Metric Structures and Probabilistic Computation

METRIC STR UCTURES AND PR OBABILISTIC COMPUT A TION WESLEY CAL VER T Abstract. Con tinuous first-order logic is used to apply mo del-theoretic anal- ysis to analytic structures ( e.g. Hil bert spaces, Banac h spaces, probabilit y spaces, etc.). Classical c omputable model theory is used to examine the al- gorithmic str ucture of mathematical ob jects that can b e describ ed i n classi- cal first-order logic. The presen t paper shows that probabilistic computation (sometimes called randomized computation) can play an analogous role f or structures describ ed in con tin uous first- or der l ogic. The m ai n r esult of this pap er is an effect ive completene ss theorem, sho w- ing that ev ery decidable contin uous first-order the ory has a probabilis tically decidable mo del. Later sections give examples of the application of this frame- wo rk to v arious classes of structures, and to some problems of computational complexit y theory . 1. Introduction Contin uous firs t-order log ic w as introduced in [3, 1] as a mo del-theoretic con- text sufficient to ha ndle stability theory for so-ca lled “metric structures.” These are man y-so rted structures in which each sort is a complete metric space of finite diameter. Key examples include Hilber t s paces, Banac h spaces, probability spaces, and probability spaces with a distinguished a utomorphism. F or cla ssical first-order mo del theory , there is a mea ningful sense of computation and effectiveness: In a gr oup, for instance , we hav e a r easonable algorithmic under- standing of a gr oup if the set of triple s constituting the Ca ley table (equiv ale n tly , the set of w ords equal to the identit y e lemen t) is decida ble. Of co urse, ther e ar e often still many algorithmic unknowns in the gro up, such as the conjugacy problem and the isomorphism problem [9 ]. The aim of the pr esent pap er is to provide a similar framework for cont inuous first-order logic. The framew ork suggested is probabilis tic computation. This mo del of computa- tion has s een wide use in complexity theory [14, 27], and there is some ro om for hop e that an understanding of the relatio nship b et ween cont inuous and classica l first-order log ic might yield insight s into the relationship b etw ee n pr obabilistic a nd deterministic computation. Section 6 gives reasons for such hope. Not to get ca rried aw ay in spe culation, though, it is still cause for con tentm ent that a wa y can be found to mea ningfully talk ab out a lgorithmic infor mation in the con text of metric structures. The imposs ibilit y of finding an algorithm to solve arbitrary Diophant ine equa tions (see [19]), the relationship of isop erimetric functions to the w ord problem [24, 5], and muc h more depend o n a notion of computation adequate to the co n text of c ount able ring s (in the case o f Diophan tine Date : No vem b er 8, 2021. Key wor ds and phr ases. Computable M odel Theory , Probabilistic Computation , Randomized Computation, Contin uous Logic, Metric Structures. 1 2 WESLEY CAL VER T equations) a nd g roups (in the case of the word pro blem). Some pr eliminary re sults on spe cific metric structures, g iven in Section 5, will sug gest that there is ground for fruitful resea rch in the effectiv e theory of these structures. A key argument that probabilistic computation is the right algo rithmic frame- work for this cont ext is that it admits an effective completeness theor em. The classical theorem is this. Theorem 1.1 (Effectiv e Completeness Theorem) . A (classic al ly) de cidable the ory has a (classic al ly) de cidable mo del. A full pro of o f this result may b e found in [12], but it was known m uch earlier, at least to Millar [20]. The main theor etical contribution of the present pap er will be to in terpret the terms of this theorem in s uc h a wa y as to apply to a contin uous first-order theory a nd probabilistic computatio n. The main result o f the present pap er is the pro of in Se ction 4 o f the theorem naturally corresp onding to Theorem 1.1. Section 2 will desc ribe the syntax a nd s emant ics for con tin uous first-order logic. The r eader familiar with [2] or [1] will find nothing new in Section 2, except a choice of a finite set of logical connectives (no such choice is yet canonical in con- tin uous first-order lo gic). Section 3 will define probabilistic T ur ing machines and the cla ss of structures they compute. Se ction 4 will con tain the pro of of the main result. Section 5 will cont ain some examples, exhibiting differen t asp ects of the information whic h is c onv eyed by the statemen t that a certain s tructure is prob- abilistically computable, a nd in Section 6 we will co nclude with some remark s o n time complexity of structures. 