Horn Clauses Research Articles

Teaching of uncorrelated compound sentencies

Sentences with consequence, concessional and  (evaluative) towards the main sentence we consider as a secondary events derived from parataxic constructions, which could not   fulfill the functions of the subordinate sentence and only resembled it in appearance and form. Therefore, they should be separated into a separate group, as a complicated form of a coherent sentence, which is functionally-semantically coherent, in form - sub-structured. It is clear that  they are not syntactically equivalent , additional, definite and circumstantial clauses. They are different   to the functional classification of syntactic members, but we   can combined them  into a group of clauses dependent on the principal.  This component  limits and specifies the main sentence meaningfully.

A Solicitous Approach to Smart Contract Verification

Smart contracts are tempting targets of attacks, as they often hold and manipulate significant financial assets, are immutable after deployment, and have publicly available source code, with assets estimated in the order of millions of dollars being lost in the past due to vulnerabilities. Formal verification is thus a necessity, but smart contracts challenge the existing highly efficient techniques routinely applied in the symbolic verification of software, due to specificities not present in general programming languages. A common feature of existing works in this area is the attempt to reuse off-the-shelf verification tools designed for general programming languages. This reuse can lead to inefficiency and potentially unsound results, as domain translation is required. In this article, we describe a carefully crafted approach that directly models the central aspects of smart contracts natively, going from the contract to its logical representation without intermediary steps. We use the expressive and highly automatable logic of constrained Horn clauses for modeling and instantiate our approach to the Solidity language. A tool implementing our approach, called Solicitous , was developed and integrated into the SMTChecker module of the Solidity compiler solc. We evaluated our approach on an extensive benchmark set containing 22,446 real-world smart contracts deployed on the Ethereum blockchain over a 27-month period. The results show that our approach is able to establish safety of significantly more contracts than comparable, publicly available verification tools, with an order of magnitude increase in the percentage of formally verified contracts.

A Complete Diagrammatic Calculus for Boolean Satisfiability

We propose a calculus of string diagrams to reason about satisfiability of Boolean formulas, and prove it to be sound and complete. We then showcase our calculus in a few case studies. First, we consider SAT-solving. Second, we consider Horn clauses, which leads us to a new decision method for propositional logic programs equivalence under Herbrand model semantics.

Proving the infeasibility of Horn formulas through read-once resolution

In this paper, we study Horn formulas from the perspective of read-once resolution refutations (RORs). A Horn formula is a Boolean formula in conjunctive normal form (CNF), in which each clause contains at most one positive literal. Horn formulas are used in a number of domains, including program verification, logic programming, and econometrics. In particular, deduction in ProLog is based on unification. Unification is based on resolution and instantiation. Resolution is a system used to prove the infeasibility of Boolean formulas. It is important to note that resolution is both sound and complete. However, resolution is inefficient in the following sense: There exist CNF formulas with resolution refutations whose lengths are bounded below by an exponential function of the input size. At the same time, these formulas admit shorter (polynomially bounded) proofs of infeasibility in other proof systems, such as Frege Systems. Despite this inefficiency, resolution is simple and easy to implement and hence used in a wide variety of theorem provers. In this paper, we study two variants of resolution. These are read-once resolution (ROR) and read-once unit resolution (UROR). Both ROR and UROR are sound. However, they are incomplete since there exist infeasible Boolean formulas which do not have either an ROR or a UROR. In this paper, we look at the problems of determining if a Horn formula has an ROR or a UROR. We also examine the problem of finding the optimal length ROR of a Horn formula from both the computational complexity and the approximation perspectives. Finally, we analyze the copy complexity of Horn formulas with respect to URORs.

Optimal CHC Solving via Termination Proofs

Motivated by applications to open program reasoning such as maximal specification inference, this paper studies optimal CHC solving , a problem to compute maximal and/or minimal solutions of constrained Horn clauses (CHCs). This problem and its subproblems have been studied in the literature, and a major approach is to iteratively improve a solution of CHCs until it becomes optimal. So a key ingredient of optimization methods is the optimality checking of a given solution. We propose a novel optimality checking method, as well as an optimization method using the proposed optimality checker, based on a computational theoretical analysis of the optimality checking problem. The key observation is that the optimality checking problem is closely related to the termination analysis of programs, and this observation is useful both theoretically and practically. From a theoretical perspective, it clarifies a limitation of an existing method and incorrectness of another method in the literature. From a practical perspective, it allows us to apply techniques of termination analysis to the optimality checking of a solution of CHCs. We present an optimality checking method based on constraint-based synthesis of termination arguments, implemented our method, evaluated it on CHCs that encode maximal specification synthesis problems, and obtained promising results.

