Using appropriate notation systems for proofs, cut-reduction can often be rendered feasible on these notations, and explicit bounds can be given. Developing a suitable notation system for Bounded Arithmetic, and applying these bounds, all the known results on definable functions of certain such theories can be reobtained in a uniform way.
Deep Dive into On the computational complexity of cut-reduction.
Using appropriate notation systems for proofs, cut-reduction can often be rendered feasible on these notations, and explicit bounds can be given. Developing a suitable notation system for Bounded Arithmetic, and applying these bounds, all the known results on definable functions of certain such theories can be reobtained in a uniform way.
Bounded Arithmetic has been introduced by Buss [Bus86] as theories of arithmetic with a strong connection to computational complexity. For sake of simplicity of this introduction, we will concentrate only on the Bounded Arithmetic theories S i 2 by Buss [Bus86]. These theories are given as first order theories of arithmetic in a language which suitably extends that of Peano Arithmetic where induction is restricted in two ways. First, logarithmic induction is considered which only inducts over a logarithmic part of the universe of discourse.
ϕ(0) ∧ (∀x)(ϕ(x) → ϕ(x + 1)) → (∀x)ϕ(|x|) .
Here, |x| denotes the length of the binary representation of the natural number x, which defines a kind of logarithm on natural numbers. Second, the properties which can be inducted on, must be described by a suitably restricted (“bounded”) formula. The class of formulae used here are the Σ b i -formulae which exactly characterise Σ p i , that is, properties of the i-th level of the polynomial time hierarchy of predicates. The theory’s S i 2 main ingredients are the instances of logarithmic induction for Σ b i formulae. Let a (multi-)function f be called Σ b j -definable in S i 2 , if its graph can be expressed by a Σ b j -formula ϕ, such that the totality of f , which renders as (∀x)(∃y)ϕ(x, y), is provable from the S i 2 -axioms in first-order logic. The main results characterising definable (multi-) functions in Bounded Arithmetic are the following.
• Buss [Bus86] has characterised the Σ b i -definable functions of S i 2 as FP Σ b i-1 , the i-th level of the polynomial time hierarchy of functions.
• Krajíček [Kra93] has characterised the Σ b i+1 -definable multi-functions of S i 2 as the class FP Σ b i [wit, O(log n)] of multi-functions which can be computed in polynomial time using a witness oracle from Σ p i , where the number of oracle queries is restricted to O(log n) many (n being the length of the input).
• Buss and Krajíček [BK94] have characterised the Σ b i-1 -definable multifunctions of S i 2 as projections of solutions to problems from PLS Σ b i-2 , which is the class of polynomial local search problems relativised to Σ p i-2 -oracles. We will re-obtain all these definability characterisations by one unifying method using the results from the first part of this report in the following way. First, we will define a suitable notation system H BA for propositional derivations which are obtained by translating Bounded Arithmetic proofs. The propositional translation used here is well-known in proof-theoretic investigations; the translation has been described by Tait [Tai68], and later was independently discovered by Paris and Wilkie [PW85]. In the Bounded-Arithmetic world it is known as the Paris-Wilkie translation.
Applying the machinery from the first part we obtain a notation system CH BA of cut-elimination for H BA . CH BA will have the property that its implicit descriptions, most notably the functions tp(h) and h[i] mentioned above, will be polynomial time computable. This allows us to formulate a general local search problem on CH BA which is suitable to characterise definable multi-functions for Bounded Arithmetic. Assume that (∀x)(∃y)ϕ(x, y), describing the totality of some multi-function, is provable in some Bounded Arithmetic theory. Fix a particularly nice formal proof p of this. Given N ∈ N we want to describe a procedure which finds some K such that ϕ(N , K) holds. Invert the proof p of (∀x)(∃y)ϕ(x, y) to a proof of (∃y)ϕ(x, y) where x is fresh a variable, then substitute N for all occurrences of x. This yields a proof of (∃y)ϕ(N , y). Adding an appropriate number of cut-reduction operators we obtain a proof with all cut-formulae of (at most) the same logical complexity as ϕ. It should be noted that a notation h(N ) for this proof can be computed in time polynomial in N .
The general local search problem which finds a witness for (∃y)ϕ(N , y) can now be characterised as follows. Its instance is given by N . The set of solutions are those notations of a suitable size, which denote a derivation having the property that the derived sequent is equivalent to (∃y)ϕ(N , y) ∨ ψ 1 ∨ • • • ∨ ψ l where all ψ i are “simple enough” and false. An initial solution is given by h(N ). A neighbour to a solution h is a solution which denotes an immediate sub-derivation of the derivation denoted by h, if this exists, and h otherwise. The cost of a notation is the height of the denoted derivation. The search task is to find a notation in the set of solutions which is a fixpoint of the neighbourhood function. Obviously, a solution to the search task must exist. In fact, any solution of minimal cost has this property. Now consider any solution to the search problem. It must have the property, that none of the immediate subderivations is in the solution space. This can only happen if the last inference derives (∃y)ϕ(N , y) from a true statement ϕ(N , K) for some K ∈ N. Thus K is a witness to (∃y)ϕ(N , y), and we can outpu
…(Full text truncated)…
This content is AI-processed based on ArXiv data.