2. Continuous First-Order Logic W e will, in keeping with the existing literature on c ont inuous first-o rder logic, adopt the slightly unusual conv ention o f using 0 as a n umerical v alue fo r T rue (or acceptance) and 1 as a n umerical v alue for F alse (or rejection). The a uthors of [3] chose this conv ention to emphasize the metric nature o f their lo gic. Contin uous first-or der lo gic is a n extension o f Luk asiewicz prop ositional log ic. The following definitions are from [2]. 2.1. Seman ti cs. Definition 2.1. A c ontinuous signatur e is an ob ject o f the for m L = ( R , F , G , n ) where (1) R and F are disjo in t and R is nonempt y , a nd (2) n is a function asso ciating to each member of R ∪ F its arit y (3) G has the form { δ s,i : (0 , 1] → (0 , 1] : s ∈ R ∪ F and i < n s } Members o f R a re called r elation symb ols , and members o f F fun ction symb ols . W e now define the class o f structures. Definition 2.2. Let L = ( R , F , G , n ) be a contin uous signatur e. A c ontinu ous L -pr e-stru ctur e is an o rdered pair M = ( M , ρ ), where M is a no n-empt y set, and ρ is a function on R ∪ F such that (1) T o each function symbol f , the function ρ assigns a mapping f M : M n ( f ) → M METRIC STR UCTURES AND PROBABILISTIC COMPUT A TION 3 (2) T o eac h rela tion symbol P , the function ρ assigns a ma pping f M : M n ( P ) → [0 , 1]. (3) The function ρ a ssigns d to a pseudo- metric d M : M × M → [0 , 1 ]. (4) F or each f ∈ F for e ach i < n f , and for ea c h ǫ ∈ (0 , 1], we have ∀ ¯ a, ¯ b, c, e  d M ( c, e ) < δ f ,i ⇒ d M  f M (¯ a, c, ¯ b ) , f M (¯ a, e, ¯ b )  ≤ ǫ  where l h (¯ a ) = i and lh ( ¯ a ) + l h ( ¯ b ) = n f − 1 . (5) F or each P ∈ R for eac h i < n P , and for ea c h ǫ ∈ (0 , 1], we have ∀ ¯ a, ¯ b, c, e  d M ( c, e ) < δ f ,i ⇒ | P M (¯ a, c, ¯ b ) − P M (¯ a, e, ¯ b ) | ≤ ǫ  where l h (¯ a ) = i and lh ( ¯ a ) + l h ( ¯ b ) = n P − 1 . Definition 2.3. A c ontinuous we ak L -struct ur e is a contin uous L -pr e-structure such that ρ assigns to d a metric. Since w e are concerned here with countable structures (i.e. those accessible to computation), we will no t use the str onger notion o f a c ontinuous L -structur e co m- mon in the litera ture, which req uires that ρ b e assigned to a c omplete metric. How ever, it is po ssible, g iven a contin uous w eak s tructure (even a pre-str ucture), to pass to a completion [2]. Definition 2.4. Le t V denote the set of v ariables , and let σ : V → M . Let ϕ be a formula. (1) The int erpr etation under σ of a term t (written t M ,σ ) is defined by r eplacing each v ariable x in t b y σ ( x ). (2) Let ϕ b e a formula. W e then define the val ue of ϕ in M under σ (written M ( ϕ, σ )) as follows: (a) M ( P ( ¯ t ) , σ ) := P M ( t M ,σ ) (b) M ( α . − β , σ ) := ma x ( M ( α, σ ) − M ( β , σ ) , 0) (c) M ( ¬ α, σ ) := 1 − M ( α, σ ) (d) M ( 1 2 α, σ ) := 1 2 M ( α, σ ) (e) M (sup x α, σ ) := sup a ∈ M M ( α, σ a x ), where σ a x is eq ual to σ except that σ a x ( x ) = a . (3) W e write ( M , σ ) | = ϕ exactly when M ( ϕ, σ ) = 0. Of co urse, if ϕ has no free v ariables , then the v alue of M ( ϕ, σ ) is independent o f σ . 2.2. Syn tax. Definition 2.5. Let S 0 be a set of distinct prop ositio nal sy m b ols. Let S b e freely generated from S 0 by the formal bina ry op eratio n . − and the unary op erations ¬ and 1 2 . Then S is said to b e a c ont inuous pr op ositional lo gic . W e now define truth a ssignments for con tinuous propos itional logic. Definition 2.6. Let S b e a contin uous prop os itional logic. (1) if v 0 : S 0 → [0 , 1] is a mapping, we can ex tend v 0 to a unique mapping v : S → [0 , 1] by setting (a) v ( ϕ . − ψ ) := max ( v ( ϕ ) − v ( ψ ) , 0) (b) v ( ¬ ϕ ) := 1 − v ( ϕ ) (c) v ( 1 2 ϕ ) = 1 2 v ( ϕ ) 4 WESLEY CAL VER T W e s ay that v is the truth assignment defined b y v 0 . (2) W e write v | = Σ for so me Σ ⊆ S whenever v ( ϕ ) = 0 for all ϕ ∈ Σ. Roughly , ϕ . − ψ has the sense of ψ → ϕ . W e can also, of co urse, define ϕ ∧ ψ as ϕ . − ( ϕ . − ψ ), a nd ϕ ∨ ψ via deMor gan’s law. W e can also define something resembling equiv alence, | ϕ − ψ | = ( ϕ . − ψ ) ∨ ( ψ . − ϕ ). Luk asiewicz prop ositiona l logic is the fra gment