Satisfiability Research Articles

GENES: An Efficient Recursive zk-SNARK and Its Novel Application in Blockchain

The rapid development of blockchain has significantly promoted research on zero-knowledge proofs (ZKPs), especially zero-knowledge succinct noninteractive arguments of knowledge (zk-SNARK). As is well known, protocol proof and verification time, as well as proof size, are the main obstacles that restrict the implementation of ZKPs in practical applications, so they have become the main concerns of researchers in recent years. This work achieves a new recursive zk-SNARK called GENES, which does not have a trusted setup and is secure under the standard discrete logarithm assumption. GENES is designed from the form of the rank-1 constraint system (R1CS) satisfiability problem. Recursive proof composition is achieved by merging multiple R1CS instances, which transforms the verification of numerous proofs into the verification of a single proof. Moreover, multi-helpers amortize proof commitments in this study, significantly reducing the computational pressure and time cost of proof generation. Compared with previous work, GENES effectively improves the proof time and verification time, but at the cost of larger proof sizes. We provide a blockchain Layer-1 scaling solution leveraging GENES to demonstrate its practicality.

SAT-GATv2: A Dynamic Attention-Based Graph Neural Network for Solving Boolean Satisfiability Problem

We propose SAT-GATv2, a graph neural network (GNN)-based model designed to solve the Boolean satisfiability problem (SAT) through graph-based deep learning techniques. SAT-GATv2 transforms SAT formulas into graph structures, leveraging message-passing neural networks (MPNNs) to propagate local information and dynamic attention mechanisms (GATv2) to accurately capture inter-node dependencies and enhance node feature representations. Unlike traditional heuristic-driven SAT solvers, SAT-GATv2 adopts a data-driven approach, learning structural patterns directly from graph representations and providing a complementary framework to existing methods. Experimental results demonstrate that SAT-GATv2 achieves an accuracy improvement of 1.75–5.51% over NeuroSAT on challenging random 3-SAT(n) instances, highlighting its effectiveness in handling difficult problem distributions, and outperforms other GNN-based models on SR(n) datasets, showcasing its scalability and adaptability. Ablation studies validate the critical roles of MPNNs and GATv2 in improving prediction accuracy and scalability. While SAT-GATv2 does not yet surpass CDCL-based solvers in overall performance, it addresses their limitations in scalability and adaptability to complex instances, offering an efficient graph-based alternative for tackling larger and more complex SAT problems. This study establishes a foundation for integrating deep learning with combinatorial optimization, emphasizing its potential for applications in artificial intelligence and operations research.

Satisfiability Modulo User Propagators

Modern SAT solvers are often integrated as sub-reasoning engines into more complex tools to address problems beyond the Boolean satisfiability problem. Consider, for example, solvers for Satisfiability Modulo Theories (SMT), combinatorial optimization, model enumeration, and model counting. There, the SAT solver can often provide relevant information beyond the satisfiability answer and the domain knowledge of the embedding system, such as symmetry properties or theory axioms, may benefit the CDCL search. However, this knowledge can often not be efficiently represented in clausal form. This paper proposes a general interface to inspect and influence the internal behaviour of CDCL SAT solvers. The aim is to capture the essential functionalities that simplify and improve use cases requiring a more fine-grained interaction with the SAT solver than provided via the standard IPASIR interface. For our experiments, the state-of-the-art SAT solver CaDiCaL is extended with the proposed interface and evaluated on two representative use cases: enumerating graphs within the SAT modulo Symmetries framework (SMS), and as the main CDCL(T) SAT engine of the SMT solver cvc5.

Making It Tractable to Detect and Correct Errors in Graphs

This article develops Hercules, a system for entity resolution (ER), conflict resolution (CR), timeliness deduction (TD), and missing value/link imputation (MI) in graphs. It proposes GCR + s, a class of graph cleaning rules (GCR) that support not only predicates for ER and CR but also temporal orders to deduce timeliness and data extraction to impute missing data. As opposed to previous graph rules, GCR + s are defined with a dual graph pattern to accommodate irregular structures of schemaless graphs and adopt patterns of a star form to reduce the complexity. We show that while the implication and satisfiability problems are intractable for GCR + s, it is in polynomial time to detect and correct errors with GCR + s. Underlying Hercules, we train a ranking model to predict the temporal orders on attributes and embed it as a predicate of GCR + s. We provide an algorithm for discovering GCR + s by combining the generations of patterns and predicates. We also develop a method for conducting ER, CR, TD, and MI in the same process to improve the overall quality of graphs by leveraging their interactions and chasing with GCR + s; we show that the method has the Church–Rosser property under certain conditions. Using real-life and synthetic graphs, we empirically verify that Hercules is 53% more accurate than the state-of-the-art graph cleaning systems and performs comparably in efficiency and scalability.