Higher-Order MSL Horn Constraints

The monadic shallow linear (MSL) class is a decidable fragment of first-order Horn clauses that was discovered and rediscovered around the turn of the century, with applications in static analysis and verification. We propose a new class of higher-order Horn constraints which extend MSL to higher-order logic and develop a resolution-based decision procedure. Higher-order MSL Horn constraints can quite naturally capture the complex patterns of call and return that are possible in higher-order programs, which make them well suited to higher-order program verification. In fact, we show that the higher-order MSL satisfiability problem and the HORS model checking problem are interreducible, so that higher-order MSL can be seen as a constraint-based approach to higher-order model checking. Finally, we describe an implementation of our decision procedure and its application to verified socket programming.

Wh-Movement in Hadhrami Arabic: A minimalist perspective

This study explores the wh-movement in Hadhrami Arabic (HA), a non-standard variety of Arabic, focusing on the simple and embedded questions as well as definite restrictive relative clauses within the framework of Chomsky’s (1995) Minimalist Program (MP). The researchers adopted a descriptive ethnographic qualitative research design collecting their data from 10 purposefully chosen HA speakers through a questionnaire, interviews and participant observation. The study reveals that the wh-phrases in HA SVO word order are base-generated and exhibit no movement. It also reveals that HA questions may have apparent optional movement of the wh-phrases in the left periphery or in the right one. Finally, it is found that HA definite restrictive relative clauses are not derived by movements.

Reconstructing a single-head formula to facilitate logical forgetting

Abstract Logical forgetting is removing some variables from a formula while maintaining its consequences on the remaining variables. This removal may require exponential time on arbitrary propositional Horn formulae, but it only takes polynomial time on single-head propositional definite Horn formulae. Single-head means that no variable is the head of multiple clauses. An algorithm to make a formula single-head if possible is shown. It enlarges the set of formulae admitting polynomial-time forgetting by making them single-head if possible and then running the usual algorithm for forgetting.

CHC-COMP 2022: Competition Report

CHC-COMP 2022 is the fifth edition of the competition of solvers for Constrained Horn Clauses. The competition was run in March 2022; the results were presented at the 9th Workshop on Horn Clauses for Verification and Synthesis held in Munich, Germany, on April 3, 2022. This edition featured six solvers, and eight tracks consisting of sets of linear and nonlinear clauses with constraints over linear integer arithmetic, linear real arithmetic, arrays, and algebraic data types. This report provides an overview of the organization behind the competition runs: it includes the technical details of the competition setup as well as presenting the results of the 2022 edition.

The Security Protocol Verifier ProVerif and its Horn Clause Resolution Algorithm

ProVerif is a widely used security protocol verifier. Internally, ProVerif uses an abstract representation of the protocol by Horn clauses and a resolution algorithm on these clauses, in order to prove security properties of the protocol or to find attacks. In this paper, we present an overview of ProVerif and discuss some specificities of its resolution algorithm, related to the particular application domain and the particular clauses that ProVerif generates. This paper is a short summary that gives pointers to publications on ProVerif in which the reader will find more details.

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.

Proceedings 9th Workshop on Horn Clauses for Verification and Synthesis and 10th International Workshop on Verification and Program Transformation

Proceedings 9th Workshop on Horn Clauses for Verification and Synthesis and 10th International Workshop on Verification and Program Transformation

A first polynomial non-clausal class in many-valued logic

The relevance of polynomial classes of formulas to deductive efficiency motivated their further research, and currently, a great number of such classes is known. Nevertheless, they have been exclusively sought in the setting of propositional logic and clausal form, which is of course expressively limiting for real-world applications.Along these lines and towards making tractability applicable beyond propositional clausal logic, firstly, we define theRegular many-valuedHorn Non-Clausal (NC) class, or RH, obtained by suitably amalgamating both regular classes: Horn and NC. Then we show that recognizing whether any NC formula is Horn-NC takes only linear time.Secondly, we demonstrate that the relationship between RH and: (1) its subclass of regular Horn formulas is that syntactically RH subsumes the Horn class but semantically both classes are equivalent; and (2) its superclass of regular non-clausal formulas is that RH contains all non-clausal formulas whose clausal form is Horn.Thirdly, we define Regular Non-Clausal Unit-Resolution, or RURNC, and prove that RURNC is complete for RH and also checks its satisfiability in polynomial time. Altogether, RH is a class recognized and solved polynomially, which shows that our intended goal is reached since RH is many-valued, non-clausal and tractable.Fourthly, based on RH, we characterize extensive classes of Horn-NC-like formulas, super-classes of RH, which are composed of formulas that are logically equivalent to some formula in RH. We furnish syntactical patterns of such Horn-NC-like formulas, being outside RH and whose satisfiability test is also decidable in polynomial time.As RH and RURNC are, both, basic in the DPLL scheme, the most efficient in propositional logic, and can be extended to some other non-classical logics, we argue that they pave the way for efficient non-clausal DPLL-based approximate reasoning.Field:Tractable Approximate Automated Reasoning.

