Logic Programming

Logic programming is a language paradigm based on logic, more specifically on resolution theorem proving in the predicate calculus as proposed in Robinson [1965]. Robinson had the foresight to distinguish the importance of two components in automatic theorem proving: a single inference rule called resolution and the testing for equality of trees called unification. Resolution is an inference step used to prove the validity of predicate calculus formulas expressed as clauses. In its simplest version: P ~ Q and -P ~ R imply Q ~ R, which is called the resolvant. Unification is the matching of terms used in a resolution step. It consists of testing the satisfiability of the equality of terms (i.e., labeled trees) whose leaves may contain variables. For example, the unification of the terms p(X, q(Z, a)) and p(b, q(a, Y)) succeeds, yielding the bindings X 5 b, Z 5 a, and Y 5 a. Prolog, the main representative of LP, consists of a sequence of Horn clauses. A Horn clause is one containing (at most) one positive literal. The term definite clause is used to denote a clause with exactly one positive literal. Prolog programs can be viewed as a set of definite clauses in which the positive literal is the head of the rule and the negative literals constitute the body or tail of the rule. From a procedural point of view, a head corresponds to the definition of a Boolean function whose body consists of conjunctions of calls to the Boolean functions representing the tail [Kowalski 1979]. A quintessential example of a Prolog program is that of append. It consists of two Horn clauses specifying that list L3 is the concatenation of two lists, L1 and L2:

The purpose of this paper is to discuss the correctness of a method for constructing parallel processing programs from a problem description. The framework we adopt for this purpose is Equivalent Transformation Framework (ETF), which regards computation as transformation of definite clauses. In the framework, a problem's domain knowledge and a query are described in definite clauses, and its meaning is defined by a model of the set of definite clauses. Then meaning-preserving transformation rules for the query are generated. We propose a parallel processing method based on “specialization”, a part of operation in the transformations, and discuss new parallel processing method based on the specialization that maintains correctness of the computation. The specialization is generalized notion of substitution in logic programming, and it allows more rich representation. We examine the correctness of our approach based the specialization.

In this paper, we discuss the computational complexity of Read-once resolution (ROR) with respect to Horn formulas. Recall that a Horn formula is a boolean formula in conjunctive normal form (CNF), such that each clause has at most one positive literal. Horn formulas find applications in a number of domains such as program verification and logic programming. It is well-known that deduction in ProLog is based on unification, which in turn is based on resolution and instantiation. Resolution is a sound and complete procedure to check whether a boolean formula in CNF is satisfiable. Although inefficient in general, resolution has been used widely in theorem provers, on account of its simplicity and ease of implementation. This paper focuses on two variants of resolution, viz., Read-once resolution and Unit Read-once resolution (UROR). Both these variants are sound, but incomplete. In this paper, the goal is to check for the existence of proofs (refutations) of Horn formulas under these variants. We also discuss the computational complexity of determining optimal length proofs where appropriate.

We propose in this paper a slight modification of the Knuth and Bendix algorithm for solving the equality problem in non-equational theories defined by a set of Horn clauses. We prove that the completeness property of the algorithm is then preserved, provided that a weak axiomatization of boolean calculus and equality has been given to the algorithm. In particular, we need only the reflexivity axiom for equality.This algorithm can also be interpreted as the extension to Horn clauses of the resolution/narrowing algorithm proposed by Lankford [LA], which applies only when the equality predicate does not occur positively in non-unit clauses. We give some examples of theorems proved by this method, which show its efficiency in comparison with other paramodulation-based methods.Another application of the Knuth and Bendix algorithm is the proof by induction (Musser [MU], Huet-Hullot [HH], and others). We show that our version of the completion algorithm can be used for proving universally quantified formulas (not only equations) in the initial model defined by a set of Horn clauses.KeywordsNormal FormInference RuleEquational TheoryCritical PairValid ConsequenceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We present a method for specialising the constraints in constrained Horn clauses with respect to a goal. We use abstract interpretation to compute a model of a query-answer transformation of a given set of clauses and a goal. The effect is to propagate the constraints from the goal top-down and propagate answer constraints bottom-up. Our approach does not unfold the clauses at all; we use the constraints from the model to compute a specialised version of each clause in the program. The approach is independent of the abstract domain and the constraints theory underlying the clauses. Experimental results on verification problems show that this is an effective transformation, both in our own verification tools (convex polyhedra analyser) and as a pre-processor to other Horn clause verification tools.

Complexity of manipulation and bribery in premise-based judgment aggregation with simple formulas

A generalization of Horn clauses to a higher-order logic is described and examined as a basis for logic programming. In qualitative terms, these higher-order Horn clauses are obtained from the first-order ones by replacing first-order terms with simply typed λ-terms and by permitting quantification over all occurrences of function symbols and some occurrences of predicate symbols. Several proof-theoretic results concerning these extended clauses are presented. One result shows that although the substitutions for predicate variables can be quite complex in general, the substitutions necessary in the context of higher-order Horn clauses are tightly constrained. This observation is used to show that these higher-order formulas can specify computations in a fashion similar to first-order Horn clauses. A complete theorem-proving procedure is also described for the extension. This procedure is obtained by interweaving higher-order unification with backchaining and goal reductions, and constitutes a higher-order generalization of SLD-resolution. These results have a practical realization in the higher-order logic programming language called λProlog.

Horn clause queries and generalizations

Given a first order language L, and a notion of a logic L w.r.t. L, we investigate the topological properties of the space of L-structures for L. We show that under a topology called the query topology which arises naturally in logic programming, the space of L-models (where L is a decent logic) of any sentence (set of clauses) in L may be regarded as a (closed, compact) T4-space. We then investigate the properties of maps from structures to structures. Our results allow us to apply various well-known results on the fixed-points of operators on topological spaces to the semantics of logic programming – in particular, we are able to derive necessary and sufficient topological conditions for the completion of covered general logic programs to be consistent. Moreover, we derive sufficient conditions guaranteeing the consistency of program completions, and for logic programs to be determinate. We also apply our results to characterize consistency of the unions of program completions.