Effectively encoding satisfiability problems into Ising models for quantum annealing

Ising Models, defined by quadratic objective functions (or Hamiltonians), enable to use quantum annealers to search for optimal or near-optimal solutions of satisfiability problems. However, current quantum annealers have limited resolution, meaning that small or closely-valued coefficients in the Hamiltonian may be obscured by flux noise, leading to a degradation in the performance of quantum annealing. In this paper, we propose a novel design methodology for encoding satisfiability problems into Ising models via Integer Linear Programming. Experimental results show that our method can effectively reduce the resolution requirements for quantum annealers.

Declarative Approaches to Outcome Determination in Judgment Aggregation

Judgment aggregation (JA) offers a generic formal framework for modeling various settings involving information aggregation by social choice mechanisms. For many judgment aggregation rules, computing collective judgments is computationally notoriously hard. The central outcome determination problem, in particular, is often complete for higher levels of the polynomial hierarchy. This complexity barrier makes it challenging to develop practical exact algorithms to outcome determination. Taking on this challenge, in this work we develop practical exact algorithms for outcome determination under a range of the most central JA rules—namely Kemeny, Slater, MaxHamming, Young, Dodgson, Reversal scoring, Condorcet, Ranked agenda, and LexiMax—by harnessing the declarative approach, in particular, Boolean satisfiability (SAT) and integer programming techniques. For the Kemeny, Slater, MaxHamming, Young, and Dodgson rules, we detail direct approaches based on maximum satisfiability (MaxSAT) and integer programming. For the Reversal scoring, Condorcet, Ranked agenda, and LexiMax rules, we develop iterative algorithms, including algorithms based on the counterexample-guided abstraction refinement (CEGAR) paradigm, making use of recent advances in incremental MaxSAT solving and preferential SAT-based reasoning. We provide an open-source implementation of the algo- rithms, and empirically evaluate them using real-world preference data. We compare the performance of our implementation to a recent approach which makes use of declarative solver technology for answer set programming (ASP). The results demonstrate that our approaches scale significantly beyond the reach of the ASP-based algorithms for all of the judgment aggregation rules considered.

A Proof Calculus for Automated Deduction in Propositional Product Logic

Propositional product logic belongs to the basic fuzzy logics with continuous t-norms using the product t-norm (defined as the ordinary product of real numbers) on the unit interval [0,1]. This paper introduces a proof calculus for the product logic which is suitable for automated deduction. The calculus provides one of possible generalisations of the family of modifications of the procedure (algorithm) of Davis, Putnam, Logemann, and Loveland (DPLL) in the context of fuzzy logics. We show that the calculus is refutation sound and finitely complete as well. The deduction, satisfiability, and validity problems are solved in the finite case. The achieved results contribute to the theoretical (logic and computational) description of multi-step fuzzy inference.

An Exact and Fast SAT Formulation for the DCJ Distance.

Reducing into a satisfiability (SAT) formulation has been proven effective in solving certain NP-hard problems. In this work, we extend this research by presenting a novel SAT formulation for computing the double-cut-and-join (DCJ) distance between two genomes with duplicate genes. The DCJ distance serves as a crucial metric in studying genome rearrangement. Previous work achieved an exact solution by transforming it into a maximum cycle decomposition (MCD) problem on the corresponding adjacency graph of two genomes, followed by reducing this problem into an integer linear programming (ILP) formulation. Using both simulated datasets and real genomic datasets, we firmly conclude that the SAT-based method scales much better and runs faster than using ILP, being able to solve a whole new range of instances which the ILP-based method cannot solve in a reasonable amount of time. This underscores the SAT-based approach as a versatile and more powerful alternative to ILP, with promising implications for broader applications in computational biology.

