205 research outputs found

    Polynomial Identity Testing and the Ideal Proof System: PIT is in NP if and only if IPS can be p-simulated by a Cook-Reckhow proof system

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    The Ideal Proof System (IPS) of Grochow & Pitassi (FOCS 2014, J. ACM, 2018) is an algebraic proof system that uses algebraic circuits to refute the solvability of unsatisfiable systems of polynomial equations. One potential drawback of IPS is that verifying an IPS proof is only known to be doable using Polynomial Identity Testing (PIT), which is solvable by a randomized algorithm, but whose derandomization, even into NSUBEXP, is equivalent to strong lower bounds. However, the circuits that are used in IPS proofs are not arbitrary, and it is conceivable that one could get around general PIT by leveraging some structure in these circuits. This proposal may be even more tempting when IPS is used as a proof system for Boolean Unsatisfiability, where the equations themselves have additional structure. Our main result is that, on the contrary, one cannot get around PIT as above: we show that IPS, even as a proof system for Boolean Unsatisfiability, can be p-simulated by a deterministically verifiable (Cook-Reckhow) proof system if and only if PIT is in NP. We use our main result to propose a potentially new approach to derandomizing PIT into NP

    The Complexity of Some Geometric Proof Systems

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    In this Thesis we investigate proof systems based on Integer Linear Programming. These methods inspect the solution space of an unsatisfiable propositional formula and prove that this space contains no integral points. We begin by proving some size and depth lower bounds for a recent proof system, Stabbing Planes, and along the way introduce some novel methods for doing so. We then turn to the complexity of propositional contradictions generated uniformly from first order sentences, in Stabbing Planes and Sum-Of-Squares. We finish by investigating the complexity-theoretic impact of the choice of method of generating these propositional contradictions in Sherali-Adams

    Separating Incremental and Non-Incremental Bottom-Up Compilation

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    The aim of a compiler is, given a function represented in some language, to generate an equivalent representation in a target language L. In bottom-up (BU) compilation of functions given as CNF formulas, constructing the new representation requires compiling several subformulas in L. The compiler starts by compiling the clauses in L and iteratively constructs representations for new subformulas using an "Apply" operator that performs conjunction in L, until all clauses are combined into one representation. In principle, BU compilation can generate representations for any subformulas and conjoin them in any way. But an attractive strategy from a practical point of view is to augment one main representation - which we call the core - by conjoining to it the clauses one at a time. We refer to this strategy as incremental BU compilation. We prove that, for known relevant languages L for BU compilation, there is a class of CNF formulas that admit BU compilations to L that generate only polynomial-size intermediate representations, while their incremental BU compilations all generate an exponential-size core

    Proof-theoretic Semantics for Intuitionistic Multiplicative Linear Logic

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    This work is the first exploration of proof-theoretic semantics for a substructural logic. It focuses on the base-extension semantics (B-eS) for intuitionistic multiplicative linear logic (IMLL). The starting point is a review of Sandqvist’s B-eS for intuitionistic propositional logic (IPL), for which we propose an alternative treatment of conjunction that takes the form of the generalized elimination rule for the connective. The resulting semantics is shown to be sound and complete. This motivates our main contribution, a B-eS for IMLL , in which the definitions of the logical constants all take the form of their elimination rule and for which soundness and completeness are established

    The impact of heterogeneity and geometry on the proof complexity of random satisfiability

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    Satisfiability is considered the canonical NP-complete problem and is used as a starting point for hardness reductions in theory, while in practice heuristic SAT solving algorithms can solve large-scale industrial SAT instances very efficiently. This disparity between theory and practice is believed to be a result of inherent properties of industrial SAT instances that make them tractable. Two characteristic properties seem to be prevalent in the majority of real-world SAT instances, heterogeneous degree distribution and locality. To understand the impact of these two properties on SAT, we study the proof complexity of random -SAT models that allow to control heterogeneity and locality. Our findings show that heterogeneity alone does not make SAT easy as heterogeneous random -SAT instances have superpolynomial resolution size. This implies intractability of these instances for modern SAT-solvers. In contrast, modeling locality with underlying geometry leads to small unsatisfiable subformulas, which can be found within polynomial time

    A DPLL(T) Framework for Verifying Deep Neural Networks

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    Deep Neural Networks (DNNs) have emerged as an effective approach to tackling real-world problems. However, like human-written software, automatically-generated DNNs can have bugs and be attacked. This thus attracts many recent interests in developing effective and scalable DNN verification techniques and tools. In this work, we introduce a NeuralSAT, a new constraint solving approach to DNN verification. The design of NeuralSAT follows the DPLL(T) algorithm used modern SMT solving, which includes (conflict) clause learning, abstraction, and theory solving, and thus NeuralSAT can be considered as an SMT framework for DNNs. Preliminary results show that the NeuralSAT prototype is competitive to the state-of-the-art. We hope, with proper optimization and engineering, NeuralSAT will carry the power and success of modern SAT/SMT solvers to DNN verification. NeuralSAT is avaliable from: https://github.com/dynaroars/neuralsat-solverComment: 27 pages, 8 figures. NeuralSAT is avaliable from: https://github.com/dynaroars/neuralsat-solve

