DOCSLIB.ORG

Why Are Proof Complexity Lower Bounds Hard?

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Why Are Proof Complexity Lower Bounds Hard? 2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS) Why are Proof Complexity Lower Bounds Hard? Jan´ Pich Rahul Santhanam Department of Computer Science Department of Computer Science University of Oxford University of Oxford Oxford, United Kingdom Oxford, United Kingdom [email protected] [email protected] Abstract—We formalize and study the question of whether its limits? Or are these problems fundamentally intractable there are inherent difficulties to showing lower bounds on in some sense? propositional proof complexity. Since complexity theory is the theory of intractability, We establish the following unconditional result: Proposi- tional proof systems cannot efficiently show that truth tables it is natural to apply it to the seeming intractability of of random Boolean functions lack polynomial size non-uniform complexity-theoretic questions themselves. Since the early proofs of hardness. Assuming a conjecture of Rudich, proposi- days of complexity theory, progress on complexity lower tional proof systems also cannot efficiently show that random bounds has gone hand-in-hand with the formalization of k-CNFs of linear density lack polynomial size non-uniform various sorts of barriers to progress. In the 70s, analogies proofs of unsatisfiability. Since the statements in question assert the average-case hardness of standard NP problems (MCSP between complexity theory and recursion theory were devel- and 3-SAT respectively) against co-nondeterministic circuits oped, with various concepts and techniques from recursion for natural distributions, one interpretation of our result is theory being adapted to the resource-bounded setting. How- that propositional proof systems are inherently incapable of ever, Baker, Gill & Solovay [7] observed in the late 70s that efficiently proving strong complexity lower bounds in our popular machine simulation and diagonalization techniques formalization. Another interpretation is that an analogue of the Razborov-Rudich ‘natural proofs’ barrier holds in proof from recursion theory relativized, i.e., continued to work complexity: under reasonable hardness assumptions, there even when machines were given access to an arbitrary are natural distributions on hard tautologies for which it is oracle. By giving an oracle relative to which P = NP and infeasible to show proof complexity lower bounds for strong another oracle relative to which P = NP [7], they proved that enough proof systems. no relativizing techniques could solve the NP vs P question. For the specific case of the Extended Frege (EF) proposi- It seemed that techniques of a fundamentally different sort tional proof system, we show that at least one of the following cases holds: (1) EF has no efficient proofs of superpolynomial were required. circuit lower bound tautologies for any Boolean function or (2) After this barrier result, attention shifted to a more finitis- There is an explicit family of tautologies of each length such tic setting. Rather than considering uniform machines that that under reasonable hardness assumptions, most tautologies work for all inputs, Boolean circuits corresponding to finite are hard but no propositional proof system can efficiently functions became the object of study. It is well-known that establish hardness for most tautologies in the family. Thus, P under reasonable hardness assumptions, either the Circuit can be simulated by polynomial-size Boolean circuits, and Lower Bounds program toward complexity separations cannot therefore super-polynomial lower bounds on Boolean circuit be implemented in EF, or there are inherent obstacles to size imply lower bounds against P. implementing the Cook-Reckhow program for EF. Perhaps the hope was that circuits are simpler and more Keywords-Computational complexity, proof complexity, cir- ‘combinatorial’ objects, and therefore more amenable to cuit lower bounds, barriers lower bounds via combinatorial and algebraic techniques. Indeed, initial results were promising. In a series of I. INTRODUCTION influential works [2], [21], [24], [48] applying the technique of random restrictions, super-polynomial lower bounds A. Motivation were shown for the Parity function against constant-depth Complexity theory is full of questions that are easy to circuits. Razborov [40] and Smolensky [47] developed the state but hard to answer. The most famous of these is the P polynomial approximation technique to give lower bounds vs NP problem [14], but there are numerous others such as against constant-depth circuits with prime modular gates. the NP vs coNP problem, the PSPACE vs P problem, and Razborov [39] used the method of approximations to show the BPP vs P problem. In all of these cases, despite decades that the Clique problem required super-polynomial size of effort, very little progress has been made. Is this because monotone circuits. This sequence of works introduced complexity theory is a young field, and we have not yet had several new lower bound techniques, and