Paqui Lucio

A cut-free and invariant-free sequent calculus for PLTL

Sequent calculi usually provide a general deductive setting that uniformly embeds other proof-theoretical approaches, such as tableaux methods, resolution techniques, goal-directed proofs, etc. Unfortunately, in temporal logic, existing sequent calculi make use of a kind of inference rules that prevent the effective mechanization of temporal deduction in the general setting. In particular, temporal sequent calculi either need some form of cut, or they make use of invariants, or they include infinitary rules. This is the case even for the simplest kind of temporal logic, propositional linear temporal logic (PLTL). In this paper, we provide a complete finitary sequent calculus for PLTL, called FC, that not only is cut-free but also invariant-free. In particular, we introduce new rules which provide a new style of temporal deduction. We give a detailed proof of completeness.

Systematic Semantic Tableaux for PLTL

The better known methods of semantic tableaux for deciding satisfiability in propositional linear temporal logic generate graphs in addition to classical trees. The test of satisfaction is made from the graph and it does not correspond with the application of rules in any calculus for PLTL. We present here a new method of semantic tableaux without using additional graphs. The method is based on a new complete finitary sequent calculus for PLTL which allows us to incorporate all the information in a tree. This approach makes our tableaux better suited for completely automatic theorem proving.

Constructive negation by bottom-up computation of literal answers

In this paper, we present a new proposal for an efficient implementation of constructive negation. In our approach the answers for a literal are bottom-up computed by solving equality constraints, instead of by handling frontiers of subsidiary computation trees. The required equality constraints are given by Shepherdsons operators which are, in a sense, similar to bottom-up immediate consequence operators. However, in order to make the procedure efficient two main techniques are applied. First, we restrict our constraints to a class of success-answers (resp. fail-answers) which are easy to manipulate and to solve (or to prove their unsatisfiability). And, second, we take advantage of the monotonic nature of Shepherdsons operators to make the procedure incremental and to avoid recalculations that are typical in frontiers-based methods. Then, goal computation is made in the usual top-down CLP scheme of collecting the answers for the selected literal into the constraint of the goal. The procedural mechanism for constructive negation is designed not only to generate every correct answer of a goal, but also to detect failure. That is, in spite of the bottom-up nature of the calculation of literal answers, goal computation is not necessarily infinite. The operational semantics that makes use of these ideas, called BCN, is sound and complete with respect to three-valued program completion for the whole class of normal logic programs. A prototype implementation of this approach has been developed and the experimental results are very promising.

Structured Sequent Calculi for Combining Intuitionistic and Classical First-Order Logic

We define a sound and complete logic, called \({\cal FO}^{\supset}\), which extends classical first-order predicate logic with intuitionistic implication.

Dual systems of tableaux and sequents for PLTL

Abstract On one hand, traditional tableau systems for temporal logic (TL) generate an auxiliary graph that must be checked and (possibly) pruned in a second phase of the refutation procedure. On the other hand, traditional sequent calculi for TL make use of a kind of inference rules (mainly, invariant-based rules or infinitary rules) that complicates their automatization. A remarkable consequence of using auxiliary graphs in the tableaux framework and invariants or infinitary rules in the sequents framework is that TL fails to carry out the classical correspondence between tableaux and sequents. In this paper, we first provide a tableau method TTM that does not require auxiliary graphs to decide whether a set of PLTL-formulas is satisfiable. This tableau method TTM is directly associated to a one-sided sequent calculus called TTC. Since TTM is free from all the structural rules that hinder the mechanization of deduction, e.g. weakening and contraction, then the resulting sequent calculus TTC is also free from this kind of structural rules. In particular, TTC is free of any kind of cut, including invariant-based cut. From the deduction system TTC, we obtain a two-sided sequent calculus GTC that preserves all these good freeness properties and is finitary, sound and complete for PTL. Therefore, we show that the classical correspondence between tableaux and sequent calculi can be extended to TL. Every deduction system is proved to be complete. In addition, we provide illustrative examples of deductions in the different systems.

A Strong Logic Programming View for Static Embedded Implications

A strong (L) logic programming language ([14, 15]) is given by two subclasses of formulas (programs and goals) of the underlying logic L, provided that: firstly, any program P (viewed as a L-theory) has a canonical model MP which is initial in the category of all its L-models; secondly, the L-satisfaction of a goal G in MP is equivalent to the L-derivability of G from P, and finally, there exists an effective (computable) proof-subcalculus of the L-calculus which works out for derivation of goals from programs. In this sense, Horn clauses constitute a strong (first-order) logic programming language. Following the methodology suggested in [15] for designing logic programming languages, an extension of Horn clauses should be made by extending its underlying first-order logic to a richer logic which supports a strong axiomatization of the extended logic programming language. A well-known approach for extending Horn clauses with embedded implications is the static scope programming language presented in [8]. In this paper we show that such language can be seen as a strong FO⊃ logic programming language, where FO⊃ is a very natural extension of first-order logic with intuitionistic implication. That is, we present a new characterization of the language in [8] which shows that Horn clauses extended with embedded implications, viewed as FO⊃-theories, preserves all the attractive mathematical and computational properties that Horn clauses satisfy as first-order-theories.

Invariant-Free Clausal Temporal Resolution

Resolution is a well-known proof method for classical logics that is well suited for mechanization. The most fruitful approach in the literature on temporal logic, which was started with the seminal paper of M. Fisher, deals with Propositional Linear-time Temporal Logic (PLTL) and requires to generate invariants for performing resolution on eventualities. The methods and techniques developed in that approach have also been successfully adapted in order to obtain a clausal resolution method for Computation Tree Logic (CTL), but invariant handling seems to be a handicap for further extension to more general branching temporal logics. In this paper, we present a new approach to applying resolution to PLTL. The main novelty of our approach is that we do not generate invariants for performing resolution on eventualities. Hence, we say that the approach presented in this paper is invariant-free. Our method is based on the dual methods of tableaux and sequents for PLTL that we presented in a previous paper. Our resolution method involves translation into a clausal normal form that is a direct extension of classical CNF. We first show that any PLTL-formula can be transformed into this clausal normal form. Then, we present our temporal resolution method, called trs-resolution, that extends classical propositional resolution. Finally, we prove that trs-resolution is sound and complete. In fact, it finishes for any input formula deciding its satisfiability, hence it gives rise to a new decision procedure for PLTL.

A Functorial Framework for Constraint Normal Logic Programming

The semantic constructions and results for definite programs do not extend when dealing with negation. The main problem is related to a well-known problem in the area of algebraic specification: if we fix a constraint domain as a given model, its free extension by means of a set of Horn clauses defining a set of new predicates is semicomputable. However, if the language of the extension is richer than Horn clauses its free extension (if it exists) is not necessarily semicomputable. In this paper we present a framework that allows us to deal with these problems in a novel way. This framework is based on two main ideas: a reformulation of the notion of constraint domain and a functorial presentation of our semantics. In particular, the semantics of a logic program P is defined in terms of three functors:

Logical foundations for more expressive declarative temporal logic programming languages

({\mathcal {OP}}_{P} ,{\mathcal {ALG}}_{P} ,{\mathcal {LOG}}_{P})

Elimination of Local Variables from Definite Logic Programs

that apply to constraint domains and provide the operational, the least fixpoint and the logical semantics of P, respectively. To be more concrete, the idea is that the application of