    Prioritized Unit Propagation and Extended Resolution Techniques for SAT Solvers

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    NP-complete problems like the Boolean Satisfiability (SAT) Problem are ubiquitous in computer science, mathematics, and engineering. Consequently, researchers have developed algorithms such as Conflict-Driven Clause-Learning (CDCL) SAT solvers, aimed at determining the satisfiability of Boolean formulas. As the result of decades of research in the development of CDCL SAT solvers, these algorithms solve real-life SAT instances surprisingly quickly, performing well despite the fact that the SAT problem is believed to be intractable in general. While modern CDCL SAT solvers are efficient for many real-world applications, there is continual demand for ever more powerful heuristics for newer applications. This demand in turn provides the impetus for research in solver heuristics. In this thesis, we address this need by proposing a new heuristic for Boolean Constraint Propagation (BCP), a key component of CDCL SAT solvers, and a novel, extensible, architectural design of an Extended Resolution (ER) SAT solver, a class of solvers that is more powerful than CDCL solvers. The impressive performance of CDCL SAT solvers on real-life Boolean instances is, in part, made possible by a combination of logical reasoning rules and heuristics integrated into different components of the solvers. Given that such combinations are currently the most successful paradigm in SAT solving, it is natural to ask how such combinations can be made even more efficient. We observe that there are two different approaches that can be taken to improve SAT solvers: one approach is to modify individual components within the SAT solving algorithm, and the other approach is to change the overall structure of the algorithm. We explore both approaches in this thesis. Following the first approach, we examine a critical component of CDCL: the Boolean Constraint Propagation (BCP) algorithm, which systematically finds implications of variable assignments made by the solver. In most implementations of BCP, variable values are propagated greedily -- the values of implied variables are set immediately after they are detected. This observation suggests that there could be a smarter way to perform BCP by prioritizing part of the search space rather than propagating implied variables immediately after they are encountered. In this work, we develop an algorithm which allows BCP to prioritize propagations, choose a heuristic priority ordering of the variables, and demonstrate a class of instances where our prioritized BCP algorithm, combined with this heuristic ordering, is able to outperform the traditional BCP algorithm. For the second approach, we note that solvers are fundamentally mathematical proof systems, and that CDCL produces proofs in the Resolution proof system, which is theoretically weaker than Extended Resolution (ER), a related proof system. Hence, it is natural to try integrating ER techniques into the CDCL algorithm, thus rendering it more powerful. However, it is well known that automating the ER proof system deterministically can be very challenging. Instead of proposing a single set of techniques to implement the ER proof system, we develop a programmatic framework (and an associated set of techniques) that enables one to upgrade CDCL solvers into an ER-based SAT solver. More precisely, we add three new major programmatic components: extension variable addition, extension variable substitution, and extension variable deletion. These components can be easily extended to test various ER ideas and heuristics. One of our considered heuristics is shown to be generally competitive with the baseline CDCL solver while improving upon the baseline for a specific class of cryptographic instances

    Computer Aided Verification

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    This open access two-volume set LNCS 13371 and 13372 constitutes the refereed proceedings of the 34rd International Conference on Computer Aided Verification, CAV 2022, which was held in Haifa, Israel, in August 2022. The 40 full papers presented together with 9 tool papers and 2 case studies were carefully reviewed and selected from 209 submissions. The papers were organized in the following topical sections: Part I: Invited papers; formal methods for probabilistic programs; formal methods for neural networks; software Verification and model checking; hyperproperties and security; formal methods for hardware, cyber-physical, and hybrid systems. Part II: Probabilistic techniques; automata and logic; deductive verification and decision procedures; machine learning; synthesis and concurrency. This is an open access book

    Tools and Algorithms for the Construction and Analysis of Systems

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    This open access book constitutes the proceedings of the 28th International Conference on Tools and Algorithms for the Construction and Analysis of Systems, TACAS 2022, which was held during April 2-7, 2022, in Munich, Germany, as part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2022. The 46 full papers and 4 short papers presented in this volume were carefully reviewed and selected from 159 submissions. The proceedings also contain 16 tool papers of the affiliated competition SV-Comp and 1 paper consisting of the competition report. TACAS is a forum for researchers, developers, and users interested in rigorously based tools and algorithms for the construction and analysis of systems. The conference aims to bridge the gaps between different communities with this common interest and to support them in their quest to improve the utility, reliability, exibility, and efficiency of tools and algorithms for building computer-controlled systems

    Monte Carlo Forest Search: UNSAT Solver Synthesis via Reinforcement learning

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    We introduce Monte Carlo Forest Search (MCFS), an offline algorithm for automatically synthesizing strong tree-search solvers for proving \emph{unsatisfiability} on given distributions, leveraging ideas from the Monte Carlo Tree Search (MCTS) algorithm that led to breakthroughs in AlphaGo. The crucial difference between proving unsatisfiability and existing applications of MCTS, is that policies produce trees rather than paths. Rather than finding a good path (solution) within a tree, the search problem becomes searching for a small proof tree within a forest of candidate proof trees. We introduce two key ideas to adapt to this setting. First, we estimate tree size with paths, via the unbiased approximation from Knuth (1975). Second, we query a strong solver at a user-defined depth rather than learning a policy across the whole tree, in order to focus our policy search on early decisions, which offer the greatest potential for reducing tree size. We then present MCFS-SAT, an implementation of MCFS for learning branching policies for solving the Boolean satisfiability (SAT) problem that required many modifications from AlphaGo. We matched or improved performance over a strong baseline on two well-known SAT distributions (\texttt{sgen}, \texttt{random}). Notably, we improved running time by 9\% on \texttt{sgen} over the \texttt{kcnfs} solver and even further over the strongest UNSAT solver from the 2021 SAT competition
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