of this logic which do es not in volv e 1 2 . T o make a firs t-order predicate v ariant of this logic, we use sup in the place of ∀ and inf in the pla ce of ∃ (with the o bvious semantics, as will b e descr ibe d in what follows). W e t y pically als o include a binar y function d , whose standar d inter- pretation is generally a metric. Now we giv e the syntactic axioms for contin uous first-order logic: (A1) ( ϕ . − ψ ) . − ϕ (A2) (( χ . − ϕ ) . − ( χ . − ψ )) . − ( ψ . − ϕ ) (A3) ( ϕ . − ( ϕ . − ψ )) . − ( ψ . − ( ψ . − ϕ )) (A4) ( ϕ . − ψ ) . − ( ¬ ψ . − ¬ ϕ ) (A5) 1 2 ϕ . − ( ϕ . − 1 2 ϕ ) (A6) ( ϕ . − 1 2 ϕ ) . − 1 2 ϕ . (A7) (sup x ψ . − sup x ϕ ) . − sup x ( ψ . − ϕ ) (A8) ϕ [ t/x ] . − sup x ϕ where no v a riable in t is bound by a quantifier in ϕ . (A9) sup x ϕ . − ϕ , wherever x is not free in ϕ . (A10) d ( x, x ) (A11) d ( x, y ) . − d ( y , x ) (A12) ( d ( x, z ) . − d ( x, y )) . − d ( y , z ) (A13) F or each f ∈ F , each ǫ ∈ (0 , 1], and each r , q ∈ D with r > ǫ and q < δ f ,i ( ǫ ), the axiom ( q . − d ( z , w )) ∧ ( d ( f ( ¯ x, z , ¯ y ) , f ( ¯ x, w , ¯ y )) . − r ), where l h ( ¯ x ) + l h ( ¯ y ) = n f − 1 . (A14) F or each P ∈ R , each ǫ ∈ (0 , 1 ], and e ach r, q ∈ D with r > ǫ and q < δ P,i ( ǫ ), the a xiom ( q . − d ( z , w )) ∧ (( P ( ¯ x, z , ¯ y ) . − P ( ¯ x, w , ¯ y )) . − r ), where l h ( ¯ x ) + l h ( ¯ y ) = n P − 1 . Axioms A1–A4 ar e those of Luk asiewic z prop ositional logic, and ax ioms A5–A6 are thos e of contin uous prop ositional lo gic. Axioms A7 –A9 des crib e the role of the qua n tifiers. Axioms A10–A12 guara n tee that d is a pseudo metric, and axio ms A13–A14 guarantee uniform con tin uity o f functions a nd r elations. W e write Γ ⊢ Q ϕ whenever ϕ is prov able from Γ in co n tinuous first-order logic. Where no confusio n is likely , w e will wr ite Γ ⊢ ϕ . 3. Probabilisticl y Comput able Structures If M is a T uring mac hine, w e wr ite M x ( n ) for the result o f applying M to input n with oracle x . Excepting the p olar it y change to matc h the conv entions ab ov e, the following definition is standar d; it ma y b e found, for instance, in [27]. Definition 3.1. Le t 2 ω be the set of infinite binary seque nces, with the usual Leb esgue probability measure µ . (1) A pr ob abilistic T uring machine is a T ur ing machine equipp ed with a n ora cle for an element of 2 ω , called the ra ndom bits , with output in { 0 , 1 } . (2) W e say that a pr obabilistic T uring ma c hine M ac c epts n with pr ob ability p if and o nly if µ { x ∈ 2 ω : M x ( n ) ↓ = 0 } = p . METRIC STR UCTURES AND PROBABILISTIC COMPUT A TION 5 (3) W e say that a probabilistic T uring mac hine M r eje cts n with pr ob ability p if and o nly if µ { x ∈ 2 ω : M x ( n ) ↓ = 1 } = p . Definition 3.2. Let L b e a computable contin uous signature. Let M be a co n- tin uous L -structure. Let L ( M ) b e the expansion of L by a cons tan t c m for each m ∈ M (i.e. a unary predicate c m ∈ R where c M m ( x ) := d ( x, m )). Then the c on- tinuous atomic dia gr am of M , written D ( M ) is the set of all pa irs ( ϕ, p ), whe re ϕ is a quantifier-free (i.e. sup- and inf -free) sentence in L ( M ) and M ( ϕ, σ ) = p . The contin uous elemen tary diagram D ∗ ( M ) is the same, ex cept that ϕ is no t req uired to be quantifi er-fre e. Note that the definition is indep enden t o f σ , since a sentence has no free v a riables. Definition 3.3. W e say that a contin uous pre-structure M is pr ob abilistic al ly c om- putable (resp ectively , pr ob abilistic al ly de cidable ) if and only if there is some prob- abilistic T uring machine T such that, for e v ery pair ( ϕ, p ) ∈ D ( M ) (r espec tiv ely , D ∗ ( M )) the machine T accepts ϕ with pr obability p . Suppo se T is a deterministic mac hine (i.e. o ne that makes no use o f its ra ndom bits; a classical T uring mac hine) and M a class ical fir st-order structure. The n this definition corresp onds exa ctly to the classical definition o f a computable structure. W e ca nnot do entirely without the pro babilistic ma c hines (that is, w e cannot thoroughly understand pro babilistically computable structures using only cla ssical T uring ma ch ines), as the fo llw oing result s hows. Lemma 3. 