (under the complexity assumptions we use abov e) theories. It is an open problem if these unprov- ability results hold also for S 1 2 (Open Problems 5.1 and 5.8(d) in [ Oli25 ]) Both of these results use the KPT theorem and the Student-T eacher Game derived from it, to ﬁnd a contradiction. Hence, the question we are trying to answer is what is the strongest Student-T eacher Game class suf ﬁcient to obtain contradiction. For the former case, concerning the unprov ability of circuit upper bounds, we can see from the proof that having more than constant rounds breaks the ar gument of composing polynomial-time functions. For the latter case, it is not clear ho w to adapt the proof for the number of rounds which is polynomial in 2 n . W e were able to extend the unprov ability as follows (see Figure 2 for summary). S 1 2 Open problem whether unprov ability holds here PV 1 + BB (Σ b 1 ) Best theory known for the unprov ability of P ⊆ SIZE [ n k ] PV 1 + LLIND ( s Σ b 1 ) Best theory known for the unprov ability of NSubExp ⊆ Avg − coNSIZE [2 n δ ] PV 1 + BB ( s Σ b 1 , log) New best theory for both unprov ability results PV 1 Best unprov ability results known until no w Figure 2: Summary of the unprov ability results. Both results were pre viously known for the theory PV 1 , while we show them for PV 1 + BB (Σ b 1 ) and PV 1 + LLIND ( s Σ b 1 ) . W e get unprov ability of both results for the theory PV 1 + BB ( s Σ b 1 , log ) which lies in the intersection. All the inclusions are strict under the hardness assumption NP ⊆ P / p oly . Theorem 1.13 (Theorem 6.7 ) . F or every k ∈ N , PV 1 + BB (Σ b 1 ) ⊢ P ⊆ SIZE [ n k ] . Theorem 1.14 (Theorem 6.8 ) . PV 1 + LLIND ( s Σ b 1 ) ⊬ NSubExp ⊆ Avg - coNSIZE [2 n δ ] . Here, we use Avg on the uniform distribution. In other words, the lo wer bound states that for any NSubExp T uring machine M and any D ∈ coNSIZE [2 n δ ] , Pr x ∈{ 0 , 1 } n [ M ( x ) = D ( x )] ≤ 1 / 2 + 1 / 2 n δ . It follows from the pre vious results that the theory PV 1 + BB ( s Σ b 1 , log ) is contained in both of these theories and is still stronger than PV 1 unless NP ⊆ P / p oly . W e thus obtain the following as a corollary . Theorem 1.15 (Corollary 6.10 ) . • F or every k ∈ N , PV 1 + BB ( s Σ b 1 , log ) ⊢ P ⊆ SIZE [ n k ] . • F or every rational δ ∈ (0 , 1) , PV 1 + BB ( s Σ b 1 , log ) ⊬ NSubExp ⊆ Avg − coNSIZE [2 n δ ] . 12 1.3 Open Problems All the separations of theories we give are conditioned upon reasonable complexity-theoretic assump- tions. The most general open problem would be to e xtend upon these results by pro ving them under weaker assumptions, and hopefully to ev entually resolve the state of the separations without any assumptions, which seems like a distant goal at this point. Is it possible to show that T i 2 ⊆ S i 2 + BB (Σ b i +1 ) , under some plausible assumptions? The conditional separation S i 2 ⊊ S i 2 + BB (Σ b i +1 ) ⊊ S i +1 2 is related to another open problem. The inclusions resemble the situation from Buss’ hierarchy , where S i 2 ⊆ T i 2 ⊆ S i +1 2 . By a result of Ressayre [ Res86 ], the theory S i 2 + BB (Σ b i +1 ) is ∀ Σ b i +1 -conserv ati ve over S i 2 . This contrasts with the e xpected situation between S i 2 and T i 2 , where different propositional proof systems correspond to the theories and thus no conservati vity is expected. Hence arises the question of the dif ference between S i 2 + BB (Σ b i +1 ) and T i 2 . W e have already showed that S i 2 + BB (Σ b i +1 ) ⊆ T i 2 , assuming Σ p i +1 ⊆ ∆ p i +1 / p oly . By [ Kra93 ] combined with the aforementioned conserv ati vity result of S i 2 + BB (Σ b i +1 ) ov er S i 2 , we obtain that the other separation T i 2 ⊆ S i 2 + BB (Σ b i +1 ) is only known to follo w from P Σ p i being distinct from P Σ p i [ O (log n )] , the class of problems decidable by a polynomial-time oracle machine with O (log n ) -many calls to a Σ p i -oracle, and this h ypothesis is not kno wn to be implied by the polynomial hierarchy not collapsing. Therefore, it is natural to ask the follo wing. Open Pr oblem. Is it possible to show that T i 2 ⊆ S i 2 + BB (Σ b i +1 ) , under the assumption that the polynomial hierarchy does not collapse? What is more, in Corollary 6.10 we sho w that both unprov abilities of Kraj ´ ı ˇ cek–Oli veira and Pich– Santhanam are true for any theory which is included in PV 1 + BB (Σ b 1 ) and PV 1 + LLIND ( s Σ b 1 ) . The authors are under the suspicion that PV 1 + BB ( s Σ b 1 , log ) might not be the greatest such theory . Open Problem. Is it possible to axiomatize the greatest common subtheory of PV 1 + BB (Σ b 1 ) and PV 1 + LLIND ( s Σ b 1 ) by some natural axiom scheme? Can we separate this theory from PV 1 + BB ( s Σ b 1 , log ) under some plausible assumptions? 1.4 Organisation The rest of the paper is org anised as follo ws: 1. In Section 2 , we state some preliminaries on Bounded Arithmetic and the dif ferent axioms we use. 2. In Section 3 , we prove Theorems 1.4 and 1.5 , which concern the main technical tool of the separations among dif ferent classes of Student-T eacher Games. 3. In Section 4 , we provide a unifying proof for witnessing theorems, and we sho w ho w it applies to the length induction axiom and the bounded replacement axiom, which correspond respecti vely to rounds and parallel queries per round in Student-T eacher Games. 4. In Section 5 , we combine the results of the pre vious sections to prov e Theorems 1.9 to 1.12 about the separations of dif ferent Bounded Arithmetic theories. 5. In Section 6 , we use the new witnessing theorems to lift the aforementioned unprov ability results to seemingly stronger theories (Theorems 1.13 and 1.14 ). 6. In Section A of the Appendix, we provide some more detailed calculations for the functions used in the generalised versions of our theorems. 13 Acknowledgements. W e would like to thank Igor Oliveira for hosting Ond ˇ rej Je ˇ zil at the Univ ersity of W arwick and initiating the discussion about witnessing theorems, Student-T eacher Games and unprovability results. W e are also grateful to Jan Kraj ´ ı ˇ cek for pointing out the pro v ability of the bounded collection axiom with double length induction on non-strict formulas, Sam Buss for pointing us to the open problem from his thesis, and Neil Thapen for discussion about the method of witnessing described in his thesis. Ond ˇ rej Je ˇ zil was supported by Charles University Research Center program No. UNCE/24/SCI/022, the project SVV -2025-260837, and by the GA UK project No. 246223. Dimitrios Tsintsilidas was supported by the UKRI Frontier Research Guarantee Grant EP/Y007999/1, the Centre for Discrete Mathematics and its Applications (DIMAP) at the Uni versity of W arwick, and Chancellor’ s Scholarship. 2 Pr eliminaries W e refer to [ Bus85 ] and [ Kra95 ] for an introduction to Bounded Arithmetic. Here, we state some facts from these sources which are useful for the purposes of the paper . W e ﬁrst need to classify formulas in different classes. Deﬁnition 2.1 (Formulas) . • The quantiﬁers of the form ∃ x ≤ | t | or ∀ y ≤ | t | for some term t , are called sharply bounded quanti- ﬁers , and the quantiﬁers of the form ∃ x ≤ t or ∀ y ≤ t for some term t , are called bounded quantiﬁers . W e sometimes use the notation ∃ b and ∀ b for bounded quantiﬁers, if we do not care to determine the term t . • The class of formulas Σ b 0 = Π b 0 are called sharply bounded formulas, and the y are the formulas where the only quantiﬁers appearing are sharply bounded. • For i ≥ 0 , the classes Σ b i +1 and Π b i +1 are the smallest formula classes that satisfy the properties: 1. Σ b i ∪ Π b i ⊆ Σ b i +1 and Σ b i ∪ Π b i ⊆ Π b i +1 . 2. If ϕ ∈ Σ b i +1 , then ∃ b ϕ ∈ Σ b i +1 , and ¬ ϕ ∈ Π b i +1 . 3. If ϕ ∈ Π b i +1 , then ∀ b ϕ ∈ Π b i +1 , and ¬ ϕ ∈ Σ b i +1 . 4. Σ b i +1 and Π b i +1 are closed under sharply bounded quantiﬁcation, disjunction and conjunction. • For i ≥ 0 , the classes strict Σ b i +1 ( s Σ b i +1 ) are deﬁned as the formulas in the form ∃ b ∀ b ∃ b . . . ϕ, where there are i + 1 alternating bounded quantiﬁers and ϕ is a sharply bounded formula. The classes strict Π b i +1 ( s Π b i +1 ) are deﬁned dually . • The class ∆ b i +1 is deﬁned by the Σ b i +1 formulas which are proved to be equi v alent with a Π b i +1 formula ov er a base theory speciﬁed (if not speciﬁed, it is implied we use S 1 2 ). W e introduce a generalisation of the sharply bounded quantiﬁers, when we have dif ferent functions of bounds. Deﬁnition 2.2 ( b -bounded Formulas) . For a bound function b , which is in the speciﬁed language, such that b ( x ) ≤ x , the quantiﬁers ∃ x ≤ b ( | t | ) and ∀ y ≤ b ( | t | ) for some term t , are called b -sharply bounded quantiﬁers. For i ≥ 0 , the classes b Σ b i +1 and b Π b i +1 are the closures of s Σ b i +1 and s Π b i +1 ov er b -sharply bounded quantiﬁcation, disjunction and conjunction. 