it seemed that the time to develop a deep understanding of computation and steady progress was being made toward the goal of 2575-8454/19/$31.00 ©2019 IEEE 1305 DOI 10.1109/FOCS.2019.00080 separating NP and P via a Circuit Lower Bounds Program. P . Equivalently, NP = coNP iff there is a hard sequence of tautologies for every propositional proof system P . Circuit Lower Bounds Program: Separate NP and Just as the Circuit Lower Bounds Program aims to make P by proving super-polynomial lower bounds for functions progress on the NP vs P question, the work of Cook and in NP against increasingly expressive classes of Boolean Reckhow suggests an approach to making progress on the circuits. NP vs coNP question. Unfortunately, the Circuit Lower Bounds Program stalled Cook-Reckhow Program: Separate NP and coNP by in the early 90s. Even now, almost 30 years later, we proving super-polynomial lower bounds on the proof size still don’t know if there are explicit functions that require of tautologies in increasingly powerful proof systems P . super-polynomial depth-two threshold circuits, or constant- depth circuits with Mod 6 gates. However, even if no We note a couple of differences between the circuit breakthrough lower bounds were proved for a while, our complexity lower bound setting and the proof complexity understanding of barriers to lower bounds did evolve, due lower bound setting. In the circuit complexity setting, it mainly to work of Razborov [41], [42] and Razborov is easy to see by a counting argument that most Boolean & Rudich [45]. Razborov & Rudich [45] developed the functions do not have small circuits. The challenge is to concept of natural proofs, which captures known circuit show a circuit lower bound for an explicit Boolean function, lower bounds proved via combinatorial and algebraic i.e., a function in NP. In the proof complexity setting, we techniques, and showed that under standard cryptographic do not care about explicitness - we are satisfied with lower assumptions, natural proofs cannot prove super-polynomial bounds for any sequence of tautologies. In this setting, lower bounds for general Boolean circuits. counting arguments do not work to get non-constructive The natural proofs barrier has not proven to be as lower bounds, since the space of small proofs is comparable damaging to the finitistic approach to complexity as the in size to the space of tautologies. relativization barrier was to the machine-based approach. Despite these differences, progress on the Cook-Reckhow Circuits are still a popular model, and combinatorial and Program has been partly inspired by analogies with the algebraic techniques are still heavily used. Various methods Circuit Lower Bounds Program. Haken [23] showed super- to evade the natural proofs barrier have been proposed, polynomial lower bounds on the size of proofs of the Pi- though none have yet resulted in any breakthrough lower geonhole Principle in the Resolution proof system. Ajtai [3] bounds for explicit functions. obtained a significant extension of this result by showing that The meta-mathematics of circuit lower bounds is one of the Pigeonhole Principle is also super-polynomially hard for our motivations for the results in this paper, but the main Bounded-Depth Frege - a restricted version of the standard motivation comes from a different approach to complex- Frege system where the lines are bounded-depth circuits. ity theory, via proof complexity. The NP vs P problem Ajtai’s super-polynomial lower bound was strengthened to essentially asks if short proofs for mathematical theorems an exponential lower bound by [9]. Both [3] and [9] use are easy to find. Thus logic and proof are inherent to the strong versions of the random restriction technique used to problem. These aspects are emphasized not just in Steve prove lower bounds on bounded-depth circuits. Cook’s seminal paper [14] stating the problem, but already There has been significant effort devoted to proving in Godel’s famous letter to von Neumann in the early 1950s. lower bounds for constant-depth Frege proofs with modular Cook and Reckhow [18] defined the notion of a proposi- counting gates - the proof-theoretic analogue of constant- tional proof system: a polynomial-time computable surjec- depth circuits with modular counting gates - but without any tive function mapping strings to tautologies. The requirement success. Thus the Cook-Reckhow program is also stalled. that the range of the function is the set of tautologies What is even more unsatisfactory, though, is that unlike models the soundness of the proof system; the surjectiveness the Circuit Lower Bounds program, we do not have a good of the function models completeness; the polynomial-time explanation for why progress is stalled. There is no known computability of the function models efficient verifiability analogue of the natural proofs barrier for proof complexity, of proofs. A proof of a tautology φ with respect to a and indeed this is one of the main open questions asked in propositional proof system P is simply a string x such that the survey of Beame and Pitassi [10] from the late 90s.