4 (No Derandomization Lemma) . Ther e is a pr ob abilistic al ly c omputable we ak struct ur e M such that the set { ( ϕ, p ) ∈ D ( M ) : p ∈ D} is not classic al ly c omputable. Pr o of. Let U b e a computably enumerable set, and let S be the co mplemen t of U . W e fir st construct a proba bilistically computable function f such tha t P ( f σ ( x ) = 0) = 1 2 if and only if x ∈ S . A t stag e t , if x ∈ U t , pic k t wo strings σ t , τ t of leng th t + 2 such that f t ( x ) do es not halt with random bits σ t or τ t . W e define the function f t +1 := f t ∪ { f σ t ( x ) = 0 , f τ t ( x ) = 1 } . On the other hand, if x ∈ U t , then we arrang e that f σ t +1 ( x ) = 0 for all σ of length at most t + 2 where f σ t ( x ) do es not halt. Let f = S t ∈ ω f t . Now if x ∈ S , we nev er see x ∈ U t , so f ( x ) = 0 with pr obability 1 2 . Otherwise, there is some t such that x ∈ U t − U t − 1 , and then f ( x ) = 0 with probability 1 − t P i =2 2 − i > 1 2 . Now we let M b e the structure ( ω , f ), where f is in terpreted as a unary predicate in the obvious way , and d is the discrete metric. If we could decide mem b ership in { ( ϕ, p ) ∈ D ( M ) : p ∈ D } with a class ical T uring machine, then we c ould also decide mem b ership in U .  Of course, this sa me a rgument could work for any other uniformly co mputable set of re als in place of D . T o some exten t, though, w e co uld do without the probabilistic machines. The following results sho w that the sta temen t of the No Derandomizatio n Lemma is the stro ngest po ssible. 6 WESLEY CAL VER T Prop ositio n 3. 5. F or any pr ob abilistic al ly c omput able pr e-structu r e M , ther e is some (classic al ly) c omputable function f , monotonic al ly incr e asing in t he se c ond variable, and some (classic al ly) c omputable function g , monotonic al ly de cr e asing in the se c ond variable, such that for any p air ( ϕ, p ) ∈ D ( M ) , we have lim s →∞ f ( ϕ, s ) = p and lim s →∞ g ( ϕ, s ) = p . Pr o of. Let M b e computed by the probabilistic T uring machine T M . Let ( σ s ) s ∈ ω be an effectiv e list of all strings in 2 <ω . Now w e define f ( ϕ, s ) := X i ≤ s  T σ i M ( ϕ ) · 2 − lh ( σ i )  . The definition of g is symmetric. These clear ly hav e the cor rect prop erties.  F unctions of the same for m as f and g are often seen in cla ssical computable mo del theory [15, 13, 6, 7]. Corollary 3.6. F or any pr ob abilistic al ly c omputable pr e-struct ur e M , the set of p airs, ( ϕ, p ) ∈ D ( M ) is the c omplement of a (classic al ly) c omput ably enumer able set. Pr o of. F ollows immediately from Prop osition 3.5.  These limitations not withsta nding, the definition via probabilistic ma chin es gives a mor e natur al contin uity with the established literature on contin uo us first-or der mo del theory [1]. In addition, this definition is in an y case no t dispensable when, for instance, time co mplexit y of co mputation is at issue (see Section 6). 4. Effective Completeness Theorem 1.1 is an imp or tan t piece of evidence that cla ssical T uring co mputation (or any of the many equiv alent conce pts) is pr op erly synchronized with classica l first-order logic. In particular, it asser ts that under the minimal, obviously neces- sary h yp otheses, a class ical first-order theory has a mo del which can be r epresented by a classica l computation. The aim of the pres en t section is to pr ov e a similar result fo r co n tinuous first-o rder lo gic a nd probabilistic co mputation. The followin g analogue to the c lassical concept o f the decidability of a theor y was proposed in [2]. Definition 4.1. Let L be a c ont inuous signature and Γ a s et of formulas of L . (1) W e define ϕ ◦ Γ := sup { M ( ϕ, σ ) : ( M , σ ) | = Γ } . (2) If T is a complete contin uous first-o rder theory , we sa y that T is de cidable if and o nly if there is a (classica lly) computable function f such that f ( ϕ ) is an index for a computable real num ber equal to ϕ ◦ T . Theorem 4.2. L et T b e a de cidable c ontinuous first-or der the ory. Then ther e is a pr ob abilistic al ly de cidable c ont inuous pr e-structur e M such that M | = T . Pr o of. The cons truction o f a model M is given in [2], by an analogue o f Henkin’s metho d. O ur mo del will b e essentially the same, exce pt that some care must b e taken with effectiveness. The principal conten t of the theorem co nsists in s howing that this structure is probabilistica lly decidable. W e will define a probabilis tic T uring ma ch ine which, for an y formula ϕ , accepts ϕ with pr obability M ( ϕ ). METRIC STR UCTURES AND PROBABILISTIC COMPUT A TION 7 W e b egin by adding Henkin witnesses. Let D denote the dy adic num b ers in the int erv a l (i.e. those of the for m k 2 n for k , n ∈ N ). Definition 4.3. Le t Γ b e a set of form ulae. Then Γ is sa id to b e Henkin c omplete if for every form ula ϕ , every v ariable x , a nd ev ery p < q ∈ D , ther e is a cons tan t c such that (sup x ϕ . − q ) ∧ ( p . − ϕ [ c/x ]) ∈ Γ . Lemma 4. 