14 Deﬁnition 2.3 (Axioms) . The theories we are interested in can be formalised by dif ferent axiom schemes. The main ones are: 1. IND ( φ ) := ∀ x h  φ (0) ∧ ∀ n < x ( φ ( n ) → φ ( n + 1))  → φ ( x ) i 2. LIND ( φ ) := ∀ x h  φ (0) ∧ ∀ n < | x | ( φ ( n ) → φ ( n + 1))  → φ ( | x | ) i 3. PIND ( φ ) := ∀ x h  φ (0) ∧ ∀ n ≤ x ( φ ( ⌊ n/ 2 ⌋ ) → φ ( n ))  → φ ( x ) i 4. MIN ( φ ) := ∀ x h φ ( x ) → ∃ n ≤ x ∀ m < n φ ( n ) ∧ ¬ φ ( m ) i 5. LMIN ( φ ) := ∀ x h φ ( x ) → ∃ n ≤ x ∀ m ≤ x φ ( n ) ∧  | m | < | n | → ¬ φ ( m )  i 6. MAX ( φ ) := ∀ x h φ (0) → ∃ n ≤ x ∀ m ≤ x φ ( n ) ∧  n < m → ¬ φ ( m )  i 7. LMAX ( φ ) := ∀ x h φ ( x ) → ∃ n ≤ x ∀ m ≤ x φ ( n ) ∧  | n | < | m | → ¬ φ ( m )  i 8. BB ( φ ) := ∀ x, t h ( ∀ j ≤ | x | ∃ y ≤ t φ ( j, y )) → ( ∃ w ∀ j ≤ | x | φ ( j, w j )) i The theories S i 2 and T i 2 of Buss’ s hierarchy are deﬁned using BASIC , which is a ﬁxed set of axioms that determine basic operations of the symbols and arithmetic, and extending it by dif ferent kinds of induction: S i 2 := BASIC + LIND (Σ b i ) , T i 2 := BASIC + IND (Σ b i ) . W e will also need the theories PV i , deﬁned in [ KPT91 , Coo75 ]. The theory PV i +1 inducti vely deﬁnes symbols for all Σ p i predicates and it is closed under deﬁnition by cases (see next subsection) and under limited recursion on notation. In fact, we can reg ard it as a univ ersal theory that contains symbols for all functions in FP Σ p i and also contains induction on open formulas. It is easy to see that PV i +1 can be re garded as a fully conservati ve extension of T i 2 ; that is why we use them interchangeably throughout the paper . The important property of PV i +1 is its universality , which enables us to use them for witnessing theorems. Let us note that for i ≥ 0 , j ≥ 1 we always consider the classes of formulas Σ b i and Π b i to be over the language of PV and we do not allo w PV j -symbols unless explicitly stated, this is to keep the complexity of the deﬁnable sets by the formulas intact. W ith the language of PV , we mean the language of the theory PV 1 , which contains symbols for all FP functions. This is why it is useful to work on this language, and from now on, apart from the case of the theories PV i +1 for i ≥ 1 , this is the language we use. In the follo wing, we see some equi v alences of dif ferent axioms. Proposition 2.4. Over the base theory S 1 2 or PV 1 , we have the following equivalences for all i ∈ N : (a) IND (Σ b i ) ⇐ ⇒ MIN (Σ b i ) ⇐ ⇒ MAX (Σ b i ) . (b) LIND (Σ b i ) ⇐ ⇒ PIND (Σ b i ) ⇐ ⇒ LMIN (Σ b i ) ⇐ ⇒ LMAX (Σ b i ) . This means that we can axiomatise the dif ferent theories in Buss’ s hierarchy in the follo wing way: S i 2 := BASIC + LIND (Σ b i ) = BASIC + PIND (Σ b i ) = BASIC + LMIN (Σ b i ) = BASIC + LMAX (Σ b i ) . 15 T i 2 := BASIC + IND (Σ b i ) = BASIC + MIN (Σ b i ) = BASIC + MAX (Σ b i ) . BASIC here is a ﬁxed set of axioms that determine basic operations of the symbols and arithmetic. In this work, we also consider the axioms with some speciﬁc bound, b on the length of the variables. This function b must be a symbol of PV , increasing and PV 1 must prov e that b ( x ) ≤ x . 1. LIND ( φ, b ) := ∀ x h  φ (0) ∧ ∀ n < b ( | x | ) ( φ ( n ) → φ ( n + 1))  → φ ( b ( | x | )) i 2. LMIN ( φ, b ) := ∀ x h φ ( x ) → ∃ n ≤ x ∀ m ≤ x φ ( n ) ∧  b ( | m | ) < b ( | n | ) → ¬ φ ( m )  i 3. LMAX ( φ, b ) := ∀ x h φ ( x ) → ∃ n ≤ x ∀ m ≤ x φ ( n ) ∧  b ( | n | ) < b ( | m | ) → ¬ φ ( m )  i 4. BB ( φ, b ) := ∀ x, t h ( ∀ j ≤ b ( | x | ) ∃ y ≤ t φ ( j, y )) → ( ∃ w ∀ j ≤ b ( | x | ) φ ( j, w j )) i In a similar way with Theorem 2.4 , we can sho w the follo wing. Proposition 2.5. Over the base theory S 1 2 or PV 1 , we have the following equivalences for all i ∈ N : LIND (Σ b i , b ) ⇐ ⇒ LMIN (Σ b i , b ) ⇐ ⇒ LMAX (Σ b i , b ) . The bounded replacement axiom scheme for Σ b i +1 formulas is used in a theory to con vert a Σ b i +1 formula with arbitrary sharply bounded quantiﬁers into an equiv alent one which is strict. This is sometimes called “sharply bounded quantiﬁer exchange property”, analogously with the property in unbounded arithmetic. The follo wing is similar as Corollary 15 in [ Bus85 ]. Proposition 2.6 (Sharply bounded quantiﬁer e xchange property) . The theory PV 1 + BB ( s Σ b i +1 ) pr oves that for every Σ b i +1 formula, φ , ther e is an equivalent s Σ b i +1 formula, φ ′ . W e generalise this property to b -sharply bounded quantiﬁers, which can be considered as the goal of deﬁning BB ( s Σ b i +1 , b ) in the ﬁrst place. Proposition 2.7 ( b -sharply bounded quantiﬁer exchange property) . The theory PV 1 + BB ( s Σ b i +1 , b ) pr oves that for every b Σ b i +1 formula, φ , ther e is an equivalent s Σ b i +1 formula, φ ′ . Pr oof. W e prov e this inducti vely , and we only consider the case of b -sharply bounded quantiﬁcation, the cases of disjunction and conjunction are tri vial. Assume that φ = ( ∀ x ≤ b ( t )) φ 0 , where φ 0 ∈ b Σ b i +1 is equi v alent ov er the theory with the s Σ b i +1 formula φ ′ 0 . W e can write φ ′ 0 = ∃ b y φ ′ 1 , so from BB ( φ ′ 1 , b ) , we get ∀ x ≤ b ( t ) ∃ b y φ ′ 1 ( x, y ) → ∃ b w ∀ x ≤ b ( t ) φ ′ 1 ( x, w x ) This is actually an equiv alence and the left-hand side is equiv alent with φ , while the right-hand side is strict, which gi ves us the desired result. An important fact for the theories we use, is that length induction can pro ve the bounded replacement scheme for the same class of formulas. The original result is the follo wing. Theorem 2.8 ([ Bus85 ],Theorem 14) . F or i ≥ 0 , S i +1 2 ⊢ BB (Σ b i +1 ) . In the generalised form, we hav e: 16 Theorem 2.9 (Length induction proves bounded replacement) . F or i ≥ 0 , and a function b , as above T i 2 + LIND ( s Σ b i +1 , b ) ⊢ BB ( b Σ b i +1 , b ) . Pr oof. Let φ ∈ s Σ b i +1 . W e want to show that LIND ( s Σ b i +1 , b ) ⊢ BB ( φ, b ) . Assume that ∀ x ≤ b ( | t | ) ∃ y ≤ s φ ( x, y ) . W e write φ = ∃ b z φ ′ ( x, y , z 1 , z 2 ) , where φ ′ is strict Π b i and we consider the s Σ b i +1 formula: ψ ( u ) := ∃ b w , z ′ ∀ b x  Seq ( w ) ∧ Seq ( z ′ ) ∧ Len ( w ) = Len ( z ′ ) = u ∧ ( x ≤ u → φ ′ ( x, w x , z ′ x ))  . Notice that this is truly strict, since the uni v ersal quantiﬁer ∀ b x can be mer ged with the ﬁrst quantiﬁer of φ ′ . W e want to sho w that ∀ u < b ( | t | ) ψ ( u ) → ψ ( u + 1) . Assume ψ ( u ) , this means that we have two sequence of length u , let w u , z ′ u , where for each x ≤ u , φ ′ ( x, w x , z x ) . By the assumption, we know that ∃ y ≤ s ∃ b z φ ( u + 1 , y , z ) , since u + 1 ≤ b ( | t | ) . W e name those y u +1 and z u +1 and we consider the sequences Ext ( w u , y u +1 ) and Ext ( z ′ u , z u +1 ) , where the function Ext ( w , x ) extends the sequence w by putting the element x in the end. It is easy to see that these satisfy the existential quantiﬁers of ψ ( u + 1) . By LIND ( ψ , b ) , we get that ∃ b w , z ′ ∀ b x ≤ b ( | t | ) φ ′ ( x, w x , z ′ x ) . By changing the two last quantiﬁers, we prove that the replacement holds for φ . Hence, T i 2 + LIND ( s Σ b i +1 , b ) ⊢ BB ( s Σ b i +1 , b ) Ho we v er , by Theorem 2.7 , T i 2 + BB ( s Σ b i +1 , b ) ⊢ BB ( b Σ b i +1 , b ) , thus we get the desired re- sult. The last result we recall, sho ws that the replacement axiom on Σ b i +1 formulas can pro ve length induction on Σ b i formulas. Later , we show that it cannot prove length induction on Σ b i +1 , ev en if it is weakened by some sublinear bound b . It is still open whether , it also proves induction on Σ b i formulas. Theorem 2.10 ([ Bus85 ],Theorem 16) . F or i ≥ 1 , S 1 2 + BB (Σ b i +1 ) ⊢ S i 2 . 3 Student-T eacher Games In this section, we prov e the two main theorems about Student-T eacher Games, where the former shows that adaptivity gi ves more computational power to the Student (Theorem 1.4 ), and the latter states the same for parallelism (Theorem 1.5 ). As a note, for the proof of the theorems, we will need to make a con version between dif ferent input sizes; the ﬁrst input is for the Student-T eacher Games search problem, and the other is the input for a decision problem about a language L . W e use m for the input size of the former , and n for the input size of the latter . In the two formulas that we use for the separations taking place in Theorems 1.4 and 1.5 , we consider p ( n ) dif ferent instances of the decision problem, x i , each of size n . This means that the total size of all these strings, which is actually the input for the Student-T eacher Game search problem, is m = n · p ( n ) . W e want the function p ( n ) to be determined according to our needs for the necessary separation. This separation will be achiev ed with the dif ference of p ( n ) and p ( n ) − 1 , but we want to con vert this result as some function of m . Suppose we are given the function g ( m ) with g ( m ) ≤ m , which may correspond to the number of rounds, or queries, etc., and we want the separation between g ( m ) and g ( m ) − 1 . This me ans that we should choose the appropriate p ( n ) , such that p ( n ) = g ( m ) (which equiv alently means that p ( n ) = g ( n · p ( n )) . This task is non-tri vial, but we show that the function p ( n ) exists in Section A.1 . From now on, in the proofs, we use a simpler form of the solution, giv en by the relations n = m g ( m ) and p ( n ) = g ( m ) , but assuming the process we described. 