Propositional proof complexity is the study of the sizes of propositional proofs, and more generally, the resources necessary to certify propositional tautologies. Questions about proof sizes have connections with computational complexity, theories of arithmetic, and satisfiability algorithms. This is article includes a broad survey of the field, and a technical exposition of some recently developed techniques for proving lower bounds on proof sizes.

We introduce new proof systems for propositional logic, simple deduction Frege systems, general deduction Frege systems, and nested deduction Frege systems, which augment Frege systems with variants of the deduction rule. We give upper bounds on the lengths of proofs in Frege proof systems compared to lengths in these new systems. As applications we give near-linear simulations of the propositional Gentzen sequent calculus and the natural deduction calculus by Frege proofs. The length of a proof is the number of lines (...) in the proof. A general deduction Frege proof system provides at most quadratic speedup over Frege proof systems. A nested deduction Frege proof system provides at most a nearly linear speedup over Frege system where by "nearly linear" is meant the ratio of proof lengths is O) where α is the inverse Ackermann function. A nested deduction Frege system can linearly simulate the propositional sequent calculus, the tree-like general deduction Frege calculus, and the natural deduction calculus. Hence a Frege proof system can simulate all those proof systems with proof lengths bounded by O). Also we show that a Frege proof of n lines can be transformed into a tree-like Frege proof of O lines and of height O. As a corollary of this fact we can prove that natural deduction and sequent calculus tree-like systems simulate Frege systems with proof lengths bounded by O. (shrink)

The bounded arithmetic theory S2 is finitely axiomatized if and only if the polynomial hierarchy provably collapses. If T2i equals S2i + 1 then T2i is equal to S2 and proves that the polynomial time hierarchy collapses to ∑i + 3p, and, in fact, to the Boolean hierarchy over ∑i + 2p and to ∑i + 1p/poly.

This article is a continuation of our search for tautologies that are hard even for strong propositional proof systems like EF, cf. [Kra-wphp,Kra-tau]. The particular tautologies we study, the τ-formulas, are obtained from any.

We investigate the possibility to characterize (multi) functions that are Σ b i -definable with small i (i = 1, 2, 3) in fragments of bounded arithmetic T 2 in terms of natural search problems defined over polynomial-time structures. We obtain the following results: (1) A reformulation of known characterizations of (multi)functions that are Σ b 1 - and Σ b 2 -definable in the theories S 1 2 and T 1 2 . (2) New characterizations of (multi)functions that are (...) Σ b 2 - and Σ b 3 -definable in the theory T 2 2 . (3) A new non-conservation result: the theory T 2 2 (α) is not ∀Σ b 1 (α)-conservative over the theory S 2 2 (α). To prove that the theory T 2 2 (α) is not ∀Σ b 1 (α)-conservative over the theory S 2 2 (α), we present two examples of a Σ b 1 (α)-principle separating the two theories: (a) the weak pigeonhole principle WPHP (a 2 , f, g) formalizing that no function f is a bijection between a 2 and a with the inverse g, (b) the iteration principle Iter (a, R, f) formalizing that no function f defined on a strict partial order ( $\{0,\dots,a\}$ , R) can have increasing iterates. (shrink)

We consider tautologies formed from a pseudo-random number generator, defined in Krajíček [11] and in Alekhnovich et al. [2]. We explain a strategy of proving their hardness for Extended Frege systems via a conjecture about bounded arithmetic formulated in Krajíček [11]. Further we give a purely finitary statement, in the form of a hardness condition imposed on a function, equivalent to the conjecture.

We describe a general method how to construct from a propositional proof system P a possibly much stronger proof system iP. The system iP operates with exponentially long P-proofs described "implicitly" by polynomial size circuits. As an example we prove that proof system iEF, implicit EF, corresponds to bounded arithmetic theory $V_{2}^{1}$ and hence, in particular, polynomially simulates the quantified propositional calculus G and the $\pi_{1}^{b}-consequences$ of $S_{2}^{1}$ proved with one use of exponentiation. Furthermore, the soundness of iEF is not (...) provable in $S_{2}^{1}$ . An iteration of the construction yields a proof system corresponding to $T_{2} + Exp$ and, in principle, to much stronger theories. (shrink)

Propositional proof complexity is the study of the sizes of propositional proofs, and more generally, the resources necessary to certify propositional tautologies. Questions about proof sizes have connections with computational complexity, theories of arithmetic, and satisfiability algorithms. This is article includes a broad survey of the field, and a technical exposition of some recently developed techniques for proving lower bounds on proof sizes.