A puzzle related to superlative modification in definite relative clauses in Jordanian Arabic

This paper addresses a puzzle involving two modes of superlative modification in definite relatives in Jordanian Arabic1: the so-called (A)ttributive and (G)enitive superlatives.2 We show that under the low interpretation of the definite relative clause, both modes of superlative modification exhibit an asymmetrical behavior: while the A-superlative is ambiguous between the absolute and relative readings, the relative reading is blocked in G-superlatives. To resolve the puzzle, we propose a compositional analysis based on the late merger of the definite relative which generates the absolute/relative ambiguity in A-superlatives and at the same time it blocks the relative reading in G-superlatives.

Verifying Catamorphism-Based Contracts using Constrained Horn Clauses

AbstractWe address the problem of verifying that the functions of a program meet their contracts, specified by pre/postconditions. We follow an approach based on constrained Horn clauses (CHCs) by which the verification problem is reduced to the problem of checking satisfiability of a set of clauses derived from the given program and contracts. We consider programs that manipulate algebraic data types (ADTs) and a class of contracts specified by catamorphisms, that is, functions defined by simple recursion schemata on the given ADTs. We show by several examples that state-of-the-art CHC satisfiability tools are not effective at solving the satisfiability problems obtained by direct translation of the contracts into CHCs. To overcome this difficulty, we propose a transformation technique that removes the ADT terms from CHCs and derives new sets of clauses that work on basic sorts only, such as integers and booleans. Thus, when using the derived CHCs there is no need for induction rules on ADTs. We prove that the transformation is sound, that is, if the derived set of CHCs is satisfiable, then so is the original set. We also prove that the transformation always terminates for the class of contracts specified by catamorphisms. Finally, we present the experimental results obtained by an implementation of our technique when verifying many non-trivial contracts for ADT manipulating programs.

On the Complexity of Inductively Learning Guarded Clauses

We investigate the computational complexity of mining guarded clauses from clausal datasets through the framework of inductive logic programming (ILP). We show that learning guarded clauses is NP-complete and thus one step below the Sigma2-complete task of learning Horn clauses on the polynomial hierarchy. Motivated by practical applications on large datasets we identify a natural tractable fragment of the problem. Finally, we also generalise all of our results to k-guarded clauses for constant k.

DeepStochLog: Neural Stochastic Logic Programming

Recent advances in neural-symbolic learning, such as DeepProbLog, extend probabilistic logic programs with neural predicates. Like graphical models, these probabilistic logic programs define a probability distribution over possible worlds, for which inference is computationally hard. We propose DeepStochLog, an alternative neural-symbolic framework based on stochastic definite clause grammars, a kind of stochastic logic program. More specifically, we introduce neural grammar rules into stochastic definite clause grammars to create a framework that can be trained end-to-end. We show that inference and learning in neural stochastic logic programming scale much better than for neural probabilistic logic programs. Furthermore, the experimental evaluation shows that DeepStochLog achieves state-of-the-art results on challenging neural-symbolic learning tasks.

Uniform Interpolants in EUF: Algorithms using DAG-representations

The concept of uniform interpolant for a quantifier-free formula from a given formula with a list of symbols, while well-known in the logic literature, has been unknown to the formal methods and automated reasoning community for a long time. This concept is precisely defined. Two algorithms for computing quantifier-free uniform interpolants in the theory of equality over uninterpreted symbols (EUF) endowed with a list of symbols to be eliminated are proposed. The first algorithm is non-deterministic and generates a uniform interpolant expressed as a disjunction of conjunctions of literals, whereas the second algorithm gives a compact representation of a uniform interpolant as a conjunction of Horn clauses. Both algorithms exploit efficient dedicated DAG representations of terms. Correctness and completeness proofs are supplied, using arguments combining rewrite techniques with model theory.

Model Checking Algorithm for Repairing Inference between Conjunctive Forms

Let K be a propositional formula and let ϕ be a query, the propositional inference problem K |= ϕ is a Co-NP-complete problem for propositional formulas without restrictions. We show the existence of polynomial-time cases for some fragments of propositional formulas K and ϕ that are different from the already known case of Horn formulas. For example, in the case that K is a formula in the fragment Krom or Horn or Monotone, then K |= ϕ can be decided in polynomial-time for any CNF ϕ. Given two conjunctive normal forms (CNF’s) K and ϕ, when K ̸|= ϕ, our proposal builds a minimal set of independent clauses S. The falsifying assignments of S are exactly the subset of models of K, which are not models for ϕ. In this way, the CNF S could extend the initial formula K in such a way that repair the inference, it is (S ∪ K) |= ϕ is held.

Extracting Rules from Neural Networks with Partial Interpretations

We investigate the problem of extracting rules, expressed in Horn logic, from neural network models.Our work is based on the exact learning model, in which a learner interacts with a teacher (the neural network model) via queries in order to learn an abstract target concept, which in our case is a set of Horn rules. We consider partial interpretations to formulate the queries. These can be understood as a representation of the world where part of the knowledge regarding the truthness of propositions is unknown. We employ Angluin’s algorithm for learning Horn rules via queries and evaluate our strategy empirically.