Minimal Herbrand models of sets of first-order clauses are useful in several areas of computer science, for example, automated theorem proving, program verification, logic programming, databases, and artificial intelligence. In most cases, the conventional model generation algorithms are inappropriate because they generate nonminimal Herbrand models and can be inefficient. This article describes an approach for generating the minimal Herbrand models of sets of first-order clauses. The approach builds upon i>positive unit hyperresolution (i>PUHR) i>tableaux, that are in general smaller than conventional tableaux. PUHR tableaux formalize the approach initially introduced with the theorem prover SATCHMO. Two minimal model generation procedures are described. The first one expands PUHR tableaux depth-first relying on a i>complement splitting expansion rule and on a form of backtracking involving constraints. A Prolog implementation, named MM-SATCHMO, of this procedure is given, and its performance on benchmark suites is reported. The second minimal model generation procedure performs a breadth-first, constrained expansion of PUHR (complement) tableaux. Both procedures are optimal in the sense that each minimal model is constructed only once, and the construction of nonminimal models is interrupted as soon as possible. They are complete in the following sense: The depth-first minimal model generation procedure computes all minimal Herbrand models of the considered clauses provided these models are all finite. The breadth-first minimal model generation procedure computes all finite minimal Herbrand models of the set of clauses under consideration. The proposed procedures are compared with related work in terms of both principles and performance on benchmark problems.

We consider constrained Horn clause solving from the more general point of view of solving formula equations. Constrained Horn clauses correspond to the subclass of Horn formula equations. We state and prove a fixed-point theorem for Horn formula equations which is based on expressing the fixed-point computation of a minimal model of a set of Horn clauses on the object level as a formula in first-order logic with a least fixed point operator. We describe several corollaries of this fixed-point theorem, in particular concerning the logical foundations of program verification, and sketch how to generalise it to incorporate abstract interpretations.

Automatic program repair modifies a faulty program to make it correct with respect to a specification. Previous approaches have typically been restricted to specific programming languages and a fixed set of syntactical mutation techniques — e.g., changing the conditions of if statements. We present a more general technique based on repairing sets of unsolvable Horn clauses. Working with Horn clauses enables repairing programs from many different source languages, but also introduces challenges, such as navigating the large space of possible repairs. We propose a conservative semantic repair technique that only removes incorrect behaviors and does not introduce new behaviors. Our proposed framework allows the user to request the best repairs — it constructs an optimization lattice representing the space of possible repairs, and uses a novel local search technique that exploits heuristics to avoid searching through sub-lattices with no feasible repairs. To illustrate the applicability of our approach, we apply it to problems in software-defined networking (SDN), and illustrate how it is able to help network operators fix buggy configurations by properly filtering undesired traffic. We show that interval and Boolean lattices are effective choices of optimization lattices in this domain, and we enable optimization objectives such as modifying the minimal number of switches. We have implemented a prototype repair tool, and present preliminary experimental results on several benchmarks using real topologies and realistic repair scenarios in data centers and congested networks.

The abstraction and over-approximation of protocols and web services by a set of Horn clauses is a very successful method in practice. It has however limitations for protocols and web services that are based on databases of keys, contracts, or even access rights, where revocation is possible, so that the set of true facts does not monotonically grow with state transitions. We extend the scope of these over-approximation methods by defining a new way of abstraction that can handle such databases, and we formally prove that the abstraction is sound. We realize a translator from a convenient specification language to standard Horn clauses and use the verifier ProVerif and the theorem prover SPASS to solve them. We show by a number of examples that this approach is practically feasible for wide variety of verification problems of security protocols and web services

Many SRL models pose logical inference as weighted satisfiability solving. Performing logical inference after completely grounding clauses with all possible constants is computationally expensive and approaches such as LazySAT [8] utilize the sparseness of the domain to deal with this. Here, we investigate the efficiency of restricting the Knowledge Base (Σ) to the set of first order horn clauses. We propose an algorithm that prunes the search space for satisfiability in horn clauses and prove that the optimal solution is guaranteed to exist in the pruned space. The approach finds a model, if it exists, in polynomial time; otherwise it finds an interpretation that is most likely given the weights. We provide experimental evidence that our approach reduces the size of search space substantially.KeywordsFirst Order LogicHorn ClausesMaxSATSatisfiability

Automatic program repair modifies a faulty program to make it correct with respect to a specification. Previous approaches have typically been restricted to specific programming languages and a fixed set of syntactical mutation techniques-e.g., changing the conditions of if statements. We present a more general technique based on repairing sets of unsolvable Horn clauses. Working with Horn clauses enables repairing programs from many different source languages, but also introduces challenges, such as navigating the large space of possible repairs. We propose a conservative semantic repair technique that only removes incorrect behaviors and does not introduce new behaviors. Our proposed framework allows the user to request the best repairs-it constructs an optimization lattice representing the space of possible repairs, and uses a novel local search technique that exploits heuristics to avoid searching through sub-lattices with no feasible repairs. To illustrate the applicability of our approach, we apply it to problems in software-defined networking (SDN), and illustrate how it is able to help network operators fix buggy configurations by properly filtering undesired traffic. We show that interval and Boolean lattices are effective choices of optimization lattices in this domain, and we enable optimization objectives such as modifying the minimal number of switches. We have implemented a prototype repair tool, and present preliminary experimental results on several benchmarks using real topologies and realistic repair scenarios in data centers and congested networks.