Closures and fairness in the semantics of programming logic

AbstractWe use the notions of closures and fair chaotic iterations to give a semantics to logic programs. The relationships between the semantics of individual rules and the semantics of the whole program are established and an application to parallel processing is mentioned. A chaotic fixed point theorem is given, which carries the non-determinism inherent to resolution. Finally, we introduce a general definition of finite failure and the concept of fair SLD resolution, and show that this procedure is sound and strongly complete with respect to both success and finite failure, thus extending a result of Apt and Van Emden (1982)

First-Order Logic Theorem Proving and Model Building via Approximation and Instantiation

In this paper we consider first-order logic theorem proving and model building via approximation and instantiation. Given a clause set we propose its approximation into a simplified clause set where satisfiability is decidable. The approximation extends the signature and preserves unsatisfiability: if the simplified clause set is satisfiable in some model, so is the original clause set in the same model interpreted in the original signature. A refutation generated by a decision procedure on the simplified clause set can then either be lifted to a refutation in the original clause set, or it guides a refinement excluding the previously found unliftable refutation. This way the approach is refutationally complete. We do not step-wise lift refutations but conflicting cores, finite unsatisfiable clause sets representing at least one refutation. The approach is dual to many existing approaches in the literature because our approximation preserves unsatisfiability

A Machine Checked Model of Idempotent MGU Axioms For Lists of Equational Constraints

We present formalized proofs verifying that the first-order unification algorithm defined over lists of satisfiable constraints generates a most general unifier (MGU), which also happens to be idempotent. All of our proofs have been formalized in the Coq theorem prover. Our proofs show that finite maps produced by the unification algorithm provide a model of the axioms characterizing idempotent MGUs of lists of constraints. The axioms that serve as the basis for our verification are derived from a standard set by extending them to lists of constraints. For us, constraints are equalities between terms in the language of simple types. Substitutions are formally modeled as finite maps using the Coq library Coq.FSets.FMapInterface. Coq's method of functional induction is the main proof technique used in proving many of the axioms.Comment: In Proceedings UNIF 2010, arXiv:1012.455

PrIC3: Property Directed Reachability for MDPs

IC3 has been a leap forward in symbolic model checking. This paper proposes PrIC3 (pronounced pricy-three), a conservative extension of IC3 to symbolic model checking of MDPs. Our main focus is to develop the theory underlying PrIC3. Alongside, we present a first implementation of PrIC3 including the key ingredients from IC3 such as generalization, repushing, and propagation

Runtime Verification Through Forward Chaining

In this paper we present a novel rule-based approach for Runtime Verification of FLTL properties over finite but expanding traces. Our system exploits Horn clauses in implication form and relies on a forward chaining-based monitoring algorithm. This approach avoids the branching structure and exponential complexity typical of tableaux-based formulations, creating monitors with a single state and a fixed number of rules. This allows for a fast and scalable tool for Runtime Verification: we present the technical details together with a working implementation

Determinacy analysis for logic programs using mode and type information

We propose an analysis for detecting procedures and goals that are deterministic (i.e. that produce at most one solution), or predicates whose clause tests are mutually exclusive (which implies that at most one of their clauses will succeed) even if they are not deterministic (because they cali other predicates that can produce more than one solution). Applications of such determinacy information include detecting programming errors, performing certain high-level program transformations for improving search efñciency, optimizing low level code generation and parallel execution, and estimating tighter upper bounds on the computational costs of goals and data sizes, which can be used for program debugging, resource consumption and granularity control, etc. We have implemented the analysis and integrated it in the CiaoPP system, which also infers automatically the mode and type information that our analysis takes as input. Experiments performed on this implementation show that the analysis is fairly accurate and efncient

A Modular Order-sorted Equational Generalization Algorithm

Generalization, also called anti-unification, is the dual of unification. Given terms t and t , a generalizer is a term t of which t and t are substitution instances. The dual of a most general unifier (mgu) is that of least general generalizer (lgg). In this work, we extend the known untyped generalization algorithm to, first, an order-sorted typed setting with sorts, subsorts, and subtype polymorphism; second, we extend it to work modulo equational theories, where function symbols can obey any combination of associativity, commutativity, and identity axioms (including the empty set of such axioms); and third, to the combination of both, which results in a modular, order-sorted equational generalization algorithm. Unlike the untyped case, there is in general no single lgg in our framework, due to order-sortedness or to the equational axioms. Instead, there is a finite, minimal and complete set of lggs, so that any other generalizer has at least one of them as an instance. Our generalization algorithms are expressed by means of inference systems for which we give proofs of correctness. This opens up new applications to partial evaluation, program synthesis, and theorem proving for typed equational reasoning systems and typed rulebased languages such as ASF+SDF, Elan, OBJ, Cafe-OBJ, and Maude. © 2014 Elsevier Inc. All rights reserved. 1.M. Alpuente, S. Escobar, and J. Espert have been partially supported by the EU (FEDER) and the Spanish MEC/MICINN under grant TIN 2010-21062-C02-02, and by Generalitat Valenciana PROMETEO2011/052. J. Meseguer has been supported by NSF Grants CNS 09-04749, and CCF 09-05584.Alpuente Frasnedo, M.; Escobar Román, S.; Espert Real, J.; Meseguer, J. (2014). A Modular Order-sorted Equational Generalization Algorithm. Information and Computation. 235:98-136. https://doi.org/10.1016/j.ic.2014.01.006S9813623