Nash meets Łukasiewicz: computing equilibria through logic

Abstract A Nash equilibrium is a strategy profile of a game in which none of the players involved has any gain by changing alone his/her own strategy. Given a two-player game, we show a codification of all of its Nash equilibria into Łukasiewicz infinitely-valued logic, that is, we derive a propositional theory in this logic whose models codify all the Nash equilibria. Based on such propositional theory, we derive a polynomial reduction from the problem of computing a Nash equilibrium to the problem of satisfiability of (sets of) formulas of Łukasiewicz infinitely-valued logic. These applications of logic to game theory lead to new methods for computing Nash equilibria.

On probabilistic and causal reasoning with summation operators

Abstract Ibeling et al. (2023, Annals of Pure and Applied Logic, 103339) axiomatize increasingly expressive languages of causation and probability, and Mossé et al. (2024, The Review of Symbolic Logic, 17, 106–131) show that reasoning (specifically the satisfiability problem) in each causal language is as difficult, from a computational complexity perspective, as reasoning in its merely probabilistic or ‘correlational’ counterpart. Introducing a summation operator to capture common devices that appear in applications—such as the $do$-calculus of Pearl (Causality. Cambridge University Press, 2009) for causal inference, which makes ample use of marginalization—van der Zander et al. (The hardness of reasoning about probabilities and causality. In Proceedings of the Thirty-Second International Joint Conference on Artificial Intelligence, IJCAI-23, E. Elkind, ed., pp. 5730–5738. International Joint Conferences on Artificial Intelligence Organization. Main Track. 2023) extend these earlier complexity results to causal and probabilistic languages with marginalization. We complete this extension, fully characterizing the complexity of probabilistic and causal reasoning with summation, demonstrating that these again remain equally difficult. Surprisingly, allowing free variables for random variable values results in a system that is undecidable, so long as the ranges of these random variables are unrestricted. We finally axiomatize these languages featuring marginalization (or more generally summation), resolving open questions posed by Ibeling et al. (2023, Annals of Pure and Applied Logic, 103339).

A DPLL Procedure with Dichotomous Branching for Propositional Product Logic

The propositional product logic is one of the basic fuzzy logics with continuous t-norms, employing the product t-norm on the unit interval [Formula: see text]. Our aim is to combine well-established automated deduction (theorem proving) with fuzzy inference. As a first step, we devise a modification of the procedure of Davis, Putnam, Logemann, and Loveland (DPLL) with dichotomous branching for inference in the product logic. We prove that the procedure is refutation sound and finitely complete. As a consequence, solutions to the deduction, satisfiability, and validity problems will be proposed for the finite case. The presented results are applicable to a design of intelligent systems, exploiting some kind of multi-step fuzzy inference.

Opening the Black Box inside Grover’s Algorithm

Grover’s algorithm is one of the primary algorithms offered as evidence that quantum computers can provide an advantage over classical computers. It involves an “oracle” (external quantum subroutine), which must be specified for a given application and whose internal structure is not part of the formal scaling of the quadratic quantum speedup guaranteed by the algorithm. Grover’’s algorithm also requires exponentially many calls to the quantum oracle (approximately 2n calls where n is the number of qubits) to succeed, raising the question of its implementation on both noisy and error-corrected quantum computers. In this work, we construct a quantum-inspired algorithm executable on a classical computer that performs Grover’s task in a linear number of calls to (simulations of) the oracle—an exponentially smaller number than Grover’s algorithm—and demonstrate this algorithm explicitly for Boolean satisfiability problems. The complexity of our algorithm depends on the cost to simulate the oracle once, which may or may not be exponential, depending on its internal structure. Indeed, Grover’s algorithm does not have an quantum speedup as soon as one is given access to the “source code” of the oracle, which may reveal an internal structure of the problem. Our findings illustrate this point explicitly, as our algorithm exploits the structure of the quantum circuit used to program the quantum computer to speed up the search. There are still problems where Grover’s algorithm would provide an asymptotic speedup if it could be run accurately for large enough sizes. Our quantum-inspired algorithm provides lower bounds, in terms of the quantum-circuit complexity, for the quantum hardware to beat classical approaches for these problems. These estimates, combined with the unfavorable scaling of the success probability of Grover’s algorithm, which in the presence of noise decays as the exponential of the exponential of the number of qubits, makes a practical speedup unrealistic even under extremely optimistic assumptions of the evolution of both hardware quality and availability. Published by the American Physical Society 2024

