Original Clause Set Research Articles

Contract Strengthening through Constrained Horn Clause Verification

The functional properties of a program are often specified by providing a contract for each of its functions. A contract of a function consists of a pair of formulas, called a precondition and a postcondition, which, respectively, should hold before and after execution of that function. It might be the case that the contracts supplied by the programmer are not adequate to allow a verification system to prove program correctness, that is, to show that for every function, if the precondition holds and the execution of the function terminates, then the postcondition holds. We address this problem by providing a technique which may strengthen the postconditions of the functions, thereby improving the ability of the verifier to show program correctness. Our technique consists of four steps. First, the translation of the given program, which may manipulate algebraic data structures (ADTs), and its contracts into a set of constrained Horn clauses (CHCs) whose satisfiability implies the validity of the given contracts. Then, the derivation, via CHC transformation performed by the VeriCaT tool, of a new set of CHCs that manipulate only basic sorts (such as booleans or integers) and whose satisfiability implies the satisfiability of the original set of clauses. Then, the construction of a model, if any, of the new, derived CHCs using the CHC solver SPACER for basic sorts. Finally, the translation of that model into the formulas that suitably strengthen the postconditions of the given contracts. We will present our technique through an example consisting of a Scala program for reversing lists. Note that the Stainless verifier is not able to prove the correctness of that program when considering the given contracts, while it succeeds when considering the contracts with the strengthened postconditions constructed by applying our technique.

SPASS-AR: A First-Order Theorem Prover Based on Approximation-Refinement into the Monadic Shallow Linear Fragment

We introduce FO-AR, an approximation-refinement approach for first-order theorem proving based on counterexample-guided abstraction refinement. A given first-order clause set N is transformed into an over-approximation \(N^{\prime }\) in a decidable first-order fragment. That means if \(N^{\prime }\) is satisfiable so is N. However, if \(N^{\prime }\) is unsatisfiable, then the approximation provides a lifting terminology for the found refutation which is step-wise transformed into a proof of unsatisfiability for N. If this fails, the cause is analyzed to refine the original clause set such that the found refutation is ruled out for the future and the procedure repeats. The target fragment of the transformation is the monadic shallow linear fragment with straight dismatching constraints, which we prove to be decidable via ordered resolution with selection. We further discuss practical aspects of SPASS-AR, a first-order theorem prover implementing FO-AR. We focus in particularly on effective algorithms for lifting and refinement.

Enhanced Membrane Computing Algorithm for SAT Problems Based on the Splitting Rule

Boolean propositional satisfiability (SAT) problem is one of the most widely studied NP-complete problems and plays an outstanding role in many domains. Membrane computing is a branch of natural computing which has been proven to solve NP problems in polynomial time with a parallel compute mode. This paper proposes a new algorithm for SAT problem which combines the traditional membrane computing algorithm of SAT problem with a classic simplification rule, the splitting rule, which can divide a clause set into two axisymmetric subsets, deal with them respectively and simultaneously, and obtain the solution of the original clause set with the symmetry of their solutions. The new algorithm is shown to be able to reduce the space complexity by distributing clauses with the splitting rule repeatedly, and also reduce both time and space complexity by executing one-literal rule and pure-literal rule as many times as possible.

CSE_E 1.0: An Integrated Automated Theorem Prover for First-Order Logic

First-order logic is an important part of mathematical logic, and automated theorem proving is an interdisciplinary field of mathematics and computer science. The paper presents an automated theorem prover for first-order logic, called C S E _ E 1.0, which is a combination of two provers contradiction separation extension (CSE) and E, where CSE is based on the recently-introduced multi-clause standard contradiction separation (S-CS) calculus for first-order logic and E is the well-known equational theorem prover for first-order logic based on superposition and rewriting. The motivation of the combined prover C S E _ E 1.0 is to (1) evaluate the capability, applicability and generality of C S E _ E , and (2) take advantage of novel multi-clause S-CS dynamic deduction of CSE and mature equality handling of E to solve more and harder problems. In contrast to other improvements of E, C S E _ E 1.0 optimizes E mainly from the inference mechanism aspect. The focus of the present work is given to the description of C S E _ E including its S-CS rule, heuristic strategies, and the S-CS dynamic deduction algorithm for implementation. In terms of combination, in order not to lose the capability of E and use C S E _ E to solve some hard problems which are unsolved by E, C S E _ E 1.0 schedules the running of the two provers in time. It runs plain E first, and if E does not find a proof, it runs plain CSE, then if it does not find a proof, some clauses inferred in the CSE run as lemmas are added to the original clause set and the combined clause set handed back to E for further proof search. C S E _ E 1.0 is evaluated through benchmarks, e.g., CASC-26 (2017) and CASC-J9 (2018) competition problems (FOFdivision). Experimental results show that C S E _ E 1.0 indeed enhances the performance of E to a certain extent.

Predicate Pairing for program verification

AbstractIt is well-known that the verification of partial correctness properties of imperative programs can be reduced to the satisfiability problem for constrained Horn clauses (CHCs). However, state-of-the-art solvers for constrained Horn clauses (or CHC solvers) based onpredicate abstractionare sometimes unable to verify satisfiability because they look for models that are definable in a given class 𝓐 of constraints, called 𝓐-definable models. We introduce a transformation technique, calledPredicate Pairing, which is able, in many interesting cases, to transform a set of clauses into an equisatisfiable set whose satisfiability can be proved by finding an 𝓐-definable model, and hence can be effectively verified by a state-of-the-art CHC solver. In particular, we prove that, under very general conditions on 𝓐, the unfold/fold transformation rules preserve the existence of an 𝓐-definable model, that is, if the original clauses have an 𝓐-definable model, then the transformed clauses have an 𝓐-definable model. The converse does not hold in general, and we provide suitable conditions under which the transformed clauses have an 𝓐-definable modelif and only ifthe original ones have an 𝓐-definable model. Then, we present a strategy, called Predicate Pairing, which guides the application of the transformation rules with the objective of deriving a set of clauses whose satisfiability problem can be solved by looking for 𝓐-definable models. The Predicate Pairing (PP) strategy introduces a new predicate defined by the conjunction of two predicates occurring in the original set of clauses, together with a conjunction of constraints. We will show through some examples that an 𝓐-definable model may exist for the new predicate even if it does not exist for its defining atomic conjuncts. We will also present some case studies showing that Predicate Pairing plays a crucial role in the verification ofrelational properties of programs, that is, properties relating two programs (such as program equivalence) or two executions of the same program (such as non-interference). Finally, we perform an experimental evaluation of the proposed techniques to assess the effectiveness of Predicate Pairing in increasing the power of CHC solving.

A reducing method for set of clauses using matrices

AbstractIn resolution theory, logical formulas of a theorem and axioms must be converted to a form of a clause set (a set of clauses).For more efficient proof, it is desirable that an original clause set is a Horn set. Renaming is one of the operations which can convert a given clause set to a logically equivalent Horn set.To perform the operation for a clause set, negation signs of all literals that have the same predicate symbol should be inverted simultaneously. However, it is difficult to perform the operation systematically and efficiently.In this paper, a symbolic matrix called “literal reverse matrix” is defined for each literal in a clause. Moreover, the definition of reverse matrix is extended for a clause.All renamings which convert a given clause to a Horn clause are derived from clause reverse matrix. Consequently, all renamings which convert a given clause set to a Horn set are synthesized from renamings for each clause in the set.