Load more

Recommended publications
  • Circuit Complexity, Proof Complexity and Polynomial Identity Testing
    Electronic Colloquium on Computational Complexity, Report No. 52 (2014) Circuit Complexity, Proof Complexity and Polynomial Identity Testing Joshua A. Grochow and Toniann Pitassi April 12, 2014 Abstract We introduce a new and very natural algebraic proof system, which has tight connections to (algebraic) circuit complexity. In particular, we show that any super-polynomial lower bound on any Boolean tautology in our proof system implies that the permanent does not have polynomial- size algebraic circuits (VNP 6= VP). As a corollary to the proof, we also show that super- polynomial lower bounds on the number of lines in Polynomial Calculus proofs (as opposed to the usual measure of number of monomials) imply the Permanent versus Determinant Conjecture. Note that, prior to our work, there was no proof system for which lower bounds on an arbitrary tautology implied any computational lower bound. Our proof system helps clarify the relationships between previous algebraic proof systems, and begins to shed light on why proof complexity lower bounds for various proof systems have been so much harder than lower bounds on the corresponding circuit classes. In doing so, we highlight the importance of polynomial identity testing (PIT) for understanding proof complex- ity. More specifically, we introduce certain propositional axioms satisfied by any Boolean circuit computing PIT. (The existence of efficient proofs for our PIT axioms appears to be somewhere in between the major conjecture that PIT2 P and the known result that PIT2 P=poly.) We use these PIT axioms to shed light on AC0[p]-Frege lower bounds, which have been open for nearly 30 years, with no satisfactory explanation as to their apparent difficulty.
    [Show full text]
  • Reflection Principles, Propositional Proof Systems, and Theories
    Reflection principles, propositional proof systems, and theories In memory of Gaisi Takeuti Pavel Pudl´ak ∗ July 30, 2020 Abstract The reflection principle is the statement that if a sentence is provable then it is true. Reflection principles have been studied for first-order theories, but they also play an important role in propositional proof complexity. In this paper we will revisit some results about the reflection principles for propositional proofs systems using a finer scale of reflection principles. We will use the result that proving lower bounds on Resolution proofs is hard in Resolution. This appeared first in the recent article of Atserias and M¨uller [2] as a key lemma and was generalized and simplified in some spin- off papers [11, 13, 12]. We will also survey some results about arithmetical theories and proof systems associated with them. We will show a connection between a conjecture about proof complexity of finite consistency statements and a statement about proof systems associated with a theory. 1 Introduction arXiv:2007.14835v1 [math.LO] 29 Jul 2020 This paper is essentially a survey of some well-known results in proof complexity supple- mented with some observations. In most cases we will also sketch or give an idea of the proofs. Our aim is to focus on some interesting results rather than giving a complete ac- count of known results. We presuppose knowledge of basic concepts and theorems in proof complexity. An excellent source is Kraj´ıˇcek’s last book [19], where you can find the necessary definitions and read more about the results mentioned here.