17 3.1 Adaptivity Pr oof of Theorem 1.4 . This theorem shows the power of adaptivity in Student-T eacher Games. W e will sho w that for any sublinear , unbounded, increasing, polynomial-time function r ( m ) and any unbounded, increasing, polynomial-time function 1 ≤ q ( m ) ≤ p oly ( m ) , S T Σ p i [ r ( m ) + 1 , 1] ⊆ S T Σ p i [ r ( m ) , q ( m )] = ⇒ Σ p i +1 ⊆ ∆ p i +1 / p oly . Fix n = m r ( m ) and p ( n ) = r ( m ) , as discussed abov e, and consider an arbitrary Σ p i +1 language, L , which has the form ∃ φ , where φ is a ∆ p i +1 predicate ( x ∈ L ⇐ ⇒ ∃ y φ ( x, y ) ). W e deﬁne the new predicate φ ′ ( x, y , z ) := φ ( x, z ) → φ ( x, y ) , also in ∆ p i +1 , and take the sentence Φ := ∀ x 1 , . . . , x p ( n ) ∈ { 0 , 1 } n ∃ y 1 , . . . , y p ( n ) ∀ i, z 1 , . . . , z p ( n ) φ ′ ( x i , y i , z i ) , which deﬁnes the follo wing TFΣ p i + 2 search problem: “Given p ( n ) n -bit strings, ﬁnd witnesses (of the ﬁrst existential quantiﬁer) for all those of them that belong to the language L . ” The problem is characterized by the relation R ( x, y ) := ∀ i, z 1 , . . . , z p ( n ) φ ′ ( x i , y i , z i ) ∈ Π p i +1 , where x = ( x 1 , . . . , x p ( n ) ) is the input of size n · p ( n ) = m and y = ( y 1 , . . . , y p ( n ) ) is the sequence of witnesses. It is easy to see that the problem can be solved with a Student-T eacher Game of p ( n ) + 1 = r ( m ) + 1 rounds with one query per round: The Student at ev ery round outputs the witnesses they know along with 0 ’ s for the witnesses of the instances that they do not hav e kno wledge of, while the T eacher must give at least one correct witness for one of the instances at ev ery round as a counterexample. After at most p ( n ) rounds, the Student kno ws all the witnesses, which they output at the last round. Assume now that R ( x, y ) ∈ S T Σ p i [ r ( m ) , q ( m )] , which means that it is solv able by a Student in p ( n ) rounds with polynomially many queries at each round. Let f ∈ FP Σ p i be the function of the Student, which takes as input the sequence x 1 , . . . , x p ( n ) , the number of round j and depending on j , j − 1 counterexamples in the form of i, z 1 , . . . , z p ( n ) , and outputs at most p oly ( n ) different sequences of candidate witnesses. If the number of the round is greater than p ( n ) , then the function rejects. The goal is to ﬁnd a poly ( n ) -time algorithm that uses this function and p oly ( n ) -size advice in order to decide the n -size instances of L . This means that L ∈ ∆ p i +1 / p oly . Before we continue, we ﬁx the T eacher’ s strategy . At e very round, the T eacher recei ves p oly ( n ) number of queries in the form y j = ( y j 1 , . . . , y j p ( n ) ) , where j indices over the queries. Then, the T eacher outputs as counterexample for all the queries, the minimum index i , such that x i ∈ L , but this is not witnessed by any y j i for all j , along with the least lexicographically true witness z of x i . Note that if there is no such minimum, then the Student could ha v e already w on, because the y could output the tuple with all the correct instances, which can be v alidated by their computational po wer . Hence, we disregard these cases. In order to construct the advice, we start by the set U of all x ∈ L of size n . Formally , U := { x : | x | = n ∧ ∃ y φ ( x, y ) } Let S be a subset of U with p ( n ) − 1 elements and x ∈ U \ S . W e can order the set S ∪ { x } (e.g. in increasing order) and make the tuple X ( S,x ) = ( x 1 , . . . , x l − 1 , x, x l , . . . , x p ( n ) − 1 ) , where x i ∈ S for all i ∈ [ p ( n ) − 1] . 6 W e say that the pair ( S , x ) is good if providing as input X ( S,x ) to the Student-T eacher game with the giv en Student, the ﬁxed T eacher ne ver outputs the inde x l and a witness for x . 6 These should not be confused with the x i ’ s of the formula Φ . 18 Claim 3.1. F or every p ( n ) -element subset of U , ther e e xists at least a good pair . T o prov e the claim, we can order the elements of the set and provide the ordered tuple as input to the Student-T eacher computation. Since the T eacher strategy is to output a single index with the corresponding witness at every round, and the T eacher provides p ( n ) − 1 counterexamples, there is at least an index l and a witness z , which the T eacher nev er outputs. The l th element of the tuple with the set of the remaining elements form the good pair . As a result, if the set U has N elements, then there are at least as many good pairs in U as p ( n ) -element sets, which is  N p ( n )  . On the other hand there are  N p ( n ) − 1  -many ( p ( n ) − 1) -element sets, hence there e xists such a set S contained in at least  N p ( n )  N p ( n ) − 1  − 1 = N − p ( n ) + 1 p ( n ) good pairs. W e recursiv ely construct the sets U i , S i starting with U 0 = U , as follows: If the set U i has N i elements, then we take a ( p ( n ) − 1) -subset S i ⊆ U i which is contained in at least N i − p ( n )+1 p ( n ) good pairs. Afterwards, we construct U i +1 := U i \ { x : ( S i , x ) is a good pair } . This means that for λ = p ( n ) − 1 p ( n ) , N i +1 = λN i + λ < λN i + 1 , thus N k < λ k N 0 + k . F or k = log( N ) log( λ − 1 ) = O ( n ) , the last relation implies that N k < k . Therefore, the set Q = S 0 ∪ · · · ∪ S k − 1 ∪ U k ⊆ U has size O ( n · p ( n )) . Finally , we describe the non-uniform algorithm that decides L on inputs of size n . The correctness of the algorithm comes from the construction of the set Q , where all x ∈ U form a good pair with some S i or belong to Q . Input : A string x ∈ { 0 , 1 } n . Advice: The set Q and the set of the least lexicographic witness for each element of Q . 1 If x ∈ Q , then check the corresponding witness and Accept; 2 Otherwise, for all S i , simulate the Student-T eacher Game with input X ( S i ,x ) using the function f and the witnesses of S i ; 3 If any of these Student-T eacher Games generates a v alid witness, then Accept; 4 If none of these Games generates a v alid witness, then Reject. Algorithm 1: The pseudocode of the ∆ p i +1 / p oly -algorithm that decides L on input size n . 3.2 Parallelism Pr oof of Theorem 1.5 . This theorem shows the power of the number of parallel queries per round in Student- T eacher Games. W e will show that for any sublinear , unbounded, increasing, polynomial-time functions r 1 ( m ) , q 1 ( m ) , r 2 ( m ) , q 2 ( m ) with r 1 ( m ) q 1 ( m ) < r 2 ( m ) q 2 ( m ) , S T Σ p i [1 + r 2 ( m ) , q 2 ( m )] ⊆ S T Σ p i [1 + r 1 ( m ) , q 1 ( m )] = ⇒ Σ p i +1 ⊆ ∆ p i +1 / p oly . W e ﬁx n = m r 2 ( m ) q 2 ( m ) and p ( n ) = r 2 ( m ) q 2 ( m ) , as discussed above, and we consider an arbitrary Σ p i +1 language, L , which has the form ∃ φ , where φ is a ∆ p i +1 predicate. No w , we take the sentence Ψ := ∀ x 1 , . . . , x p ( n ) ∈ { 0 , 1 } n ( ∃ i ≤ p ( n ) ∀ y ¬ φ ( x i , y )) ∨  ∃ y 1 , . . . , y p ( n ) ∀ j ≤ p ( n ) φ ( x j , y j )  19 which deﬁnes the follo wing TFΣ p i + 2 search problem: “Given p ( n ) n -bit strings, ﬁnd an index of an element whic h is not in the language L or ﬁnd a p ( n ) -long sequence that includes witnesses (of the ﬁrst e xistential quantiﬁer) for all of the elements in the input. ” The problem is characterized by the relation R ( x, y ) := ∀ i, z 1 , . . . , z p ( n ) φ ′ ( x i , y i , z i ) ∈ Π p i +1 , where x = ( x 1 , . . . , x p ( n ) ) is the input of size n · p ( n ) = m and the search terms are i and the sequence ( y 1 , . . . , y p ( n ) ) . It is easy to see that the problem can be solved with a Student-T eacher Game of r 2 ( m ) + 1 rounds with q 2 ( m ) queries per round: The Student splits the numbers of [ p ( n )] into r 2 ( m ) sets of q 2 ( m ) elements, and at each round guesses that the index where there is no witness, is one of the elements of the set. They achiev e that by the q 2 ( m ) parallel queries. After r 2 ( m ) rounds, the Student has either found an element without a witness or has been provided witnesses for all the elements, which they combine in a sequence and output in the last round. Assume no w that R ( x, y ) ∈ S T Σ p i [1 + r 1 ( m ) , q 1 ( m )] , which means that it is solv able by a Student that has receiv ed at most r 1 ( m ) q 1 ( m ) counterexamples from the T eacher (at the last round the Student should be successful). From the given inequality , this means that the Student has recei v ed less than p ( n ) counterex- amples. Let f ∈ FP Σ p i be the function of the Student, which takes as input the sequence x 1 , . . . , x p ( n ) , the number of round j and depending on j , j − 1 counterexamples in the form of y , j . If the number of the round is greater than p ( n ) , then the function rejects. The goal is, as before, to ﬁnd a p oly ( n ) -time algorithm that uses this function and p oly ( n ) -size advice in order to decide the n -size instances of L , so that we hav e L ∈ ∆ p i +1 / p oly . Assume now that R ( x, y ) ∈ S T Σ p i [ r ( m ) , q ( m )] , which means that it is solv able by a Student in p ( n ) rounds with polynomially many queries at each round. Let f ∈ FP Σ p i be the function of the Student, which takes as input the sequence x 1 , . . . , x p ( n ) , the number of round j and depending on j , j − 1 counterexamples in the form of i, z 1 , . . . , z p ( n ) , and outputs at most p oly ( n ) different sequences of candidate witnesses. If the number of the round is greater than p ( n ) , then the function rejects. The goal is to ﬁnd a poly ( n ) -time algorithm that uses this function and p oly ( n ) -size advice in order to decide the n -size instances of L . This means that L ∈ ∆ p i +1 / p oly . Again, we ﬁx the T eacher’ s strategy . For the ﬁrst part of the disjunction, if the Student has failed, which means that x i ∈ L , the T eacher provides the least lexicographically witness for x i . W e can actually disregard the second disjunct, since the uni v ersal quantiﬁer , which the T eacher is supposed to gi v e counterexamples for , is sharply bounded, thus it can be computed efﬁciently by the Student. This means that in this case the T eacher provides no ne w information to the Student. As in the previous proof, we start by the set U of all x ∈ L of size n , and for a subset S ⊆ U with p ( n ) − 1 elements and an element x ∈ U \ S , we form the ordered p ( n ) -tuple X ( S,x ) , where x is in the l th position. Then, the pair ( S, x ) is good if providing X ( S,x ) as input to the Student-T eacher game with the gi ven Student, the T eacher is ne ver queried and ne ver provides witness for the inde x l . It is easy to see that ev ery p ( n ) -element subset of U has at least a good pair, because if we order its elements and gi ve it as input to the Student-T eacher game with 1 + r 1 ( m ) rounds and q 1 ( m ) queries per round, then there must be an index which is ne v er queried, since the number of queries is r 1 ( m ) q 1 ( m ) < p ( n ) . The remaining proof follo ws exactly the proof of Theorem 1.4 , in order to construct a ∆ p i +1 / p oly algo- rithm for deciding L . 