4. We c an effe ct ively extend T t o a c onsistent set Γ of formulae which is Henkin c omplete. Pr o of. Let L 0 = L . F or each n , let L n +1 be the res ult of adding, for each for m ula ϕ in L n , and for each x, p, q as in the previous definition, a new consta n t c ( ϕ,x,p,q ) . W e can als o extend the theory T , beginning with Γ 0 = T . F o r ea c h n , the set Γ n +1 is pro duced by adding to T , fo r each formula ϕ in L n and each x, p, q as in the previous definition, the formula (sup x ϕ . − q ) ∧ ( p . − ϕ [ c ( ϕ,x,p,q ) /x ]). Let Γ = S n Γ n . The consistency of Γ is demonstrated in [2]. Note that this co nstruction is in every w ay effective. In pa rticular, there is a (classically) c omputable function which will, g iven p, q ∈ D and G¨ o del n umbers for ϕ and x , give us a G¨ odel n umber for c ( ϕ,x,p,q ) . Moreov er, the set Γ is (clas sically) computable.  W e wr ite L ∗ = S n L n , and C = { c ( ϕ,x,p,q ) } . Lemma 4.5. We c an effe ctively ex tend Γ t o a c onsistent set ∆ 0 such that for al l formulae ϕ, ψ in L ∗ we have ϕ . − ψ ∈ ∆ or ψ . − ϕ ∈ ∆ 0 . Pr o of. W e set ∆ 0 = Γ . At stage s + 1, for ea ch pair ψ , ϕ of sentences from L ∗ such that neither ψ . − ϕ nor ϕ . − ψ is in ∆ s , we proc eed a s follows. Let θ b e the conjunction of all elements of ∆ s , and let ¯ c be the c onstants fro m C which o c cur in ( ψ . − ϕ ) . − θ . W e then c heck (effectiv e ly , s ince the theory is decidable), whether ( ∀ ¯ x (( ψ . − ϕ ) . − θ ) ( ¯ x / ¯ c )) ◦ T = 0. If so, then we add ψ . − ϕ to for m ∆ s +1 . Otherwise, we do so with ϕ . − ψ . Now ∆ 0 = S s ∆ s is a s required. That this extension is consistent is established in [2].  Definition 4.6. Let ∆ b e a set of formulas. W e say that ∆ is maximal c onsistent if ∆ is cons isten t and for a ll formulae ϕ, ψ w e have (1) If ∆ ⊢ ϕ . − 2 − n for all n , then ϕ ∈ ∆, and (2) ϕ . − ψ ∈ ∆ or ψ . − ϕ ∈ ∆. Now let ∆ 0 = S s ∆ s , and Λ = { ϕ : ∀ n [ δ 0 ⊢ ϕ . − 2 − n ] } . Now ∆ = ∆ 0 ∪ Λ is maximal consisten t, b y construction of ∆ 0 . Let M be the mo del of T whose universe is the set o f closed terms in C , as in [2]. W e now define the probabilistic T uring machine G which will witness that M is probabilistically computable. W e set K A 0 = K R 0 = A 0 = R 0 = ∅ . W e define the functions E ( S ) = { σ ⊇ τ : τ ∈ S } and P ( S ) = X σ ∈ S 1 2 lh ( a ) . 8 WESLEY CAL VER T A t stage s , if ∆ s ⊢ ϕ . − k 2 n , t hen we will arr ange that G a ccepts ϕ with probabilit y at least 1 − k 2 n . If K A s = ∅ , then we find 2 n − k nodes σ 1 , . . . , σ 2 n − k of length n in 2 <ω − E ( K R s ), and le t K A s +1 = { σ 1 , . . . , σ 2 n − k } . If K A s is no nempt y and P ( K A s ) ≥ 1 − k 2 n , then we do nothing with K A . If K A s is nonempty a nd P ( K A s ) < 1 − k 2 n , then we find so me s et Σ o f elements of 2 <ω − E ( K R s ) with leng th n so that P ( K A s ∪ Σ) = 1 − k 2 n , and let K A s +1 = K A s ∪ Σ. If ∆ s ⊢ k 2 n . − ϕ then we will ar range tha t G rejects ϕ with probability at least k 2 n . If K R s = ∅ , then we find k no des σ 1 , . . . , σ k of length n in 2 <ω − E ( K A s ), and let K R s +1 = { σ 1 , . . . , σ k } . I f K R s is no nempt y a nd P ( K R s ) ≥ k 2 n , then we do nothing with K R . If K R s is nonempt y and P ( K R s ) < k 2 n , then w e find some set Σ of elements of 2 <ω − E ( K A s ) with length n so that P ( K R s ∪ Σ) = k 2 n , and let K R s +1 = K R s ∪ Σ. A t this point, it is necessar y to verify that certain asp ects o f the cons truction describ ed so far are actually p ossible. In particula r, w e need to s how that when we s earch for elements of 2 <ω − E ( K A s ), for instance, there will be some. Now if E ( K A s ) contains more than 2 n − k 1 elements, w e must hav e P ( K A s ) < 1 − k 1 2 n , so that w e must have had ∆ s ⊢ ϕ . − k 1 2 n (Note that if ∆ s ⊢ ϕ . − p a nd q > p , then also ∆ s ⊢ ϕ . − q ). Lemma 4 . 7. If ther e is some s su ch that ∆ s ⊢ ϕ . − k 1 2 n and ∆ s ⊢ k 2 n . − ϕ , then (1 − k 1 2 n ) + k 2 n ≤ 1 . Pr o of. Supp ose no t. Then 2 n − k 1 + k > 1, so that k − k 1 > 0 and k > k 1 . How ever, we als o ha ve k 2 n . − k 1 2 n = 0, so that k 1 ≥ k , a contradiction.  