    [Show full text]
  • Contents Part 1. a Tour of Propositional Proof Complexity. 418 1. Is There A
    The Bulletin of Symbolic Logic Volume 13, Number 4, Dec. 2007 THE COMPLEXITY OF PROPOSITIONAL PROOFS NATHAN SEGERLIND Abstract. 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. Contents Part 1. A tour of propositional proof complexity. 418 1. Is there a way to prove every tautology with a short proof? 418 2. Satisfiability algorithms and theories of arithmetic 420 3. A menagerie of Frege-like proof systems 427 4. Reverse mathematics of propositional principles 434 5. Feasible interpolation 437 6. Further connections with satisfiability algorithms 438 7. Beyond the Frege systems 441 Part 2. Some lower bounds on refutation sizes. 444 8. The size-width trade-off for resolution 445 9. The small restriction switching lemma 450 10. Expansion clean-up and random 3-CNFs 459 11. Resolution pseudowidth and very weak pigeonhole principles 467 Part 3. Open problems, further reading, acknowledgments. 471 Appendix. Notation. 472 References. 473 Received June 27, 2007. This survey article is based upon the author’s Sacks Prize winning PhD dissertation, however, the emphasis here is on context. The vast majority of results discussed are not the author’s, and several results from the author’s dissertation are omitted. c 2007, Association for Symbolic Logic 1079-8986/07/1304-0001/$7.50 417 418 NATHAN SEGERLIND Part 1.
    [Show full text]
  • A Short History of Computational Complexity
    The Computational Complexity Column by Lance FORTNOW NEC Laboratories America 4 Independence Way, Princeton, NJ 08540, USA [email protected] http://www.neci.nj.nec.com/homepages/fortnow/beatcs Every third year the Conference on Computational Complexity is held in Europe and this summer the University of Aarhus (Denmark) will host the meeting July 7-10. More details at the conference web page http://www.computationalcomplexity.org This month we present a historical view of computational complexity written by Steve Homer and myself. This is a preliminary version of a chapter to be included in an upcoming North-Holland Handbook of the History of Mathematical Logic edited by Dirk van Dalen, John Dawson and Aki Kanamori. A Short History of Computational Complexity Lance Fortnow1 Steve Homer2 NEC Research Institute Computer Science Department 4 Independence Way Boston University Princeton, NJ 08540 111 Cummington Street Boston, MA 02215 1 Introduction It all started with a machine. In 1936, Turing developed his theoretical com- putational model. He based his model on how he perceived mathematicians think. As digital computers were developed in the 40's and 50's, the Turing machine proved itself as the right theoretical model for computation. Quickly though we discovered that the basic Turing machine model fails to account for the amount of time or memory needed by a computer, a critical issue today but even more so in those early days of computing. The key idea to measure time and space as a function of the length of the input came in the early 1960's by Hartmanis and Stearns.
    [Show full text]
  • Proof Complexity in Algebraic Systems and Bounded Depth Frege Systems with Modular Counting
    PROOF COMPLEXITY IN ALGEBRAIC SYSTEMS AND BOUNDED DEPTH FREGE SYSTEMS WITH MODULAR COUNTING S. Buss, R. Impagliazzo, J. Kraj¶³cek,· P. Pudlak,¶ A. A. Razborov and J. Sgall Abstract. We prove a lower bound of the form N ­(1) on the degree of polynomials in a Nullstellensatz refutation of the Countq polynomials over Zm, where q is a prime not dividing m. In addition, we give an explicit construction of a degree N ­(1) design for the Countq principle over Zm. As a corollary, using Beame et al. (1994) we obtain ­(1) a lower bound of the form 2N for the number of formulas in a constant-depth N Frege proof of the modular counting principle Countq from instances of the counting M principle Countm . We discuss the polynomial calculus proof system and give a method of converting tree-like polynomial calculus derivations into low degree Nullstellensatz derivations. Further we show that a lower bound for proofs in a bounded depth Frege system in the language with the modular counting connective MODp follows from a lower bound on the degree of Nullstellensatz proofs with a constant number of levels of extension axioms, where the extension axioms comprise a formalization of the approximation method of Razborov (1987), Smolensky (1987) (in fact, these two proof systems are basically equivalent). Introduction A propositional proof system is intuitively a system for establishing the validity of propo- sitional tautologies in some ¯xed complete language. The formal de¯nition of propositional proof system is that it is a polynomial time function f which maps strings over an alphabet § onto the set of propositional tautologies (Cook & Reckhow 1979).