20 4 W itnessing Theor ems In this section, we prove the witnessing theorems needed for our separations. W e start by giving a general witnessing theorem, which relates a theory S with a suitable relati vized uni versal theory T such that the terms of T describe the computational model for the student in the related Student-T eacher game. 4.1 General Witnessing Theor em Deﬁnition 4.1. Assume there are L -structures M , N | = T such that M ⊆ N . W e say N is an ∃ -elementary extension if for e v ery open formula φ ( x, y ) it holds that for e very m ∈ M we hav e M | = ( ∃ y ) φ ( m, y ) ⇐ ⇒ N | = ( ∃ y ) φ ( m, y ) . Theorem 4.2. [ A vi02 , Theor ems 3.2 and 3.3] Let L be a language, T be a universal L -theory and let M | = T . Then M has an ∃ -elementary extension ˆ M such that for e very open L -formula if ˆ M | = ( ∀ x )( ∃ y )( φ ( x, y )) , then ther e ar e L -terms t 1 , . . . , t k such that ˆ M | = ( ∀ x )(( φ ( x, t 1 ) ∨ · · · ∨ φ ( x, t k )) . Deﬁnition 4.3. Let L be a language, T an L -theory . W e say a theory T has terms closed under deﬁnition by cases by open formulas 7 , if for e very open L -formula φ ( x ) and L -terms t 1 ( x ) , t 2 ( x ) there is an L -term t ( x ) such that T ⊢ ( ∀ x )(( φ ( x ) ∧ t ( x ) = t 1 ( x )) ∨ ( ¬ φ ( x ) ∧ t ( x ) = t 2 ( x ))) . W e use that the languages we will w ork in are rich enough to combine the dif ferent terms in the disjunc- tion in the pre vious theorem into a single term. The follo wing is a suf ﬁcient condition to do this. Lemma 4.4. Let L be a languag e extending PV and T be an L -theory extending PV 1 . Then T has terms closed by deﬁnition by cases by open formulas. Pr oof. Assume that we hav e an open L -formula φ ( x, y ) and L -terms t 1 and t 2 . It is not hard to construct a term f ( x, y ) with the property PV 1 ⊢ ( ∀ x )( ∀ y )( f ( x, y ) = 1 ↔ φ ( x, y )) (*) starting with atomic formulas, taking the PV -symbol which checks for equality of tw o inputs, we can create a term which can check whether an equality of PV -terms is valid. Using PV -symbols for disjunction and negation, we can obtain (*) for arbitrary open formulas by induction on the comple xity of φ . The term t can be constructed as IfThenElse ( f ( x, y ) , t 1 ( x, y ) , t 2 ( x, y )) , where IfThenElse ( x, y , z ) is a PV -symbol which returns y if x = 1 otherwise it outputs z . 7 The name ‘closed under deﬁnition by cases by open formulas’ comes from [ Kra10 ]. 21 The following theorem is a general theorem for witnessing of ∀∃∀ consequences of a theory . If one starts with a theory S , e xtends it with a ne w binary function symbol α ( x, y ) , which represents the counterexample function of the T eacher in the Student-T eacher protocol, to obtain a theory T ′ , and ﬁnds a univ ersal sub- theory T ⊆ T ′ such that T ′ is ∀∃ -conserv ati ve over it and T has terms closed by deﬁnitions by open formulas, then there will be a single term in the language of T such that it T -prov ably computes the witness for the existentially quantiﬁed v ariable. In our applications of this theorem, the theory S we does not pose any restriction on the concrete beha viour of α and as such we may assume that α behaves as an appropriate teacher for the gi ven Student-T eacher game. Deﬁnition 4.5. Let L 1 , L 2 be languages, T 1 be an L 1 -theory and let T 2 be an L 2 -theory , let Γ be a class of L 1 -formulas, we say that T 2 is Γ -conservati ve over T 1 if for ev ery φ ∈ Γ which is also an L 2 -formula we hav e that T 1 ⊢ φ if and only if T 2 ⊢ φ . Theorem 4.6. Let α be a binary function symbol, let L w be a langua ge containing α and let L ⊆ L w \ { α } be another langua ge . Consider a univer sal L w -theory T with terms closed under deﬁnition by cases by open formulas, its ∀∃ -conservative extension T ′ and an L -theory S ⊆ T ′ . Let ψ be an open L -formula. If S ⊢ ( ∀ x )( ∃ y )( ∀ z )( ψ ( x, y , z )) then ther e is an L w -term t such that T ⊢ ( ∀ x )( ψ ( x, t ( x ) , α ( x, t ( x )))) . Pr oof. Assume that S ⊢ ( ∀ x )( ∃ y )( ∀ z )( ψ ( x, y , z )) . Then we ha ve by Herbrandization on the variable z that S ⊢ ( ∀ x )( ∃ y )( ψ ( x, y , α ( x, y ))) , by S ⊆ T ′ we also hav e T ′ ⊢ ( ∀ x )( ∃ y )( ψ ( x, y , α ( x, y ))) , by the ∀∃ -conserv ati vity of T ′ ov er T we ha ve T ⊢ ( ∀ x )( ∃ y )( ψ ( x, y , α ( x, y ))) , and since T is a uni versal theory we obtain by Herband’ s theorem L w -terms t 1 , . . . , t k such that T ⊢ ( ∀ x )( _ 1 ≤ i ≤ k ( ψ ( x, t i ( x ) , α ( x, t i ( x )))) , and by the closedness by deﬁnition by cases there is a single L w -term t such that T ⊢ ( ∀ x )( ψ ( x, t ( x ) , α ( x, t ( x )))) . 4.2 Sharply Bounded Replacement Scheme BB (Σ b j , b ) W e will ﬁx a PV -symbol b such that PV 1 prov es it is upper bounded by the identity , that is PV 1 ⊢ b ( x ) ≤ x . W e will no w introduce a theory which axiomatizes parallel acces to the oracle α . 22 Deﬁnition 4.7. W e deﬁne the theory PV ′ j ( ∥ , α , b ) as the theory extending PV j by a ne w function symbol α ( x, y ) which may hav e more parameters which we will not display , and for e very c ≥ 1 , we also add a function Read c with ne w axioms: ( ∀ l )( ∀ x )(( Seq ( l ) ∧ Len ( l )  = 0 ∧ Len ( l ) ≤ b ( | x | ) c ) → ( ∀ i ≤ Len ( l ))(( Read c ( l, x )) i = α (( l ) 1 i , ( l ) 2 i ))) , ( ∀ l )( ∀ x )(( ¬ Seq ( l ) ∨ Len ( l ) = 0 ∨ Len ( l ) > b ( | x | ) c ) → Read c ( l, x ) = 0) , where Seq ( l ) is the open PV -formula stating that l codes a sequence of numbers, Len is the function which gi ves the number of elements in a sequence and ( − ) i is the PV -function symbol which returns the i -th element of a sequence, and the superscripts 1 and 2 are the PV -symbols which return the ﬁrst and the second component of a pair . W e will sometimes omit the subscript and the second parameter of Read if we want to keep them implicit. There are no other axioms about α , but in the cases we care about it will represent a function oracle whose output length is bounded by a polynomial in the input length, once a speciﬁc bound is ﬁxed we will denote by PV j ( ∥ , α , b ) a theory PV ′ j ( ∥ , α , b ) with the extra axiom α ( x, y ) ≤ t ( x, y ) where t is any PV j -symbol. The function Read then allows one to query polynomially many v alues of α at once. Note that PV j ( ∥ , α , b ) is a sub-theory of PV j ( α ) , the relati vized PV j , as Read c can be identiﬁed with a speciﬁc PV j ( α ) -symbol with unrestricted oracle access. The theory PV j ( ∥ , α , b ) only reasons with func- tions which can access oracle α for constantly many times and each time ask for p oly ( b ) -many queries, as the function α cannot be iterated inside a limited recursion on notation rule. Lemma 4.8. Let α be a function satisfying | α ( x, y ) | ≤ | t ( x, y ) | for some PV j -term t . Then for every c ≥ 1 , every term s in the language of PV j ( ∥ , α , b ) and every open formula φ we have that PV j ( ∥ , α , b ) ⊢ ( ∀ i ≤ b ( | x | ) c )( φ ( i, s ( i ))) → ( ∃ w )( ∀ i ≤ b ( | x | ) c )( φ ( i, ( w ) i ) . Pr oof. W e will proceed by induction on the comple xity of s , where we will recursiv ely assign to ev ery PV j ( ∥ , α , b ) -term u another term w u which computes the witness w . If s is a PV -symbol, then the theorem is clear . Let s be of the form α ( u ( i ) , v ( i )) , for a simpler PV j ( ∥ , α , b ) -term u . It is then possible to compute the v alue for w by the follo wing term: Read c ( PairUp ( w u , w v )) , where w u and w v are the terms obtained from the induction hypothesis, and PairUp is PV -symbol which obtains two sequences of the same length on the input and outputs a sequence of pairs where the i -th component is the pair of the i -th components of the input sequenes. Let s be of the form Read ( u ( i )) , for a simpler PV j ( ∥ , α , b ) -term u . It is then possible to compute the v alue for w by the follo wing term: Regroup ( Read c + d ( Concat ( w u )) , w u ) , where w u is the term obtained from the induction hypothesis of i being bounded by b ( | x | ) d , Concat(-) concatenates a sequence of sequences into a single sequence and Regroup is a function which groups sub- sequences into elements coding the subsequences so that the resulting sequence of sequences is of the same type as the second argument of Re group. 