Paul Cohen’s method of forcing, together with Saul Kripke’s related semantics for modal and intuitionistic logic, has had profound effects on a number of branches of mathematical logic, from set theory and model theory to constructive and categorical logic. Here, I argue that forcing also has a place in traditional Hilbert-style proof theory, where the goal is to formalize portions of ordinary mathematics in restricted axiomatic theories, and study those theories in constructive or syntactic terms. I will discuss the aspects (...) of forcing that are useful in this respect, and some sample applications. The latter include ways of obtaining conservation results for classical and intuitionistic theories, interpreting classical theories in constructive ones, and constructivizing model-theoretic arguments. (shrink)

T i 2 = S i +1 2 implies ∑ p i +1 ⊆ Δ p i +1 ⧸poly. S 2 and IΔ 0 ƒ are not finitely axiomatizable. The main tool is a Herbrand-type witnessing theorem for ∃∀∃ П b i -formulas provable in T i 2 where the witnessing functions are □ p i +1.

We give combinatorial and computational characterizations of the NP search problems definable in the bounded arithmetic theories $T_{2}^{2}$ and $T_{3}^{2}$.

We prove the following results: (i) PV proves NP ⊆ P/poly iff PV proves coNP ⊆ NP/O(1). (ii) If PV proves NP ⊆ P/poly then PV proves that the Polynomial Hierarchy collapses to the Boolean Hierarchy. (iii) $S_{2}^{1}$ proves NP ⊆ P/poly iff $S_{2}^{1}$ proves coNP ⊆ NP/O(log n). (iv) If $S_{2}^{1}$ proves NP ⊆ P/poly then $S_{2}^{1}$ proves that the Polynomial Hierarchy collapses to PNP[log n]. (v) If $S_{2}^{2}$ proves NP ⊆ P/poly then $S_{2}^{2}$ proves that the Polynomial Hierarchy (...) collapses to PNP. Motivated by these results we introduce a new concept in proof complexity: proof systems with advice, and we make some initial observations about them. (shrink)

We consider the problem about the length of proofs of the sentences $\operatorname{Con}_S(\underline{n})$ saying that there is no proof of contradiction in S whose length is ≤ n. We show the relation of this problem to some problems about propositional proof systems.

We study the extension 123) of the theory S21 by instances of the dual weak pigeonhole principle for p-time functions, dWPHPx2x. We propose a natural framework for formalization of randomized algorithms in bounded arithmetic, and use it to provide a strengthening of Wilkie's witnessing theorem for S21+dWPHP. We construct a propositional proof system WF , which captures the Π1b-consequences of S21+dWPHP. We also show that WF p-simulates the Unstructured Extended Nullstellensatz proof system of Buss et al. 256). We prove that (...) dWPHP is equivalent to a statement asserting the existence of a family of Boolean functions with exponential circuit complexity. Building on this result, we formalize the Nisan–Wigderson construction in a conservative extension of S21+dWPHP. (shrink)

The four authors present their speculations about the future developments of mathematical logic in the twenty-first century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.

We consider tautologies formed form a pseudo-random number generator, defined in Krajicek [11] and in Alekhnovich et al. [2]. We explain a strategy of proving their hardness for Extended Frege systems via a conjecture about bounded arithmetic formulated in Krajicek [11]. Further we give a purely finitary statement, in the form of a hardness condition imposed on a function, equivalent to the conjecture. This is accompanied by a brief explanation, aimed at non-specialists, of the relation between prepositional proof complexity and (...) bounded arithmetic. (shrink)

Cook and Reckhow defined a propositional formulation of the pigeonhole principle. This paper shows that there are Frege proofs of this propositional pigeonhole principle of polynomial size. This together with a result of Haken gives another proof of Urquhart's theorem that Frege systems have an exponential speedup over resolution. We also discuss connections to provability in theories of bounded arithmetic.