    [Show full text]
  • On Monotone Circuits with Local Oracles and Clique Lower Bounds
    On monotone circuits with local oracles and clique lower bounds Jan Kraj´ıˇcek Igor C. Oliveira Faculty of Mathematics and Physics Charles University in Prague December 17, 2019 Abstract We investigate monotone circuits with local oracles [K., 2016], i.e., circuits containing additional inputs yi = yi(~x) that can perform unstructured computations on the input string ~x. Let µ [0, 1] be the locality of the circuit, a parameter that bounds the combined ∈ m strength of the oracle functions yi(~x), and Un,k, Vn,k 0, 1 be the set of k-cliques and the set of complete (k 1)-partite graphs, respectively⊆ (similarly { } to [Razborov, 1985]). Our results can be informally− stated as follows. (i) For an appropriate extension of depth-2 monotone circuits with local oracles, we show that the size of the smallest circuits separating Un,3 (triangles) and Vn,3 (complete bipartite graphs) undergoes two phase transitions according to µ. (ii) For 5 k(n) n1/4, arbitrary depth, and µ 1/50, we prove that the monotone ≤ ≤ ≤ Θ(√k) circuit size complexity of separating the sets Un,k and Vn,k is n , under a certain restrictive assumption on the local oracle gates. The second result, which concerns monotone circuits with restricted oracles, extends and provides a matching upper bound for the exponential lower bounds on the monotone circuit size complexity of k-clique obtained by Alon and Boppana (1987). 1 Introduction and motivation arXiv:1704.06241v3 [cs.CC] 16 Dec 2019 We establish initial lower bounds on the power of monotone circuits with local oracles (mono- tone CLOs), an extension of monotone circuits introduced in [Kra16] motivated by problems in proof complexity.
    [Show full text]
  • Propositional Proof Complexity Past Present and Future
    Prop ositional Pro of Complexity Past Present and Future Paul Beame and Toniann Pitassi Abstract Pro of complexity the study of the lengths of pro ofs in prop ositional logic is an area of study that is fundamentally connected b oth to ma jor op en questions of computational complexity theory and to practical prop erties of automated theorem provers In the last decade there have b een a number of signicant advances in pro of complexity lower b ounds Moreover new connections b etween pro of complexity and circuit complexity have b een uncovered and the interplay b etween these two areas has b ecome quite rich In addition attempts to extend existing lower b ounds to ever stronger systems of pro of have spurred the introduction of new and interesting pro of systems adding b oth to the practical asp ects of pro of complexity as well as to a rich theory This note attempts to survey these developments and to lay out some of the op en problems in the area Introduction One of the most basic questions of logic is the following Given a uni versally true statement tautology what is the length of the shortest pro of of the statement in some standard axiomatic pro of system The prop osi tional logic version of this question is particularly imp ortant in computer science for b oth theorem proving and complexity theory Imp ortant related algorithmic questions are Is there an ecient algorithm that will pro duce a pro of of any tautology Is there an ecient algorithm to pro duce the shortest pro of of any tautology Such questions of theorem proving and
    [Show full text]
  • Proof Complexity I: Introduction to Propositional Proof Complexity