23 It remains to consider s of the form g ( u ( i )) , for a simpler PV j ( ∥ , α ) -term u and a PV -symbol g . It is then possible to compute the v alue for w by the follo wing term: Apply g ( w u ) , where Apply g is a PV -symbol which receiv es as an input a sequence and outputs a sequence which is obtained by applying g to ev ery element of the input. Deﬁnition 4.9. The axiom scheme BB (Σ b j ( α ) , b ) consists of the set { BB ( φ, b ); φ ∈ Σ b j ( α ) } , where Σ b j ( α ) consists of all formulas which can be obtained by starting with a Σ b j -formula and substituting PV j ∪ { α } - terms for any of the free v ariables. Lemma 4.10. Let α be a function satisfying | α ( x ) | ≤ k | x | k for some k . Then the theory PV j ( ∥ , α , b ) + BB (Σ b j ( α ) , b ) is ∀∃ -conservative over PV j ( ∥ , α , b ) possibly e xtended by any universal sentences. Pr oof. Let M be an arbitrary model of PV j ( ∥ , α , b ) . Let ˆ M be the extension of M from Theorem 4.2 . W e will sho w that ˆ M | = BB (Σ b j ( α ) , b ) . Assume that for some φ ∈ Σ b j , c ≥ 1 and a PV -term b we have ˆ M | = ( ∀ i ≤ b ( | x | ) c )( ∃ w )( φ ( i, w )) , by Theorem 4.2 there are PV j ( ∥ , α , b ) -terms t 1 , . . . , t k such that ˆ M | = ( ∀ i ≤ b ( | x | ) c )( φ ( i, t 1 ) ∨ φ ( i, t 2 ) ∨ · · · ∨ φ ( i, t k )) , by PV 1 ⊆ PV j ( ∥ , α , b ) and Lemma 4.4 we obtain a single PV j ( ∥ , α , b ) -term s such that ˆ M | = ( ∀ i ≤ b ( | x | ) c )( φ ( i, s )) . By Lemma 4.8 we get that ˆ M | = ( ∃ w )( ∀ i ≤ b ( | x | ) c )( φ ( x, ( w ) i )) . Therefore, ˆ M | = BB (Σ b j , α , b ) , and since it is ∃ -elementary ov er M , which was chosen as an arbitrary model of PV j ( ∥ , α ) , we obtain the theorem. Theorem 4.11. Let PV j + BB (Σ b j , b ) ⊢ ( ∀ x )( ∃ y )( ∀ z ≤ t )( φ ( x, y , z )) , wher e φ is open and t is a PV j -term, then the TFΣ p j + 1 pr oblem corresponding to φ is in S T Σ p j − 1 [ O (1) , p oly ( b )] . Pr oof. W e will obtain PV 1 ( ∥ , α , b ) by accepting the axiom α ( x, y ) ≤ t ( x, y ) . By Lemma 4.4 , Lemma 4.10 and Theorem 4.6 we obtain that there is a single PV j ( ∥ , α , b ) term s such that PV j ( ∥ , α , b ) ⊢ ( ∀ x )( φ ( x, s ( x ) , α ( x, s ( x )))) , the v alue of the term s can be computed by a student which sends constantly many times poly ( b ( | x | )) -many answers to the teacher and obtains a counterexample whenever one e xists for each of the answers. The student runs in polynomial-time with Σ p j − 1 oracle access, as all of the v alues are computed by PV j -terms and PV j ( ∥ , α , b ) terms where the length of α is polynomially bounded. Since there is no ﬁxed interpretation for α except for a bound on the length of its outputs, we can interpret α as any function satisfying ( ∃ z ) ¬ φ ( x, s ( x ) , z ) → ¬ φ ( x, s ( x ) , α ( x, s ( x ))) , that is α is a function which is a teacher correcting every wrong answer . Now any oracle call (or parallel oracle call using Read d for some d ≥ 1 ) can either by answered with a counter-e xample, or α ‘admits’ that it is correct. 24 4.3 Length b -induction LIND (Σ b j , b ) W e will ﬁx a PV -symbol b such that PV 1 it is upper bounded by the identity ,, that is PV 1 ⊢ b ( x ) ≤ x . Deﬁnition 4.12. W e deﬁne the theory PV ′ j ( → , α , b ) as the theory e xtending PV j by a new function symbol α ( x, y ) which may ha ve more parameters we will not display and then proceeding in countably man y steps, as described in the follo wing. The construction is analogous to the deﬁnition of the equational theory PV . The theory PV ′ j ( → , α , b ) 0 is in the language PV j ∪ { α } and contains only the axioms of PV j . Assume, we hav e deﬁned the theory PV ′ j ( → , α , b ) k for some k , we will deﬁne PV ′ j ( → , α , b ) k +1 as follows. Consider function symbols g , h 0 , h 1 , l 0 , l 1 from the language of PV ′ j ( → , α , b ) k for which we ha ve that for i ∈ { 0 , 1 } : PV ′ j ( → , α , b ) k ⊢ | h i ( x, y , z ) | ′ ≤ | z | + | l i ( x, y ) | , then for ev ery c ≥ 1 a new function symbol f c is added to the language of PV j ( → , α , b ) k and along with it a ne w axiom ( ∀ x )( ∀ y )(( | y | ≤ b ( | x | ) c → ( f c ( x, 0) = g ( x, y )) ∧ ( f c ( x, 2 · y ) = h 0 ( x, y , f c ( x, y ))) ∧ ( f c ( x, 2 · y + 1) = h 1 ( x, y , f c ( x, y )))) ∧ | y | > b ( | x | ) c → ( f c ( x, y ) = 0)) , we say such an f c is deﬁned using limited recursion on notation for p oly ( b ) -many steps. For each formula in the ne w language φ ( x ) we also include the length b -induction axiom, that is for every c we accept ¬ φ (0) ∨ φ ( b ( | x | ) c ) ∨ ( φ ( h ( x )) ∧ ¬ φ ( h ( x ) + 1) ∧ h ( x ) + 1 ≤ b ( | x | ) c ) , where h is deﬁned depending on φ and c using limited recursion on notation for p oly ( b ) -many steps, which emulates searching on the interval from 0 to b ( | x | ) c , going from one end step by step to the other, until one of the disjuncts is satisﬁed. W e deﬁne PV ′ j ( → , α , b ) = S k ≥ 0 PV ′ j ( → , α , b ) k . There are no other axioms about α , but in the cases we care about it will represent a function oracle whose output length is bounded by a polynomial in the input length, once a speciﬁc bound is ﬁxed we will denote by PV 1 ( → , α , b ) a theory PV ′ 1 ( → , α , b ) with the extra axiom α ( x, y ) ≤ t ( x, y ) where t is a PV j -term which is implicitly chosen to reﬂect the upper bound on the gro wth-rate of α . Note that PV j ( → , α , b ) is a sub-theory of PV j ( α ) , which contains only those symbols with at most p oly ( b ) -many queries to α , as the function α cannot be iterated inside a limited recursion on notation rule more than p oly ( b ) -many times. Deﬁnition 4.13. The axiom scheme LIND ( s Σ b j ( α ) , b ) consists of the set { LIND ( φ, b ); φ ∈ s Σ b j ( α ) } , where s Σ b j ( α ) consists of all formulas which can be obtained by starting with a s Σ b j -formula and substituting some PV j ∪ { α } -terms for the free v ariables. Lemma 4.14. Let α be a function satisfying | α ( x, y ) | ≤ | t ( x, y ) | for some PV j +1 -term t . Then the theory PV j ( → , α , b ) + LIND ( s Σ b j +1 , α , b ) is ∀∃ -conservative over PV j ( → , α , b ) possibly e xtended by any set of true universal sentences. 25 Pr oof. Let M be an arbitrary model of PV j ( → , α , b ) . Let ˆ M be the e xtension of M from Theorem 4.2 . W e will sho w that ˆ M | = LIND ( s Σ b j , α , b ) . Assume, that for some open PV j -formula φ ( i, w ) possibly with extra parameters and a PV -term t we hav e ˆ M | = ( ∃ w ≤ t ) φ (0 , w ) ∧ ( ∀ i ≤ b ( | x | ) c )(( ∃ w )( φ ( i, w )) → ( ∃ w )( φ ( i + 1 , w ))) , by Theorem 4.2 there are PV 1 ( → , α , b ) -terms t 1 , . . . , t k such that ˆ M | = ( ∀ i ≤ b ( | x | ) c )( ∀ w ≤ t )( _ 1 ≤ i ≤ k (( φ ( i, w )) → ( φ ( i + 1 , t i )))) , and it is possible to only obtain a single PV j ( → , α , b ) -term s such that ˆ M | = ( ∀ i ≤ b ( | x | ) c )( ∀ w ≤ t )(( φ ( i, w )) → ( φ ( i + 1 , s ))) , by limited recursion on notation on s for p oly ( b ) -many steps it is possible to obtain a function s ′ such that PV j ( → , α , b ) prov es ( ∀ x )( ∀ w )( ∀ y , | y | ≤ b ( | x | ) c )( s ′ ( w , 0) = s ( w , 0) ∧ ( ∀ r ≤ 1)( s ′ ( w , 2 · y + r ) = s ( s ′ ( w , y ) , y ))) , and thus by length b -induction on | y | inside ˆ M we obtain ˆ M | = ( ∀ w ≤ t )( φ (0 , w ) → φ ( b ( | x | ) c , s ′ ( w , b ( | x | ) c )) , and thus ˆ M | = ( ∃ w ) φ ( b ( | x | ) c , w ) . Therefore, ˆ M | = LIND ( s Σ b j , α , b ) , and since it is ∃ -elementary over M , which was chosen as an arbi- trary model of PV j ( → , α , b ) , we obtain the theorem. Theorem 4.15. Let PV j + LIND (Σ b j , b ) ⊢ ( ∀ x )( ∃ y )( ∀ z ≤ t )( φ ( x, y , z )) , where φ is open and t is an PV j -term. Then the TFΣ p j + 1 pr oblem corresponding to φ is in S T Σ b j − 1 [ p oly ( b ) , 1] . Pr oof. W e use the bound α ( x, y ) ≤ t ( x, y ) to obtain the theory PV j ( → , α , b ) . Now , by Lemma 4.4 , Lemma 4.14 and Theorem 4.6 we obtain that there is a PV j ( → , α , b ) -term s such that PV j ( → , α , b ) ⊢ ( ∀ x )( φ ( x, s ( x ) , α ( x, s ( x )))) , the value of the term s can be computed by a student which sends p oly ( b ) -many times a single answer to the teacher and obtains a counterexample whene ver one exists. Since all of the terms are either PV j -terms or PV j ( → , α , b ) terms which are bounded by PV j -terms the student runs in polynomial-time in the length of x with Σ p j − 1 -oracle access. As in the previous section, there is no ﬁxed interpretation for α except for a bound on the length of its outputs and we can interpret α as any function satisfying ( ∃ z ) ¬ φ ( x, s ( x ) , z ) → ¬ φ ( x, s ( x ) , α ( x, s ( x ))) , that is α is a function which is a teacher correcting every wrong answer , then any oracle call can either by answered with a counter-e xample, or α ‘admits’ that it is correct. 