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11 References# Satisfiability of Systems of Ordinal Notations with the Subterm Property is Decidable.

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- Encouraged by this result, several researchers started in the late 80s the program to show decidability of the first-order theory of term algebras with different predicates than just syntactic equality. Research basically went into several directions: one direction was to add relations other than equality to the theory, in particular ordering relations that were useful for ordered rewrite calculi [6, 15] , or for typing of programming languages [22, 16]. Another direction was the addition of predicates that can be recognized by various classes of tree automata [5, 8].

[Show abstract] [Hide abstract]**ABSTRACT:**We investigate the problem of deciding first-order theories of finite trees with several distinguished congruence relations, each of them given by some equational axioms. We give an automata-based solution for the case where the different equational axiom systems are linear and variable-disjoint (this includes the case where all axioms are ground), and where the logic does not permit to express tree relations x=f(y,z). We show that the problem is undecidable when these restrictions are relaxed. As motivation and application, we show how to translate the model-checking problem of Apil, a spatial equational logic for the applied pi-calculus, to the validity of first-order formulas in term algebras with multiple congruence relations.- HILOG [3], which involves the second-order expression of meta-level predicates, has been developed as a decidable higher-order language for logic programming, and it may be more complex than the EXPTIME complexity of DATALOG. Unfortunately, in most cases, higher-order logic pro- gramming [12] makes reasoning increasingly difficult because complex structures of higher-order terms need to be treated. To overcome the aforementioned difficulties related to expressiveness and complexity, we introduce meta-predicates and their hierarchy in a restricted and decidable fragment for combining ontologies and rules.

[Show abstract] [Hide abstract]**ABSTRACT:**This paper presents a decidable fragment for combining on- tologies and rules in order-sorted logic programming. We describe order- sorted logic programming with sort, predicate, and meta-predicate hier- archies for deriving predicate and meta-predicate assertions. Meta-level predicates (predicates of predicates) are useful for representing relation- ships between predicate formulas, and further, they conceptually yield a hierarchy similar to the hierarchies of sorts and predicates. By extend- ing the order-sorted Horn-clause calculus, we develop a query-answering system that can answer queries such as atoms and meta-atoms gener- alized by containing predicate variables. We show that the expressive query-answering system computes every generalized query in single ex- ponential time, i.e., the complexity of our query system is equal to that of DATALOG.- HILOG [3], which involves the second-order expression of meta-level predicates, has been developed as a higher-order language with a first-order semantics for logic programming, and it seems to be more complex than the EXPTIME complexity of DATALOG. Unfortunately, in most cases, higher-order logic programming [12] makes reasoning increasingly difficult because complex structures of higher-order terms need to be treated. To overcome the aforementioned difficulties related to expressiveness and complexity, we introduce meta-predicates and their hierarchy in a restricted and decidable fragment for combining ontologies and rules.

[Show abstract] [Hide abstract]**ABSTRACT:**Decidable reasoning between ontologies and rules is required for the Semantic Web. This paper presents a decidable fragment for combining ontologies and rules in order-sorted logic programming. We describe order-sorted logic programming with sort, predicate, and meta-predicate hierarchies for deriving predicate and meta-predicate asser-tions. Meta-level predicates (predicates of predicates) are useful for rep-resenting relationships between predicate formulas, and further, they conceptually yield a hierarchy similar to the hierarchies of sorts and predicates. By extending the order-sorted Horn-clause calculus, we de-velop a query-answering system that can answer queries such as atoms and meta-atoms generalized by containing predicate variables. We show that the expressive query-answering system computes every generalized query in single exponential time, i.e., the complexity of our query system is equal to that of DATALOG.- In practice, for selection functions on non-ground clauses (and, as we will see, for restricting more the non-ground inference rules), it will be useful to be able to approximate at the non-ground level, that is, to check for non-ground Q-atoms Q and Q , whether there exists some grounding substitution σ such that Qσ Q σ , or, more generally, whether for a Boolean formula F over relations Q Q or Q Q , there exists some grounding σ such that Fσ evaluates to true. This kind of ordering constraint satisfiability problem can indeed be decided for LPOs and RPOs (Comon, 1990; Jouannaud and Okada, 1991; Nieuwenhuis, 1993; Nieuwenhuis and Rivero, 1999) and KBOs (Korovin and Voronkov, 2000).

[Show abstract] [Hide abstract]**ABSTRACT:**We introduce a calculus of stratified resolution, in which special attention is paid to clauses that “define” relations. If such clauses are discovered in the initial set of clauses, they are treated using the rule of definition unfolding, i.e. the rule that replaces defined relations by their definitions. Stratified resolution comes with a powerful notion of redundancy: a clause to which definition unfolding has been applied can be removed from the search space. To prove the completeness of stratified resolution with redundancies, we use a novel combination of Bachmair and Ganzinger’s model construction technique and a hierarchical construction of orderings and least fixpoints.- The question of decidability of the theory of a total simplification ordering has been posed in Comon (1988) . The decidability of the existential fragment of a total lexicographic path ordering (lpo for short) is shown in Comon (1990), the analogous result for a total recursive path ordering has been given in Jouannaud & Okada (1991) . We prove in Example (C) the undecidability of the E4 fragment of a partial lpo .

[Show abstract] [Hide abstract]**ABSTRACT:**We claim that the reduction of Post's Correspondence Problem to the decision problem of a theory provides a useful tool for proving undecidability of first order theories given by some interpretation. The goal of this paper is to define a framework for such reduction proofs. The method proposed is illustrated by proving the undecidability of the theory of a term algebra modulo the axioms of associativity and commutativity and of the theory of a partial lexicographic path ordering.- If f ∈ lex this is a simple problem over the natural numbers, but for f ∈ mul this seems not to be the case. Hence for the moment we propose to use the algorithm of [7] or the NP one of [13] for S N , which is normally a minor part of S.

[Show abstract] [Hide abstract]**ABSTRACT:**We introduce new algorithms for deciding the satisfiability of constraints for the full recursive path ordering with status (RPO), and hence as well for other path orderings like LPO, MPO, KNS and RDO, and for all possible total precedences and signatures. The techniques are based on a new notion of solved form, where fundamental properties of orderings like transitivity and monotonicity are taken into account. Apart from simplicity and elegance from the theoretical point of view, the main contribution of these algorithms is on efficiency in practice. Since guessing is minimized, and, in particular, no linear orderings between the subterms are guessed, a practical improvement in performance of several orders of magnitude over previous algorithms is obtained, as shown by our experiments.

Conference Paper

Conventional algebraic specifications are first-order. Using higher-order equations in combination with first-order ones raises several fundamental model-theoretic and proof-theoretic questions. The model theory of higher-order equations is well understood (see [20] for a survey of algebraic specifications). The proof theory of higher-order equations is equally well understood, it requires... [Show full abstract]

Article

. Rewriting is a general paradigm for expressing computations in various logics, and we focus here on rewriting techniques in equational logic. When used at the proof level, rewriting provides with a very powerful methodology for proving completeness results, a technique that is illustrated here. We also consider whether important properties of rewrite systems such as confluence and... [Show full abstract]

Article

syntax trees, congruence laws and rewrite rules are used to define the semantics. A computation step is modeled as the application of a rewrite rule to an abstract syntax tree modulo structural congruence. Using the semantics, the critical interaction between sequential execution (including backtracking and cut pruning) and coroutining are made precise. In particular cases where this... [Show full abstract]

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