The situation for finding elemen ts o f 2 <ω − E ( K R s ) is sy mmetric. Returning to the construction, at stage s , w e will add mor e instructions. W e will guara n tee that for any σ ∈ E ( K A s ), we will have G σ ( ϕ ) ↓ = 0, and for a n y σ ∈ E ( K R s ) we will hav e G σ ( ϕ ) ↓ = 1. Let ϕ b e a sentence in L ∗ , and supp ose M ( ϕ ) = p . W e need to show that G accepts ϕ with proba bilit y p . Since ∆ is maximal consistent, for each q 0 , q 1 ∈ D with q 0 ≤ M ( ϕ ) ≤ q 1 , there was some s for w hic h ∆ s ⊢ ϕ . − q 1 and for which ∆ s ⊢ q 0 . − ϕ , and at that sta ge, we ensured that G would accept ϕ with pr obability betw een q 0 and q 1 . Since this is true for all q 0 ≤ p ≤ q 1 ∈ D , it must fo llow that G accepts ϕ with pr obability p .  W e can strengthen Theorem 4.2 to pro duce a contin uo us weak structure if we require T to b e c omplete . Definition 4. 8. Let M b e a con tinuous L -s tructure. W e w rite T h ( M ) for the set of contin uous L -sentences ϕ s uc h that M ( ϕ ) = 0. W e say that T is c omplete if T = T h ( M ) for some M . Corollary 4. 9. L et T b e a c omplete de cidable c ontinuous first-or der the ory. Then ther e is a pr ob abilistic al ly de cidable we ak stru ctur e M su ch that M | = T . Pr o of. If the signature has no metric, then Theo rem 4.2 suffices. Otherwis e, we note that T must contain the sentence sup x,y (( x = y ) . − d ( x, y )), so that when we apply Theorem 4.2, the function d M is a metric o n M .  METRIC STR UCTURES AND PROBABILISTIC COMPUT A TION 9 5. Examples A full trea tmen t o f ea ch of the following clas ses of examples suggests a pap er — or many pa per s — of its own. How ever, in each ca se some suggestion is given of the kind of data giv en by the assumption that an elemen t of the class is probabilistically computable. 5.1. Hilb ert Spaces. A pre -Hilber t space ov er a top olo gical field F is a vector space with an inner pro duct meeting all requirements of being a Hilbert space except p erhaps that it ma y not b e co mplete with r espe ct to the no rm. The a uthors of [1] identif y a pre-Hilb ert space H with the many-sorted weak structure M ( H ) = (( B n ( H ) : n ≥ 1) , 0 , { I mn } m 0 , a pr ob abilistic al ly c omputable norme d ve ctor sp ac e X and a c omputable nonline ar mapping A : X → X such that d ( A ( u ) − A ( v )) ≤ γ d ( u, v ) for s ome γ < 1 , pr o duc e an element x ∗ such that d ( x ∗ , A ( x ∗ )) < ǫ . Pr o of. Fix so me u 0 ∈ X , and let i > log γ ǫ d ( A ( u 0 ) ,u 0 ) . Such a n i can b e found effectively by standard approximations. W r ite u k for A k ( u 0 ). Then d ( A ( u i +1 ) , u i +1 ) = d ( A ( u i +1 ) , A ( u i )) ≤ γ d ( u i +1 , u i ) ≤ · · · ≤ γ i d ( A ( u 0 ) , u 0 ) < ǫ. Let u ∗ = u i +1 , and the r esult holds.  This shows that appr oximate weak solutions to certain differen tia l equations (for instance, s ome o f r eaction-diffusion type, see [10]) can b e found effectively in prob- abilistically computable structures. The key insight of this fixed p oint r esult is this. The traditiona l view of e ffectiv e mo del theory has asked whether a pa rticular the- orem is “effectiv ely true.” By contrast, the impor tant question for applications is more typically whether t he theo rem is “effectively ne arly true.” That is, approxima- tions ar e go o d enough, and often all that is necessa ry . F or solutio ns of differential equations mo deling applications, for insta nce, practitioners o ften “would rather hav e an a ccurate numerical solution of the correct mo del than an explicit s olution of the wr ong mo del. Explicit s olutions are so rare that fast acc urate n umerical METRIC STR UCTURES AND PROBABILISTIC COMPUT A TION 11 analysis is essential” [28]. While it is not new to obse rve that classica lly unsolv able problems can b e effectively approximated, the framework of proba bilistically com- putable structures in contin uous first-orde r logic is one that calls attent ion to the po ssibility of approximate solutions, rather than to the imp oss ibilit y of exact ones . 