26 5 Separations of Theories W e will no w use the Theorems 1.4 and 1.5 and their proofs to sho w the separations in Theorems 1.9 , 1.10 and 1.12 . First, we re visit the sentences Φ and Ψ from the proofs of the Theorems 1.4 and 1.5 . W e will parametrise them by the number of elements in the sequence of the input. Φ p := ∀ n ∈ Log ∀ x 1 , . . . , x p ( n ) ∈ { 0 , 1 } n ∃ y 1 , . . . , y p ( n ) ∀ i, z 1 , . . . , z p ( n ) φ ′ ( x i , y i , z i ) Ψ p := ∀ n ∈ Log ∀ x 1 , . . . , x p ( n ) ∈ { 0 , 1 } n ( ∃ i ≤ p ( n ) ∀ y ¬ φ ( x i , y )) ∨  ∃ y 1 , . . . , y p ( n ) ∀ j ≤ p ( n ) φ ( x j , y j )  T o write these in the language of PV , we can change the n ∈ Log with ∀ N ∀ n = | N | , and the sequences, instead of for example ∀ x 1 , . . . , x p ( n ) ∈ { 0 , 1 } n , having ∀ X ( Seq ( X ) ∧ Len ( X ) = p ( n ) ∧ Size ( X ) = n ) , where the functions Seq , Len and Size are the same as in Section 4 . Proposition 5.1. F or φ ∈ Σ b i +1 and Φ p deﬁned as above with p ( n ) = b ( n ) where b is a symbol of the language , T i 2 + LIND ( s Σ b i +1 , b ) ⊢ Φ b . Pr oof. Instead of using the length induction axiom, we use the equiv alent length maximisation axiom on Σ b i +1 formulas, LMAX ( s Σ b i +1 , b ) , as seen in Theorem 2.5 . W e deﬁne the follo wing Σ b i +1 formula with parameters x 1 , . . . , x b ( n ) (we can re write it as strict): ψ ( t ) := ∃ y , i ( Seq ( y ) ∧ Seq ( i ) ∧ Len ( y ) = Len ( i ) = t ∧ ∀ j ≤ t φ ( x i j , y j )) . This is that there e xist a sequence of correct witnesses y j of length t for the instances indexed by a sequence of indices i . The sentence ψ (0) is trivially true, and since n is a length, by LMAX ( s Σ b i +1 , b ) , we get: ∃ t ≤ b ( n )  ψ ( t ) ∧ ∀ u ≤ b ( n ) , u > t → ¬ ψ ( u )  (1) W e can argue in PV 1 , that there is a function Compose ( y , i, l ) , where y and i are sequences of equal length and the elements of i are between 1 and the natural number l . This function will construct a sequence Y of length l , where Y i j = y j for all elements of the sequence i , and for j ∈ [ l ] \ i , Y j = 0 . From the ﬁrst part of Equation ( 1 ), there exist y t and i t , such that if Y = Comp ose ( y , i, b ( n )) , ∀ j ∈ i t φ ( x j , Y j ) . Ho we ver , the second part giv es us that if we extend y t and i t , the sentence will not be true any more. W e use the function Ext ( w, x ) which extends the sequence w by putting the element x in the end. Then, by u > t → ¬ ψ ( u ) , we get ∀ j ∈ i t , ∀ z , Len ( Ext ( y t , z )) = Len ( Ext ( i t , j )) > t = ⇒ ∀ j ∈ i t , ∀ z , ∃ k ∈ i t ∪ { j } ¬ φ ( x Ext ( i t ,j ) k , Ext ( y t , z ) k ) = ⇒ ∀ j ∈ i t , ∀ z , ¬ φ ( x j , z ) . Combining the two results about j ∈ i t and j ∈ i t , we conclude that ∀ j, ∀ z ≤ b ( n ) ¬ φ ( x j , z ) ∨ φ ( x j , Y j ) , which implies Φ b . 27 Proposition 5.2. F or φ ∈ Σ b i +1 and Ψ p deﬁned as above with p ( n ) = b ( n ) wher e b is a symbol of the language , T i 2 + BB ( s Σ b i +1 , b ) ⊢ Ψ b . Pr oof. This is an immediate application of the bounded replacement axiom. For X = ( x 1 , . . . , x b ( n ) ) , we deﬁne ψ ( X , i, y ) := φ ( x i , y ) , where ψ is also Σ b i +1 . Then by the axiom on ψ , we get: BB ( ψ , b ) = ∀ n ∈ Log ∀ X ( ∀ i ≤ b ( n ) ∃ y ψ ( X , i, y )) → ∃ Y ∀ j ≤ b ( n ) ψ ( X , j, Y j ) , but this is equi valent with Ψ b . W e will now go to the proofs Theorems 1.9 , 1.10 and 1.12 . Assume we have two increasing functions b 1 ( x ) , b 2 ( x ) ≤ x , such that for any k ∈ N and for large enough x , b 1 ( x ) > b 2 ( x ) k . The input size of Φ b 1 and Ψ b 1 is n · b 1 ( n ) . But we ha ve b 2 ( n · b 1 ( n )) k < b 1 ( n · b 1 ( n )) 1 /k 0 ≤ b 1 ( n 2 ) 1 /k 0 < b 1 ( n ) (2) The ﬁrst inequality is from the assumptions, the second from the fact that b 1 is increasing and sublinear . For the third inequality , we claim that there is some k 0 such that it holds, hence we get the desired result. It is easy to check the v alidity of the inequality b 1 ( n 2 ) < b 1 ( n ) k 0 for large enough n , if b 1 is one of the usual sublinear functions, e.g. n 1 − ε , log n, etc. For a full proof, see Section A.2 . After we hav e ﬁx ed the abov e, we can continue to the three proofs. Pr oof of Theorem 1.9 . Assume that T i 2 + BB ( s Σ b i +1 , b 1 ) ⊆ T i 2 + BB ( s Σ b i +1 , b 2 ) . From Theorem 5.2 , we kno w that T i 2 + BB ( s Σ b i +1 , b 1 ) ⊢ Ψ b 1 , which implies that T i 2 + BB ( s Σ b i +1 , b 2 ) ⊢ Ψ b 1 , as well. This means by the witnessing theorem (Theorem 4.11 ) that Ψ b 1 ∈ S T Σ b i [ O (1) , p oly ( b 2 ( m ))] , where m = n · b 1 ( n ) is the input size. Ho we ver , from ( 2 ), we have that poly ( b 2 ( m )) < b 1 ( n ) , and, from Theorem 1.5 , Ψ b 1 ∈ S T Σ b i [ r ( m ) + 1 , q ( m )] if r ( m ) q ( m ) < b 1 ( n ) , which shows that Ψ b 1 ∈ S T Σ b i [ O (1) , p oly ( b 2 ( m ))] , thus we get a contradiction. Pr oof of Theorem 1.10 . W e know from Theorem 5.1 that T i 2 + LIND ( s Σ b i +1 , b 1 ) ⊢ Φ b 1 . On the other hand, by the proof of Theorem 1.4 , if r ( m ) < b 1 ( n ) for m = n · b 1 ( n ) , then Φ b 1 ∈ S T Σ b i [ r ( m ) , p oly ( m )] , which gives us from ( 2 ) that Φ b 1 ∈ S T Σ b i [ p oly ( b 2 ( m )) , 1] . This comes in contradiction with T i 2 + LIND ( s Σ b i +1 , b 2 ) ⊢ Φ b 1 by the witnessing theorem (Theorem 4.15 ). Pr oof of Theorem 1.11 . From Theorem 5.1 , we hav e that T i 2 + LIND ( s Σ b i +1 , b ) ⊢ Φ b , while from Theo- rem 1.4 , Φ b ∈ S T Σ b i [ O (1) , p oly ( m )] . Thus, if we assume that T i 2 + BB ( s Σ b i +1 , p oly ) ⊢ Φ b , we get a contradiction by Theorem 4.11 . Pr oof of Theorem 1.12 . From Theorem 5.2 , we ha ve that T i 2 + BB ( s Σ b i +1 , b 1 ) ⊢ Ψ b 1 , while from Theo- rem 1.5 and from ( 2 ), Ψ b 1 ∈ S T Σ b i [ p oly ( b 2 ( m )) , 1] . Thus, if we assume that T i 2 + LIND ( s Σ b i +1 , b 2 ) ⊢ Ψ b 1 , we get a contradiction by Theorem 4.15 . 28 6 Unpr ov ability Results 6.1 Unpro vability of cir cuit upper bounds In this section we will extend the result of Kraj ´ ı ˇ cek and Oliv eira [ KO17 ] to the theory PV 1 + BB (Σ b 1 ) , we will proceed as they do, b ut with some tweaks to adapt the proof for the stronger theory . 8 Deﬁnition 6.1. Let c, k ≥ 1 and let p be a unary PV -symbol. W e say a PV -symbol f (possibly with some ﬁxed parameters) describes a uniform sequence of | p (1 ( n ) ) | -size families of circuits of size c · n k if on the input 1 ( n ) , it produces a sequence of 5 -tuples ( i, 1 ( n ) , u, v , w ) , where i is the index of the circuit bounded by | p (1 ( n ) ) | , u, v are the indices of the nodes all of which hav e length polynomial in n , w is the description for the gate type of v (or the description that it is an input node). W e assume that the output node is determined by a special tuple. The number of the 5 -tuples which start with any giv en i is bounded by c · n k . And for each i ≤ | p (1 ( n ) ) | we will use f i (1 ( n ) ) to denote the sequence of all tuples starting with i . The follo wing is straightforward, so we omit the proof. Lemma 6.2. Let c, k ≥ 1 , let p be a unary PV -symbol and let f be a PV -symbol that describes a uniform sequence of | p (1 ( n ) ) | -size families of cir cuits of size c · n k . Then ther e is a PV -symbol ˜ f that decides the language L ′ succ deﬁned below . Assume that L ′ dc is the dir ect connectivity language of the sequence of | p (1 ( n ) ) | -sized families of cir cuits described by f , that is the languag e of all the 5 -tuples in the description of the family for any input size . W e let L ′ succ be the language that for eac h ( i, 1 ( n ) , u, v , w ) ∈ L ′ dc contains the 6 -tuple ( i, B in ( n )01 ( n 1 / (3 k ) ) , u, v , w , 1 ( t ) ) , wher e B in ( n ) is the dyadic numeral for n which is at most constantly longer than | n | and t is c hosen to pad the length of the 6 -tuple to length exactly ⌈ n 1 / (2 k ) ⌉ . Deﬁnition 6.3. Let c, k ≥ 1 and let f be a unary PV -symbol. W e deﬁne the sentence UP k,c ( f ) as ( ∀ 1 ( n ) )( ∃ C n , a circuit of size at most c · n k )( ∀ x, | x | = n )( f ( x ) = 1 ↔ CircuitV al ( C n , x ) = 1) , where CircuitV al ( C , x ) is a PV -symbol which ev aluates the circuit encoded by C on the input x . The follo wing lemma was already pro ved by Kraj ´ ı ˇ cek and Oliv eira in the strength which is suf ﬁcient for our purpose, so we omit its proof. Lemma 6.4 ([ K O17 , Lemma 3.1]) . F or every d ≥ 1 there is a PV -symbol g d +1 deciding a language in DTIME [ n d +1 ] , such that for every PV -symbol h deciding a language in DTIME [ n d ] with advice of length n 2 / 3 ther e is c h ≥ 1 , such that PV 1 ⊢ ( ∀ 1 ( n ) , n ≥ c h )( ∀ a, | a | = n 2 / 3 )( ∃ x, | x | = n )( h ( x, a )  = g d +1 ( x )) . 8 Let us note, that a more direct proof of this unprov ability was dev eloped in the follo w-up work of Carmosino, Kabanets, K olokolo va and Oliveira [ CKKO22 ]. The authors were able to adapt the proof from [ K O17 ], which in volv es a greater degree of manipulation of the theories in question. Considering the more direct proof in the context of our witnessing theorems could be of further interest. 