On the Foundations of Conflict-Driven Solving for Hybrid MKNF Knowledge Bases

Abstract Hybrid MKNF Knowledge Bases (HMKNF-KBs) constitute a formalism for tightly integrated reasoning over closed-world rules and open-world ontologies. This approach allows for accurate modeling of real-world systems, which often rely on both categorical and normative reasoning. Conflict-driven solving is the leading approach for computationally hard problems, such as satisfiability (SAT) and answer set programming (ASP), in which MKNF is rooted. This paper investigates the theoretical underpinnings required for a conflict-driven solver of HMKNF-KBs. The approach defines a set of completion and loop formulas, whose satisfaction characterizes MKNF models. This forms the basis for a set of nogoods, which in turn can be used as the backbone for a conflict-driven solver.

SAT solver-driven approach for validating local electron counting rule

Determining large, complex surface structures is essential for understanding and modeling crystal growth. For semiconductor surfaces, the electron counting (EC) rule is known to be useful for predicting surface stability. In this study, a scheme is proposed to automatically determine if a sampled surface structure locally satisfies the EC rule using a Boolean Satisfiability Problem (SAT) solver. This automatic determination is demonstrated on the GaN(0001)-(6 × 6) surface system with H and Ga adsorption as an example. The scheme is also applicable to the automatic generation of surface structures that are expected to be relatively stable, and is expected to accelerate the study of large and complex surface structures of various semiconductor materials.

Proof Complexity and Beyond

Proof complexity is a multi-disciplinary research area that addresses questions of the general form “how difficult is it to prove certain mathematical facts?” The current workshop focussed on recent advances in our understanding that the analysis of an appropriately tailored concept of “proof” underlies many of the arguments in algorithms, geometry or combinatorics research that make the core of modern theoretical computer science. These include the analysis of practical Boolean satisfiability (SAT) solving algorithms, the size of linear or semidefinite programming formulations of combinatorial optimization problems, the complexity of solving total NP search problems by local methods, and the complexity of describing winning strategies in two-player round-based games, to name just a few important examples.

Probabilistic computing enabled by continuous random numbers sampled from in-plane magnetized stochastic magnetic tunnel junctions

Hardware acceleration of probabilistic computing has recently attracted significant attention in the slowing down of Moore's law. A randomly fluctuating bit called as p-bit constitutes a fundamental building block for this type of physics-inspired computing scheme, which can be efficiently built out of emerging devices. Here, we report a probabilistic computing set-up, where random numbers are sampled from stochastic magnetic tunnel junctions with in-plane magnetic anisotropy. Although the sampled data have largely bipolar-like probability distributions compared to the ideally uniform ones, the results show a reasonable performance in a standard simulated annealing process on Boolean satisfiability problems up to 100 variables. The systematic simulations suggest the importance of probability distribution where some additional intermediate states help to increase the performance.

Counting the number of solutions in satisfiability problems with tensor-network message passing.

We investigate how to count the number of solutions in the Boolean satisfiability (SAT) problem, a fundamental problem in theoretical computer science that has applications in various domains. We convert the problem to a spin-glass problem in statistical physics and approximately compute the entropy of the problem, which corresponds to the logarithm of the number of solutions. We propose a new method for the entropy computing problem based on a combination of the tensor network method and the message-passing algorithm. The significance of the proposed method is its ability to consider the effects of both long loops and short loops present in the factor graph of the SAT problem. We validate the efficacy of our approach using 3-SAT problems defined on the random graphs and structured graphs, and show that the proposed method gives more accurate results on the number of solutions than the standard belief propagation algorithms. We also discuss the applications of our method across a wide range of combinatorial optimization problems.