5.3. Probabilit y Spaces. Let X = ( X , B , µ ) b e a probability space. W e say that B ∈ B is an atom if µ ( B ) > 0 and there is no B ′ ∈ B with B ′ ⊆ B and 0 < µ ( B ′ ) < µ ( B ). W e say that X is atomless if and only if B co n tains no atoms. Let ˆ B b e the quo tien t of B by the relation B 1 ∼ B 2 if and only if µ ( B 1 △ B 2 ) = 0. The authors of [1] iden tified X w ith the structure  ˆ B , 0 , 1 , · c , ∩ , ∪ , µ  with the metric d ( A, B ) = µ ( A △ B ). Theorem 5.5 ([1]) . Ther e is a c ontinuous first-or der the ory AP A su ch t hat (1) AP A is finitely axiomatiza ble. (2) AP A is c omplete. (3) AP A admits quantifier eliminatio n. (4) The c ontinuous pr e-structu r es satisfying AP A ar e exactly the atomless pr ob- ability sp ac es, r epr esente d as ab ove. Since AP A is finitely axioma tizable a nd complete, it is als o decida ble. Thus, Theorem 4.9 g ives us a probabilistically decidable model. Of course, any sepa rable probability str ucture can b e approximated b y a co un table dense set. A standard issue in effectiv e model theory is whether t wo is omorphic structures m ust b e iso morphic via a computable function. T he following result shows that the answer for probability structur es is affirma tiv e. Prop ositio n 5.6. L et B and C b e isomorph ic, pr ob abilistic al ly c omputable atom- less pr ob ability structur es with u niverses ˆ B and ˆ C , re sp e ctively. Then ther e is a (classic al ly) c omputable function witn essing the isomorphism. Pr o of. The iso morphism is constr ucted by a sta ndard ba ck-and-forth ar gument . Suppo se that f : B → C is a finite partial iso morphism, and that x ∈ B − dom ( f ). W e wish to find some y ∈ C suc h that f ∪ { ( x, y ) } is still a partial isomorphism. Without loss of genera lit y , we may ass ume that x is not in the substructure o f B gener ated by dom ( f ). Let a 0 , . . . , a n be the atoms of the substructure of B generated by dom ( f ). W e ma y ass ume, without los s of gener ality , that ea ch a i is in dom ( f ). Now the isomorphism type of x is determined by the v alues µ ( x ∩ a i ). Since B is atomless, there is an element y ∈ ˆ C such that for each i w e have µ B ( x ∩ a i ) = µ C ( y ∩ f ( a i )), and since B and C ar e proba bilistically computable, we can effectiv e ly find this y . The extens ion to a new element of C is entirely symmetric. The union of the pa rtial isomorphisms co nstructed in this way will be a computable function, and will b e an iso morphism from B to C .  5.4. Probabilit y Spaces with a Distingui s hed Autom o rphism. A standard sort o f enric hment in stabilit y theory is to expand a k nown structure b y adding a new function symbol to define a new function, and to specify that this function b e generic. Fix a n in terv al I under the Lebesg ue measure λ , and let L b e the a lgebra of measur able sets. L et G denote the group o f measure preserv ing automorphisms of ( I , L , λ ), mo dulo the re lation of almost everywhere agr eement . Let τ ∈ G . Now 12 WESLEY CAL VER T τ induces an automorphism on ( ˆ L, 0 , 1 , · c , ∩ , ∪ , λ ) in a straightforw ard way (see [4, 1]). W e say that τ ∈ G is ap erio dic if for every positive int eger n we hav e λ { x ∈ I : τ n ( x ) = x } = 0 . T o hav e a countable structure of this type, we could take a coun table dense s ubset I ′ ⊆ I , and for X ⊆ I ′ , set λ ( X ) = λ ( cl ( X )). In [4] and [1], an a xiomatization is g iven for the theory of a tomless probability spaces with a distinguis hed ap erio dic a utomorphism. This theory is complete and admits elimination of quantifiers. The authors of [4] show that entrop y aris es as a mo del-theoretic rank . The r esult b elow pa rtially descr ibes the deg ree of algor ithmic co n trol we can exp ect on iteratio ns o f suc h an automorphism. Befor e sta ting the r esult, though, a probabilistic analog ue to computable enumerabilit y should be g iven: Definition 5 . 7. W e say that a set is pro babilistically computably enumerable if and only if there is some probabilistic T uring machine M such that • If x ∈ S , then for any q < 1 the machine M a ccepts x with probability at least q , a nd • If x / ∈ S , then there is s ome q < 1 s uc h that M accepts x with probabilit y at most q . In particular (and esp ecially in lig h t of the time complexity considerations in Section 6), if we specify an error tolerance q , there is some s such that M ( s, x ) is below q whenev er x ∈ S , and (assuming the tolerance is sufficien tly small) no such s otherwise. Theorem 5.8. L et I = ( ˆ L, 0 , 1 , · c , ∩ , ∪ , λ, τ ) b e a pr ob abilistic al ly c omputable pr ob- ability st ructur e b ase d on a dense subset of the u n it interval, with a me asur e- pr eserving tra nsformation τ . L et A ⊆ I b e a set of p ositive me asur e, define d without quantifiers in c ontinuous first- or der lo gic. Write A for the set S n ∈ ω τ n ( A ) . Then fo r any isomorphism f : I → J to a pr ob abilistic al ly c omputable st r u ctur e J , the set f ( A ) is pr ob abilistic al ly c omputably enumer able. Pr o of. T ow a rd par t 1, note that A is defined by the infinitary disjunction ϕ ( x ) = _ n ∈ ω τ − n ( x ) ∈ A and tha t the set A is defined by a quantifier-free contin uous first-or der formula. The isomo rphism f must prese rve sa tisfaction o f ϕ — that is , I | = ϕ ( x ) if and only if J | = ϕ ( f ( x )). Now let M b e a pro babilistic T ur ing machine such that M ( x, s ) is the minimum v alue of V n ≤ k τ − n ( x ) ∈ A , where k r anges ov er all num b er s less than or equal to s . Then M witnesses that f ( A ) is probabilistica lly computably enum erable.  