29 Lemma 6.5. Let c, k ≥ 1 , let p be a unary PV -symbol and let f be a PV -symbol that describes a uniform sequence of | p (1 ( n ) ) | -size families of cir cuits of size c · n k and let ˜ f be the PV -symbol fr om the statement of Lemma 6.2 . Assume that PV 1 + BB (Σ b 1 ) ⊢ UP k,c ( ˜ f ) and let g denote g 3 k +3 fr om Lemma 6.4 . Then ther e e xists c f such that PV 1 + BB (Σ b 1 ) ⊢ ( ∀ 1 ( n ) , n ≥ c f )( ∀ i ≤ | p (1 ( n ) ) | )( ∃ x, | x | = n )( g ( x )  = Cir cuitV al ( f i (1 ( n ) ) , x )) . Pr oof. Assume that in a model of PV 1 + BB (Σ b 1 ) the provability f ails, that is ( ∀ c f )( ∃ 1 ( n ) , n ≥ c f )( ∃ i ≤ | p (1 ( n ) ) | )( ∀ x, | x | = n )( g ( x ) = CircuitV al ( f i (1 ( n ) ) , x )) (*) we will argue that in such a model there is a polynomial-time algorithm h which will certify that g is inﬁnitely often in DTIME [ n 3 k +2 ] /n 2 / 3 . Let c f be arbitrary , and let i and n be such that (*) is valid. By our assumption UP k,c ( ˜ f ) , we know that there are circuits recognizing the language of the 6 -tuples ( i, B in ( n )01 , u, v , w , 1 ( t ) ) , decided by circuits D m on the input size m = ⌈ n 1 / (2 k ) ⌉ . On the input x of length n the algorithm h tries all possible 3 -tuples ( u, v , w ) and for each of them runs the D m on the corresponding 6 -tuple where i is taken from ( ∗ ) , this is at most O ( n 2 k +1 ) possibilities and it takes time O ( n 2 k +2 ) to run D m on all of them. Thus by UP k,c ( ˜ f ) , the algorithm h obtains a description of a circuit of size at most c · n k and uses it to compute a value for g ( x ) in time O ( n 2 k ) . Thus, g can be computed by h ∈ DTIME [ n 2 k +2 ] /n 2 / 3 where the advice contains the description of D m . This is in contradiction with Lemma 6.4 . Lemma 6.6. Let f , ˜ f , g and p be as in the pr e vious lemma. Then there exists c f and a PV -symbol e such that PV 1 + BB (Σ b 1 ) ⊢ ( ∀ 1 ( n ) , n ≥ c f )( ∀ i ≤ | p (1 ( n ) ) | )( g ( e ( i, 1 ( n ) ))  = Cir cuitV al ( f i (1 ( n ) ) , e ( i, 1 ( n ) ))) . Pr oof. Follows straightforw ardly from Herbrand’ s theorem. Theorem 6.7. F or every k ≥ 1 ther e is a unary PV -symbol h such that for every constant c ≥ 1 PV 1 + BB (Σ b 1 ) ⊬ UP k,c ( h ) . Pr oof. W e will proceed by contradiction. Assume that there is a k where for an y unary h there is a constant c such that PV 1 + BB (Σ b 1 ) ⊢ UP k,c ( h ) . In particular , there is a constant c ≥ 1 such that PV 1 + BB (Σ b 1 ) ⊢ UP k,c ( g ) . By Theorem 4.11 , there is a polynomial-time student s which generates sets of polynomially many answers for the candidate circuit computing the symbol g and after r -many rounds of correction from the teacher it always outputs at least one circuit which computes g on the input length 1 ( n ) . 30 Assume that s 1 , . . . , s r are the PV -symbols computing the set of answers of the student from the teachers answer for each of the r -many rounds, these can be straightforwardly constructed from the subterms of s . That is, there is a PV -symbol p such that N | = ( ∃ i ≤ | p (1 ( n ) ) | )( g (( x 1 ) i ) = CircuitV al ( s 1 i (1 ( n ) ) , ( x 1 ) i )) ∨ ( ∃ i ≤ | p (1 ( n ) ) | )( g (( x 2 ) i ) = CircuitV al ( s 2 i (1 ( n ) , x 1 ) , ( x 2 ) i )) . . . ∨ ( ∃ i ≤ | p (1 ( n ) ) | )( g (( x r ) i ) = CircuitV al ( s r i (1 ( n ) , x 1 , . . . , x r − 1 ) , ( x r ) i )) , where ( − ) i denotes the PV -symbol which e xtracts the i -th element of a sequence. In the rest of the proof we will sho w that from the assumption, that there is a ˜ c 1 such that PV 1 + BB (Σ b 1 ) ⊢ UP k, ˜ c 1 ( s 1 (1 ( n ) )) , we can in vok e Lemma 6.6 to falsify the ﬁrst disjunct. Repeating this r -many times, we obtain a contradic- tion. By our assumption, there is a ˜ c 1 ≥ 1 such that PV 1 + BB (Σ b 1 ) ⊢ UP k, ˜ c 1 ( s 1 (1 ( n ) )) , by Lemma 6.6 there is a PV -symbol e 1 and c 1 , such that PV 1 + BB (Σ b 1 ) ⊢ ( ∀ 1 ( n ) , n ≥ c 1 )( ∀ i ≤ | p (1 ( n ) ) | )( g ( e 1 ( i, 1 ( n ) ))  = CircuitV al ( s 1 i (1 ( n ) ) , e 1 ( i, 1 ( n ) ))) , by substituting ˜ e 1 (1 ( n ) ) for x 1 , where ˜ e 1 (1 ( n ) ) is the PV -symbol computing the sequence ( e 1 ( i, 1 ( n ) )) i ≤| p (1 ( n ) ) | , we falsify the ﬁrst disjunct and obtain N | = ( ∃ i ≤ | p (1 ( n ) ) | )( g (( x 2 ) i ) = CircuitV al ( s 2 i (1 ( n ) , ˜ e 1 (1 ( n ) )) , ( x 2 ) i )) ∨ ( ∃ i ≤ | p (1 ( n ) ) | )( g (( x 3 ) i ) = CircuitV al ( s 3 i (1 ( n ) , ˜ e 1 (1 ( n ) ) , x 2 ) , ( x 3 ) i )) . . . ∨ ( ∃ i ≤ | p (1 ( n ) ) | )( g (( x r ) i ) = CircuitV al ( s r i (1 ( n ) , ˜ e 1 (1 ( n ) ) , x 2 , . . . , x r − 1 ) , ( x r ) i )) . T o proceed further , we again use our assumption to obtain ˜ c 2 ≥ 1 such that PV 1 + BB (Σ b 1 ) ⊢ UP k, ˜ c 2 ( s 2 (1 ( n ) , ˜ e 1 (1 ( n ) ))) , and apply Lemma 6.6 again. After r -many in vokation of Lemma 6.6 , we obtain that the empty disjunction is true in N which is a contradiction. 6.2 Unpro vability of a verage-case circuit lo wer bounds The following result closely follo ws the result in [ PS21 ], where we do all the modiﬁcations needed for the corresponding theory and the Student-T eacher Game. Another very helpful exposition for the proof can be found in [ LO23 ]. W e express the av erage-case lo wer bound NSubExp ⊆ Avg − coNSIZE [2 n δ ] in an instance manner . That is we assume we have a nondeterministic T uring machine M whose running time is a constructiv e function 31 t ( n ) with t ( n ) = 2 o ( n ) . The lo wer bound states that there is no co-nondeterministic circuit D of size s ( n ) , where we will hav e s ( n ) = 2 n δ , which approximates ef ﬁciently M , in the sense that it outputs the same for at least 1 / 2 + 2 − n δ fraction of the inputs. W e can express that with the follo wing sentence in the language of PV : LB ( M , s, m, n 0 ) := ∀ n ≥ n 0 ∈ LogLog ∀ D ∈ coNSIZE [ s ( n )] ∃ m = m ( n ) , ( x 1 , . . . , x m ) ∈ ( { 0 , 1 } n ) m ∀ i ≤ m ( n ) Error M ,D ( x i ) where we deﬁne the formula Erro r M ,D ( x ) =  ∃ y , z M ( x, y ) = 1 ∧ D ( x, z ) = 0  ∨  ∀ y ′ , z ′ M ( x, y ) = 0 ∧ D ( x, z ) = 1  , which means that M ( x )  = D ( x ) . Note, that the size of y , z , y ′ , z ′ are at most 2 n δ . Theorem 6.8. F or every n 0 ∈ N , δ ∈ Q ∩ (0 , 1) , and a non-deterministic T uring machine M with running time t ( n ) = 2 o ( n ) , it holds that PV 1 + LLIND ( s Σ b 1 ) ⊢ LB ( M , 2 n δ , 2 n / 2 − 2 n / 2 n δ , n 0 ) . Pr oof. Assume that PV 1 + LLIND ( s Σ b 1 ) ⊢ LB ( M , 2 n δ , 2 n / 2 − 2 n / 2 n δ , n 0 ) . W e hav e two consequences: 1. PV 1 + LLIND ( s Σ b 1 ) ⊢ LB ( M , 2 n δ , 1 , n 0 ) , which means that PV 1 also shows the worst case lo wer bound; for all co-nondeterministic circuits D , there is at least one input, where M and D do not agree. 2. N | = LB ( M , 2 n δ , 2 n / 2 − 2 n / 2 n δ , n 0 ) , which means that in the standard model there is no co- nondeterministic circuit D that can approximate M with success rate greater than 1 / 2 + 2 − n δ . In this proof, we assume the ﬁrst point to ﬁnd a contradiction with the second point. From the witnessing theorem for the theory PV 1 + LLIND ( s Σ b 1 ) and the ﬁrst point, we get that the formula LB ( M , 2 n δ , 1 , n 0 ) can be witnessed by a Student-T eacher Game with poly-logarithmic number of rounds. Howe ver , since n ∈ LogLog , this means that the length of the input is 2 O ( n ) , which gi ves us n k rounds for some k ∈ N . This k is ﬁxed from now on, since it depends only from the proof in PV 1 + LLIND ( s Σ b 1 ) , and we will refer to the number of rounds as r := n k . W e simulate the Student by the polynomial-time function f , which takes as input the number 2 n in unary , the description of a co-nondeterministic circuit D , the number of round i and depending on i , i − 1 counterexamples in the form of y ′ , z ′ . On every round, f runs in time 2 O ( n ) . After each round, f outputs a triplet ( x, y , z ) , where x is the candidate instance for M ( x )  = D ( x ) , and y , z are the candidate witnesses for the case that M ( x ) = 1 and D ( x ) = 0 , respectiv ely . W e will denote the outputs for the respecti ve round i ≤ r as ( x i , y i , z i ) . W e use as input for the Student-T eacher Game, a speciﬁc type of co-nondeterministic circuit, D w ( x ) , which are deﬁned using the Nisan-W igderson generator [ NW94 ]. W e ﬁx some c ≥ max(2 δ − 1 ( k + 1) , 4) and for some m ′ with n c ≤ m ′ ≤ 2 n c , we consider a boolean 2 n × m ′ matrix A = { a i,j } . For ev ery i ∈ [2 n ] , we deﬁne the set J i ( A ) of all j ∈ [ m ′ ] such that a i,j = 1 (the 1 ’ s in the i -th ro w). Then, A is a ( n, n c/ 2 ) -design if for all i , | J i ( A ) | = n c/ 2 and, for all i  = j , | J i ( A ) ∩ J j ( A ) | ≤ n . In [ NW94 ], they construct such a design, and they show that there is n 9 c/ 2 -size circuit that given w ∈ { 0 , 1 } m ′ and i ∈ [2 n ] , outputs the restriction of w on the bits index ed by J i ( A ) . Then, we deﬁne the circuits D w as D w ( x ) = ¬ M ( w | J x ( A )) . 