    Proof Complexity I: Introduction to propositional proof complexity Sam Buss Caleidoscope Research School Institute Henri Poincar´e, Paris June 17-18, 2019 Sam Buss Proof Complexity I This talk discusses: Proof systems: Frege proofs, extended Frege proofs, abstract proof systems, resolution, cutting planes, nullstellensatz, the polynomial calculus. The extension rule: Frege versus extended resolution (equivalent to extended Frege). Resolution versus extended resolution Interpolation and lower bounds: Resolution. Cutting planes. Automatizability and conditional lower bounds. Sam Buss Proof Complexity I Propositional logic, satisfiability, tautologies Propositional formulas: Variables: Range over True/False. Literal: Variables and negated variables. Formulas: Formed from variables/literals, and propositional connectives, such as ∧, ∨, →, ¬. CNF, DNF - Conjunctive/Disjunctive Normal Form Formulas. Satisfiability and Validity: A formula ϕ is a tautology iff every truth assignment makes ϕ true. A formula ϕ is satisfiable iff some truth assignment makes ϕ true. ϕ is unsatisfiable iff ¬ϕ is a tautology. It is NP-hard to determine satisfiability/validity of ϕ. One way to establish unsatisfiability is to give a proof of ¬ϕ. Sam Buss Proof Complexity I The Frege proof system F is a “textbook-style” propositional proof system with Modus Ponens as its only rule of inference. Variables: x, y, z,... range over True/False. Connectives: ¬, ∧, ∨, →. Modus Ponens: ϕ ϕ → ψ . ψ Axiom Schemes: ϕ → ψ → ϕ (ϕ → ψ) → (ϕ → ψ → χ) → (ϕ → χ) ϕ → ψ → ϕ ∧ ψ ϕ ∧ ψ → ϕ ϕ ∧ ψ → ψ and 5 more axiom schemes. Defn: The size of a Frege proof is the number of symbols in the proof. F m ϕ means ϕ has an F proof of size m.
    [Show full text]
  • A Reduction of Proof Complexity to Computational Complexity for $ AC^ 0 [P] $ Frege Systems
    A reduction of proof complexity to computational complexity for AC0[p] Frege systems Jan Kraj´ıˇcek Faculty of Mathematics and Physics Charles University in Prague Abstract We give a general reduction of lengths-of-proofs lower bounds for con- stant depth Frege systems in DeMorgan language augmented by a connec- tive counting modulo a prime p (the so called AC0[p] Frege systems) to computational complexity lower bounds for search tasks involving search trees branching upon values of maps on the vector space of low degree polynomials over Fp. In 1988 Ajtai [2] proved that the unsatisfiable set (¬PHPn) of propositional formulas _ pij and ¬pi1j ∨ ¬pi2j and ¬pij1 ∨ ¬pij2 j∈[n] for all i ∈ [n +1] = {1,...,n +1} , all i1 6= i2 ∈ [n + 1], j ∈ [n], and all i ∈ [n + 1], j1 6= j2 ∈ [n] respectively, expressing the failure of the pigeonhole principle (PHP), has for no d ≥ 1 a polynomial size refutation in a Frege proof system operating only with DeMorgan formulas of depth at most d. Subse- quently Kraj´ıˇcek [18] established an exponential lower bound for these so called AC0 Frege proof systems (for different formulas) and Kraj´ıˇcek, Pudl´ak and Woods [23] and Pitassi, Beame and Impagliazzo [26] improved independently (and announced jointly in [7]) Ajtai’s bound for PHP to exponential. All these papers employ some adaptation of the random restriction method arXiv:1311.2501v4 [math.LO] 22 Aug 2014 that has been so successfully applied earlier in circuit complexity (cf. [1, 14, 31, 15]). Razborov [28] invented already in 1987 an elegant method, simplified and generalized by Smolensky [30], to prove lower bounds even for AC0[p] circuits, p a prime.