6. Time Complexity of Structures One of the most important applications of proba bilistic T uring mac hines is their role in computational complexity theo ry (see [16, 21, 14]). Let P be so me decision problem. W e say that Q is of class RP if and only if there is a probabilistic T uring machine M Q , halting in time p olynomia l in the length of the input, such that if METRIC STR UCTURES AND PROBABILISTIC COMPUT A TION 13 x ∈ Q , then M Q accepts x with pr obability a t leas t 3 4 , and if x / ∈ Q , then M Q rejects x with pro bability 1 . This clas s has the prop erty that P ⊆ RP ⊆ NP . 1 Another complex it y clas s of interest is the clas s BPP . W e say tha t Q is o f class BPP if and only if ther e is a pr obabilistic T uring machine M Q , halting in time po lynomial in the length of the input, such that if x ∈ Q , then M Q accepts x with probability at leas t 3 4 , and if x / ∈ Q , then M Q rejects x with probability a t le ast 3 4 . Here we k now that RP ⊆ BPP ⊆ Σ p 2 ∩ Π p 2 . Definition 6 .1 (Cenzer–Remmel [8]) . Let A b e a computable structure. W e say that A is uniformly po lynomial time if the atomic diagram of A is a p oly nomial time set. Clearly , A ∈ P if a nd only if the structure ( ω , A ) is p olynomia l time. Also, fo r any p olynomial time structure M a nd any quantifier-free definable A ⊆ M n , we hav e A ∈ P . W e can extend Definition 6.1 in a r outine wa y for probabilistically computable structures. Definition 6.2. W e sa y that a pro babilistically computable s tructure is polynomial time if and only if there is some pr obabilistic T uring ma c hine T such that, for every pair ( ϕ, p ) ∈ D ( M ) the machine T halts in p olynomial time and accepts ϕ with probability p . Now we can c haracter ize the members of BPP in terms of con tinuous weak struc- tures. Theorem 6.3. The class BPP c an b e identifie d with the class of quantifier-fr e e definable sets in p olynomial time pr ob abilistic al ly c omputable s t ructur es in the fol- lowing way: (1) L et A ∈ BPP b e a su bset of ω . Th en ther e is a p olynomial time pr ob a- bilistic al ly c omputable we ak s t ructur e M and a p olynomial t ime c omputable function f : ω → M such that ther e is a quantifier-fr e e formula ϕ ( x ) such that ϕ ( x ) ≤ 1 4 for x ∈ f ( A ) and ϕ ( x ) ≥ 3 4 for x ∈ M − f ( A ) . (2) L et M b e a p olynomial time pr ob abilistic al ly c omputable we ak structu re , and let A, B b e quantifier-fr e e disj oint definable subsets of M n , wher e inf { d ( x, y ) : x ∈ A, y ∈ B } > 0 and A ∪ B is classic al ly c omputably enumer able. Then A and B ar e e ach of class BPP . Pr o of. T ow a rd the fir st p oint, let M b e a proba bilistic T uring machine witnessing that A ∈ BPP . W e let M b e the structure ( ω , A ), where A is a una ry pr edicate and A ( x ) is the pr obability tha t M accepts x . W e give M the discrete metric. F or th e seco nd p oint, let A b e defined by ϕ ( ¯ x ), and B by ψ ( ¯ x ). Now for ¯ a ∈ A ∪ B , to chec k whether ¯ a ∈ A , we c ompute M ( ϕ (¯ a ) . − ψ ( ¯ a )). The computatio n runs in po lynomial time, and ¯ a is accepted with pro bability at least 1 2 + inf { d (¯ a, y ) : y ∈ B } when ¯ a ∈ A and with probability at mos t 1 2 − inf { d (¯ a, y ) : y ∈ B } when ¯ a ∈ B .  1 In b oth this and the succeeding paragraph, th e particular fraction 3 4 is not critical. Using a so-called “Amplification Lemma, ” an y fr action ab o ve and b ounded a wa y from 1 2 will do [27]. 14 WESLEY CAL VER T References 1. I. Ben Y aaco v, A. Berenstein, C. W. Henson, and A . Usvy atso v, Mo del the ory for metric structur es , T o app ear in a Newton Institute volume in the Lecture N otes series of the London Mathematical Society , 2007. 2. I. Ben Y aaco v and A. P . P edersen, A pr o of of c ompleteness for c ontinuous first-or der lo g i c , preprint, 2008. 3. I. Ben Y aaco v and A. 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