32 If we assume that M is computable by a nondeterministic circuit of size 2 n ϵ , then D w ( x ) can be computed in co-nondeterministic size 2 n cϵ , which is less than 2 n δ , if we choose ϵ < δ / 2 c . From now on, we work with a speciﬁc size n , so there is no ambiguity with indexing the rounds with numbers which are a function of n . Particularly , r = n k is also ﬁxed. W e will use some more deﬁnitions. For a ∈ { 0 , 1 } m ′ − n c/ 2 and b ∈ { 0 , 1 } n c/ 2 , we deﬁne r x ( a, b ) as the bit-string of length m ′ with the bits of b into the positions index ed by J x ( A ) and the bits of a in the remaining positions. W e also deﬁne the trace of a string w ∈ { 0 , 1 } m ′ as the sequence of the x -outputs of the Student during the Student-T eacher Game with input the circuit D w . For this, we need to specify the outputs of the T eacher (e.g. the T eacher outputs the minimum counterexample lexicographically). Then, tr ( w ) = ( x 1 , x 2 , . . . , x t ) with t ≤ r , where for x t we ha ve that the Student succeeds, in other words M ( x t )  = D ( x t ) . Lemma 6.9. Ther e is a trace T = ( X 1 , X 2 , . . . , X t ) with t ≤ r and some a ∈ { 0 , 1 } m ′ − n c/ 2 , such that considering all the tr aces of the form tr ( r X t ( a, y )) for all y ∈ { 0 , 1 } n c/ 2 , a fr action of s ≥ 1 / ( n 3 t − 1 2 nt ) of them start with T and a fraction of at least (2 / 3 − 1 /n ) s of them ar e equal to T . W e do not revise the proof of this lemma, since there are already proofs in [ Kra11 , Pic15a , PS21 ] and a detailed proof of the av eraging ar gument in Appendix E of [ LO23 ]. W e ﬁx the trace T and the string a from the lemma, and with their help, we will deﬁne a new co- nondeterministic circuit, D that approximates M for the input size n c/ 2 . W e want this D to simulate the Student-T eacher Game for the input n , but in order to achie ve this, we need to simulate the T eacher’ s answer , as well. This is where we use the property of the Nisan-W igderson design. W e only care about instances of the Game, where the input co-nondeterministic circuit is in the form D w with w = r X t ( a, b ) for some b ∈ { 0 , 1 } n c/ 2 . From the structure of the initial sentence, we see that for any x ∈ { 0 , 1 } n , the T eacher must determine countere xamples y ′ x , z ′ x such that M ( x, y ′ x ) = 1 or D w ( x, z ′ x ) = 0 . If there is a an appropriate y ′ x , then this has size at most 2 n and it is independent from the input of the Game, D w . If there is no such y ′ x , then the T eacher must ﬁnd some z ′ x , such that D w ( x, z ′ x ) = 0 , or equiv alently , M ( w | J x ( A ) , z ′ x ) = 1 . Ho we v er , there are only n c/ 2 bits of w which are not ﬁxed by a , and from those only | J x ( A ) ∩ J X t ( A ) | ≤ n actually contribute for the function, which we get by the ( n, n c/ 2 ) -design. This means we can describe the function x 7→ z ′ x with 2 O ( n ) bits, too. In total, the answer of the T eacher for each x can be described by some advice Y x of size 2 O ( n ) . Thus, the total advice we need to simulate the T eacher for polynomial number of dif ferent inputs is still of size 2 O ( n ) . Next, we describe a co-nondeterministic algorithm with advice of size 2 O ( n ) , which is equi v alent with a co-nondeterministic circuit of size 2 O ( n ) , denoted D . The input must ha ve size n c/ 2 . Input : A string y ∈ { 0 , 1 } n c/ 2 . Advice: The trace T = ( X 1 , . . . , X t ) , a ∈ { 0 , 1 } m ′ − n c/ 2 , b 0 , b 1 ∈ { 0 , 1 } and Y x for all x ∈ T . 1 Deﬁne w = r X t ( a, y ) ; 2 Simulate the Student-T eacher Game on input D w ( x ) := ¬ M ( w | J x ( A )) until the round t ≤ r using the T eacher’ s answers Y x .; 3 If at any point of the simulation, the trace of w does not match T , then output b 0 ; // We take b 0 to be the majority bit among the values of M on all the inputs that do not have T as prefix to their trace. 4 If the simulation works correctly until round t , output b 1 . // We take b 1 = M ( X t ) . Algorithm 2: The pseudocode of the algorithm D that approximates M on input sizes n c/ 2 . 33 W e will sho w that D correctly approximates M on input sizes n c/ 2 . F or input y , if the trace of r X t ( a, y ) does not hav e T as a preﬁx, we know that the algorithm outputs the majority bit b 0 . In this case, the algorithm is correct with probability at least 1 / 2 . On the other hand, we know from Theorem 6.9 and the selection of the advice, that for at least 1 / ( n 3 t − 1 2 nt ) fraction of the inputs, the corresponding trace has T as preﬁx, and for (2 / 3 − 1 /n ) fraction of them, their trace is e xactly T . When the latter is the case, the algorithm correctly computes M ( y ) , since from the deﬁnition of the trace, the last element is when the Student succeeds in the Game, which means that M ( X t )  = D w ( X t ) . From the advice beat b 1 = M ( X t ) , we get that D ( y ) = b 1 = ¬ D w ( X t ) = M ( w | J X t ( A )) w = r X t ( a,y ) = = = = = M ( y ) . Therefore, in the case of inputs that hav e T as preﬁx in their trace, the probability that D is correct is at least 2 / 3 − 1 /n . Overall, Pr[ D ( y ) = M ( y )] ≥ 1 2 +  2 3 − 1 n  1 n 3 t − 1 2 nt ≥ 1 2 + Ω  1 2 n k +1  . The second inequalty is due to the fact that t ≤ r = n k . Since the input size is n c/ 2 , we get a contradiction if 2 n k +1 ≤ 2 ( n c/ 2 ) δ , which means that we must have c ≥ 2 δ − 1 ( k + 1) , which is the case for our choice of c . 6.3 Unpro vability of both in a single theory Both of the previous section used the structure of the Student-T eacher protocol related to the theory . On one hand, it seems to be challenging to adapt unprovability of Kraj ´ ı ˇ cek and Oliv eira for logarithmically many adaptiv e rounds of the student. On the other hand it seems do wnright impossible to use any known witnessing theorems to extend the unprovability reuslt of Pich and Santhanam to polynomially many queries per round, as these queries can cover all possible values for the witness. It is not hard to see that the theory PV 1 + BB ( s Σ b 1 , log ) is contained in both PV 1 + BB (Σ b 1 ) and PV 1 + LLIND ( s Σ b 1 ) and thus both unprovability results hold for this theory , which is by our results stronger than PV 1 assuming NP ⊆ P /pol y . Corollary 6.10. 1. F or every k ≥ 1 there is a unary PV -symbol h such that for every constant c ≥ 1 PV 1 + BB ( s Σ b 1 , log ) ⊬ UP k,c ( h ) . 2. F or every n 0 ∈ N , δ ∈ Q ∩ (0 , 1) , and a non-deterministic T uring machine M with running time t ( n ) = 2 o ( n ) , it holds that PV 1 + BB ( s Σ b 1 , log ) ⊢ LB ( M , 2 n δ , 2 n / 2 − 2 n / 2 n δ , n 0 ) . 3. Assuming NP ⊆ P /pol y , we have PV 1 ≡ PV 1 + BB ( s Σ b 1 , log ) . Pr oof. Part 1 follows immediately from Theorem 6.7 . By Theorem 2.9 we obtain that PV 1 + LLIND ( s Σ b 1 ) ⊢ BB ( s Σ b 1 , log ) , and thus part 2 follo ws from Theorem 6.8 . P art 3 is a direct application of Theorem 1.9 . 34 Refer ences [All91] Bill Allen. Arithmetizing uniform NC . Annals of Pur e and Applied Logic , 53(1):1–50, 1991. 9 , 11 [A vi02] Jeremy A vigad. Saturated models of uni versal theories. Annals of Pur e and Applied Logic , 118(3):219–234, 2002. 10 , 21 [Bus85] Samuel R Buss. Bounded arithmetic . 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Journal of Symbolic Logic , 61(3):942–966, 1996. 1 , 4 A Function Calculations A.1 T ransformation of Input Sizes Here, we describe ho w to compute the function p ( n ) gi v en the input n and show that it is actually a solution to the relations m = n · p ( n ) and p ( n ) = g ( m ) , for a function g : N → N , which has sublinear gro wth rate, it is unbounded and increasing. For the general statements of Theorems 1.4 and 1.5 , where no logical theory is in volved, it is enough for us to sho w that there is an algorithm that determines the v alue p ( n ) for e very input n . In the theories we consider , we will specify p ( n ) by some function in the language. This does not need to satisfy p ( n ) = g ( m ) , since for the separation of theories we do not need the ﬁne-grained separation we prov e in these general theorems. W e ﬁx the input n , which we assume it is big enough. W e need to ﬁnd a natural number k , such that k = g ( n · k ) . Then, p ( n ) = k . For that, we deﬁne the function h n ( k ) = g ( n · k ) − k , and no w our goal is to sho w that for big enough n , the equation h n ( k ) = 0 always has a solution. For k = 1 , we ha ve that h n (1) = g ( n ) − 1 , so it is unbounded and increasing, like g ( n ) . This means that for some n 0 , for all n ≥ n 0 , h n (1) > 0 . On the other hand, we want to sho w that for all these n , h n ( k ) will be ne gati ve at some point. From the deﬁnition of h n ( k ) , we hav e that h n ( k ) k = g ( n · k ) k − 1 = n · g ( n · k ) n · k − 1 . 37 Ho we ver , if k tends to inﬁnity , then g ( n · k ) n · k tends to 0 , since g is sublinear . This means that there is some big enough k , such that h n ( k ) e ventually becomes ne gati v e. W e hav e sho wed that h n (1) > 0 and for some big value of k , h n ( k ) < 0 . The only thing remaining is that the v alue 0 is actually achiev ed. For that, we will sho w that when h n ( k ) > 0 , then h n ( k + 1) ≥ 0 , which means that before the change to negati ve values, we actually get the solution to h n ( k ) = 0 . Since h n ( k ) > 0 and h n ( k ) is natural, we get h n ( k ) ≥ 1 = ⇒ g ( n · k ) ≥ k + 1 . This means that h n ( k + 1) = g ( n · ( k + 1)) − ( k + 1) ≥ g ( n · k ) − ( k + 1) ≥ 0 , as we wanted. The ﬁrst inequality comes from the fact that g is increasing. A.2 Gro wth of Sublinear Functions Suppose that we have two increasing functions b 1 ( x ) , b 2 ( x ) ≤ x , such that for any k ∈ N and for large enough x , b 1 ( x ) > b 2 ( x ) k . Then, we want to sho w the inequality , for large enough n 0 : ∀ n ≥ n 0 ∀ k b 2 ( n · b 1 ( n )) k < b 1 ( n ) From the assumption we hav e that ∀ n ≥ n 0 ∀ k b 2 ( n · b 1 ( n )) k < b 1 ( n · b 1 ( n )) . W e can change the abov e to sho w that also ∀ n ≥ n 0 ∀ k ∀ k 0 b 2 ( n · b 1 ( n )) k < b 1 ( n · b 1 ( n )) 1 /k 0 ≤ b 1 ( n 2 ) 1 /k 0 , where for the second inequality we use the fact that b 1 ( n ) ≤ n . For the last step, we combine the abo ve relation with ∀ n ≥ n 0 ∃ k 0 b 1 ( n 2 ) 1 /k 0 < b 1 ( n ) , which is true, because otherwise, for b ( n 0 ) > 1 , ∃ n ≥ n 0 ∀ k 0 b 1 ( n 2 ) > b 1 ( n ) k 0 , which is a contradiction, since the right-hand side tends to inﬁnity . 38

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