We consider exponentially large finite relational structures (with the universe {0.1}ⁿ) whose basic relations are computed by polynomial size (nO(1)) circuits. We study behaviour of such structures when pulled back by P/poly maps to a bigger or to a smaller universe. In particular, we prove that: 1. If there exists a P/poly map g: {0.1} → {0.1}m, n < m, iterable for a proof system then a tautology (independent of g) expressing that a particular size n set is dominating in (...) a size 2ⁿ tournament is hard for the proof system. 2. The search problem WPHP, decoding RSA or finding a collision in a hashing function can be reduced to finding a size m homogeneous subgraph in a size 22m graph. Further we reduce the proof complexity of a concrete tautology (expressing a Ramsey property of a graph) in strong systems to the complexity of implicit proofs of implicit formulas in weak proof systems. (shrink)
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)
We define the notion of approximate Euler characteristic of definable sets of a first order structure. We show that a structure admits a non-trivial approximate Euler characteristic if it satisfies weak pigeonhole principle $WPHP_{n}^{2n}$ : two disjoint copies of a non-empty definable set A cannot be definably embedded into A, and principle CC of comparing cardinalities: for any two definable sets A. B either A definably embeds in B or vice versa. Also, a structure admitting a non-trivial approximate Euler characteristic (...) must satisfy $$WPHP_{n}^{2n}$ . Further we show that a structure admits a non-trivial dimension function on definable sets if and only if it satisfies weak pigeonhole principle $WPHP_{n}^{2n}$ : for no definable set A with more than one element can $A^2$ definably embed into A. (shrink)
This article is a continuation of our search for tautologies that are hard even for strong propositional proof systems like EF, cf. [14, 15]. The particular tautologies we study, the τ-formulas, are obtained from any ᵊ/poly map g; they express that a string is outside of the range of g. Maps g considered here are particular pseudorandom generators. The ultimate goal is to deduce the hardness of the τ-formulas for at least EF from some general, plausible computational hardness hypothesis. In (...) this paper we introduce the notions of pseudo-surjective and iterable functions (related to free functions of [15]). These two properties imply the hardness of the τ-formulas from the function but unlike the hardness they are preserved under composition and iteration. We link the existence of maps with these two properties to the provability of circuit lower bounds, and we characterize maps g yielding hard τ-formulas in terms of a hitting set type property (all relative to a propositional proof system). We show that a proof system containing EF admits a pseudo-surjective function unless it simulates a proof system WF introduced by Jerabek [11]. an extension of EF. We propose a concrete map g as a candidate function possibly pseudo-surjective or free for strong proof systems. The map is defined as a Nisan-Wigderson generator based on a random function and on a random sparse matrix. We prove that it is iterable in a particular way in resolution, yielding the output/input ratio $n^{3-\varepsilon}$ (that improves upon a direct construction of Alekhnovich et al. [2]). (shrink)
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)
We recall the notions of weak and strong Euler characteristics on a first order structure and make explicit the notion of a Grothendieck ring of a structure. We define partially ordered Euler characteristic and Grothendieck ring and give a characterization of structures that have non-trivial partially ordered Grothendieck ring. We give a generalization of counting functions to locally finite structures, and use the construction to show that the Grothendieck ring of the complex numbers contains as a subring the ring of (...) integer polynomials in continuum many variables. We prove the existence of a universal strong Euler characteristic on a structure. We investigate the dependence of the Grothendieck ring on the theory of the structure and give a few counter-examples. Finally, we relate some open problems and independence results in bounded arithmetic to properties of particular Grothendieck rings. (shrink)
We construct a faithful interpretation of ukasiewicz's logic in product logic (both propositional and predicate). Using known facts it follows that the product predicate logic is not recursively axiomatizable.We prove a completeness theorem for product logic extended by a unary connective of Baaz [1]. We show that Gödel's logic is a sublogic of this extended product logic.