    [Show full text]
  • Bounded Arithmetic and Propositional Proof Complexity
    Bounded Arithmetic and Propositional Proof Complexity Samuel R. Buss¤ Departments of Mathematics and Computer Science University of California, San Deigo La Jolla, CA 92093-0112 Abstract This is a survey of basic facts about bounded arithmetic and about the relationships between bounded arithmetic and propositional proof i i complexity. We introduce the theories S2 and T2 of bounded arithmetic and characterize their proof theoretic strength and their provably total functions in terms of the polynomial time hierarchy. We discuss other axiomatizations of bounded arithmetic, such as minimization axioms. It is shown that the bounded arithmetic hierarchy collapses if and only if bounded arithmetic proves that the polynomial hierarchy collapses. We discuss Frege and extended Frege proof length, and the two translations from bounded arithmetic proofs into propositional proofs. We present some theorems on bounding the lengths of propositional interpolants in terms of cut-free proof length and in terms of the lengths of resolution refutations. We then de¯ne the Razborov- Rudich notion of natural proofs of P =6 NP and discuss Razborov's theorem that certain fragments of bounded arithmetic cannot prove superpolynomial lower bounds on circuit size, assuming a strong cryptographic conjecture. Finally, a complete presentation of a proof of the theorem of Razborov is given. 1 Review of Computational Complexity 1.1 Feasibility This article will be concerned with various \feasible" forms of computability and of provability. For something to be feasibly computable, it must be computable in practice in the real world, not merely e®ectively computable in the sense of being recursively computable. ¤Supported in part by NSF grants DMS-9503247 and DMS-9205181.
    [Show full text]
  • Generalizing Boolean Satisfiability II
    Journal of Artificial Intelligence Research 22 (2004) 481-534 Submitted 08/04; published 12/04 Generalizing Boolean Satisfiability II: Theory Heidi E. Dixon [email protected] On Time Systems, Inc. 1850 Millrace, Suite 1 Eugene, OR 97403 USA Matthew L. Ginsberg [email protected] On Time Systems, Inc. 1850 Millrace, Suite 1 Eugene, OR 97403 USA Eugene M. Luks [email protected] Computer and Information Science University of Oregon Eugene, OR 97403 USA Andrew J. Parkes [email protected] CIRL 1269 University of Oregon Eugene, OR 97403 USA Abstract This is the second of three planned papers describing zap, a satisfiability engine that substantially generalizes existing tools while retaining the performance characteristics of modern high performance solvers. The fundamental idea underlying zap is that many problems passed to such engines contain rich internal structure that is obscured by the Boolean representation used; our goal is to define a representation in which this structure is apparent and can easily be exploited to improve computational performance. This paper presents the theoretical basis for the ideas underlying zap, arguing that existing ideas in this area exploit a single, recurring structure in that multiple database axioms can be obtained by operating on a single axiom using a subgroup of the group of permutations on the literals in the problem. We argue that the group structure precisely captures the general structure at which earlier approaches hinted, and give numerous examples of its use. We go on to extend the Davis-Putnam-Logemann-Loveland inference procedure to this broader setting, and show that earlier computational improvements are either subsumed or left intact by the new method.
    [Show full text]
  • Proof Complexity Lower Bounds from Algebraic Circuit Complexity∗
    Proof Complexity Lower Bounds from Algebraic Circuit Complexity∗ Michael A. Forbes1, Amir Shpilka2, Iddo Tzameret3, and Avi Wigderson4 1 Department of Computer Science, Princeton University, Princeton, USA [email protected] 2 Department of Computer Science, Tel Aviv University, Tel Aviv, Israel [email protected] 3 Department of Computer Science, Royal Holloway, University of London, Egham, UK [email protected] 4 School of Mathematics, Institute for Advanced Study, Princeton, USA [email protected] Abstract We give upper and lower bounds on the power of subsystems of the Ideal Proof System (IPS), the algebraic proof system recently proposed by Grochow and Pitassi [26], where the circuits comprising the proof come from various restricted algebraic circuit classes. This mimics an established research direction in the boolean setting for subsystems of Extended Frege proofs whose lines are circuits from restricted boolean circuit classes. Essentially all of the subsystems considered in this paper can simulate the well-studied Nullstellensatz proof system, and prior to this work there were no known lower bounds when measuring proof size by the algebraic complexity of the polynomials (except with respect to degree, or to sparsity). Our main contributions are two general methods of converting certain algebraic lower bounds into proof complexity ones. Both require stronger arithmetic lower bounds than common, which should hold not for a specific polynomial but for a whole family defined by it. These may be likened to some of the methods by which Boolean circuit lower bounds are turned into related proof-complexity ones, especially the “feasible interpolation” technique.
    [Show full text]