Computing high-degree polynomial gradients in memory

Specialized function gradient computing hardware could greatly improve the performance of state-of-the-art optimization algorithms. Prior work on such hardware, performed in the context of Ising Machines and related concepts, is limited to quadratic polynomials and not scalable to commonly used higher-order functions. Here, we propose an approach for massively parallel gradient calculations of high-degree polynomials, which is conducive to efficient mixed-signal in-memory computing circuit implementations and whose area scales proportionally with the product of the number of variables and terms in the function and, most importantly, independent of its degree. Two flavors of such an approach are proposed. The first is limited to binary-variable polynomials typical in combinatorial optimization problems, while the second type is broader at the cost of a more complex periphery. To validate the former approach, we experimentally demonstrated solving a small-scale third-order Boolean satisfiability problem based on integrated metal-oxide memristor crossbar circuits, with competitive heuristics algorithm. Simulation results for larger-scale, more practical problems show orders of magnitude improvements in area, speed and energy efficiency compared to the state-of-the-art. We discuss how our work could enable even higher-performance systems after co-designing algorithms to exploit massively parallel gradient computation.

ReSG: A Data Structure for Verification of Majority-based In-memory Computing on ReRAM Crossbars

Recent advancements in the fabrication of Resistive Random Access Memory (ReRAM) devices have led to the development of large-scale crossbar structures. In-memory computing architectures relying on ReRAM crossbars aim to mitigate the processor-memory bottleneck that exists with current complementary metal-oxide semiconductor technology. With this motivation, several synthesis and mapping approaches focusing on the realizations of Boolean functions in the ReRAM crossbars have been proposed earlier. Thus far, the verification of the designs realized on ReRAM crossbars is done either through manual inspection or using simulation-based approaches. Since manual inspections and simulation-based approaches are limited to smaller designs, they cannot be applied to the verification of complex designs on large-scale ReRAM crossbars. Motivated by this, we propose, for the first time, an automatic equivalence checking flow that determines the equivalence between the original function specification (e.g., Majority-inverter Graph ) and the crossbar micro-operations file formats. We consider two crossbar structures, zero-transistor, one-memristor (0T1R) and one-transistor, one-memristor (1T1R) to implement the micro-operations. While the micro-operations file format exists for 0T1R crossbar structures, no representations for micro-operations to be executed in 1T1R crossbars exist yet. In this work, we introduce the micro-operation file format for 1T1R crossbar structures to efficiently represent the micro-operations as ReRAM crossbar netlists. Afterwards, we introduce two intermediate data structures, ReRAM Sequence Graph for 0T1R crossbars (ReSG-0T1R) and for 1T1R crossbars (ReSG-1T1R) , that are derived from the 0T1R and 1T1R crossbar micro-operations file formats, respectively. These ReSGs are then translated into Boolean Satisfiability (SAT) formula, and then the verification is done by checking the generated SAT formulae against the golden functional specification (represented in Verilog) using Z3 Satisfiability solver. Experimental evaluations confirm the effectiveness of the proposed verification methodology on MCNC and ISCAS benchmarks.

Solving Boolean Satisfiability Problems With The Quantum Approximate Optimization Algorithm

One of the most prominent application areas for quantum computers is solving hard constraint satisfaction and optimization problems. However, detailed analyses of the complexity of standard quantum algorithms have suggested that outperforming classical methods for these problems would require extremely large and powerful quantum computers. The quantum approximate optimization algorithm (QAOA) is designed for near-term quantum computers, yet previous work has shown strong limitations on the ability of QAOA to outperform classical algorithms for optimization problems. Here we instead apply QAOA to hard constraint satisfaction problems, where both classical and quantum algorithms are expected to require exponential time. We analytically characterize the average success probability of QAOA on a constraint satisfaction problem commonly studied using statistical physics methods: random k-SAT at the threshold for satisfiability, as the number of variables n goes to infinity. We complement these theoretical results with numerical experiments on the performance of QAOA for small n, which match the limiting theoretical bounds closely. We then compare QAOA with leading classical solvers. For random 8-SAT, we find that for more than 14 quantum circuit layers, QAOA achieves more efficient scaling than the highest-performance classical solver we tested, WalkSATlm. Our results suggest that near-term quantum algorithms for solving constraint satisfaction problems may outperform their classical counterparts. Published by the American Physical Society 2024