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 define a first-order extension LK(CP) of the cutting planes proof system CP as the first-order sequent calculus LK whose atomic formulas are CP-inequalities ∑ i a i · x i ≥ b (x i 's variables, a i 's and b constants). We prove an interpolation theorem for LK(CP) yielding as a corollary a conditional lower bound for LK(CP)-proofs. For a subsystem R(CP) of LK(CP), essentially resolution working with clauses formed by CP- inequalities, we prove a monotone interpolation theorem (...) obtaining thus an unconditional lower bound (depending on the maximum size of coefficients in proofs and on the maximum number of CP-inequalities in clauses). We also give an interpolation theorem for polynomial calculus working with sparse polynomials. The proof relies on a universal interpolation theorem for semantic derivations [16, Theorem 5.1]. LK(CP) can be viewed as a two-sorted first-order theory of Z considered itself as a discretely ordered Z-module. One sort of variables are module elements, another sort are scalars. The quantification is allowed only over the former sort. We shall give a construction of a theory LK(M) for any discretely ordered module M (e.g., LK(Z) extends LK(CP)). The interpolation theorem generalizes to these theories obtained from discretely ordered Z-modules. We shall also discuss a connection to quantifier elimination for such theories. We formulate a communication complexity problem whose (suitable) solution would allow to improve the monotone interpolation theorem and the lower bound for R(CP). (shrink)
A proof of the (propositional) Craig interpolation theorem for cut-free sequent calculus yields that a sequent with a cut-free proof (or with a proof with cut-formulas of restricted form; in particular, with only analytic cuts) with k inferences has an interpolant whose circuit-size is at most k. We give a new proof of the interpolation theorem based on a communication complexity approach which allows a similar estimate for a larger class of proofs. We derive from it several corollaries: (1) Feasible (...) interpolation theorems for the following proof systems: (a) resolution (b) a subsystem of LK corresponding to the bounded arithmetic theory S 2 2 (α) (c) linear equational calculus (d) cutting planes. (2) New proofs of the exponential lower bounds (for new formulas) (a) for resolution ([15]) (b) for the cutting planes proof system with coefficients written in unary ([4]). (3) An alternative proof of the independence result of [43] concerning the provability of circuit-size lower bounds in the bounded arithmetic theory S 2 2 (α). In the other direction we show that a depth 2 subsystem of LK does not admit feasible monotone interpolation theorem (the so called Lyndon theorem), and that a feasible monotone interpolation theorem for the depth 1 subsystem of LK would yield new exponential lower bounds for resolution proofs of the weak pigeonhole principle. (shrink)
This book presents an up-to-date, unified treatment of research in bounded arithmetic and complexity of propositional logic, with emphasis on independence proofs and lower bound proofs. The author discusses the deep connections between logic and complexity theory and lists a number of intriguing open problems. An introduction to the basics of logic and complexity theory is followed by discussion of important results in propositional proof systems and systems of bounded arithmetic. More advanced topics are then treated, including polynomial simulations and (...) conservativity results, various witnessing theorems, the translation of bounded formulas (and their proofs) into propositional ones, the method of random partial restrictions and its applications, direct independence proofs, complete systems of partial relations, lower bounds to the size of constant-depth propositional proofs, the method of Boolean valuations, the issue of hard tautologies and optimal proof systems, combinatorics and complexity theory within bounded arithmetic, and relations to complexity issues of predicate calculus. Students and researchers in mathematical logic and complexity theory will find this comprehensive treatment an excellent guide to this expanding interdisciplinary area. (shrink)
LK is a natural modification of Gentzen sequent calculus for propositional logic with connectives ¬ and $\bigwedge, \bigvee$ (both of bounded arity). Then for every d ≥ 0 and n ≥ 2, there is a set Td n of depth d sequents of total size O(n3 + d) which are refutable in LK by depth d + 1 proof of size exp(O(log2 n)) but such that every depth d refutation must have the size at least exp(nΩ(1)). The sets Td n (...) express a weaker form of the pigeonhole principle. (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.
