Basic Narrowing Revisited

In this paper we study basic narrowing as a method for solving equations in the initial algebra specified by a ground confluent and terminating term rewriting system. Since we are interested in equation solving, we don’t study basic narrowing as a reduction relation on terms but consider immediately its reformulation as an equation solving rule. This reformulation leads to a technically simpler presentation and reveals that the essence of basic narrowing can be captured without recourse to term unification. We present an equation solving calculus that features three classes of rules. Resolution rules, whose application is don’t care nondeterministic, are the basic rules and suffice for a complete solution procedure. Failure rules detect inconsistent parts of the search space. Simplification rules, whose application is don’t care nondeterministic, enhance the power of the failure rules and reduce the number of necessary don’t know steps. Three of the presented simplification rules are new. The rewriting rule allows for don’t care nondeterministic rewriting and thus yields a marriage of basic and normalizing narrowing. The safe blocking rule is specific to basic narrowing and is particulary useful in conjunction with the rewriting rule. Finally, the unfolding rule allows for a variety of search strategies that reduce the number of don’t know alternatives that need to be explored

A debugging model for functional logic programs

This paper presents a box-oriented debugging model for the functional logic language ALF. Due to the sophisticated operational semantics of ALF which is based on innermost basic narrowing with simplification, the debugger must reflect the application of the different computation rules during program execution. Hence our debugging model includes not only one box type as in Byrd's debugging model for logic programs but several different kinds of boxes corresponding to the various computation rules of the functional logic language (narrowing, simplification etc.). Moreover, additional box types are introduced in order to allow skips over (sometimes) uninteresting program parts like proofs of the condition in a conditional equation. Since ALF is a genuine amalgamation of functional and logic languages, our debugging model subsumes operational aspects of both kinds of languages. As a consequence, it can be also used for pure logic languages, pure functional languages with eager evaluation, or functional logic languages with a less sophisticated operational semantics like SLOG or eager BABEL

A Finite Representation of the Narrowing Space

The final publication is available at Springer via http://dx.doi.org/10.1007/978-3-319-14125-1_4Narrowing basically extends rewriting by allowing free variables in terms and by replacing matching with unification. As a consequence, the search space of narrowing becomes usually infinite, as in logic programming. In this paper, we introduce the use of some operators that allow one to always produce a finite data structure that still represents all the narrowing derivations. Furthermore, we extract from this data structure a novel, compact equational representation of the (possibly infinite) answers computed by narrowing for a given initial term. Both the finite data structure and the equational representation of the computed answers might be useful in a number of areas, like program comprehension, static analysis, program transformation, etc.Nishida, N.; Vidal, G. (2013). A Finite Representation of the Narrowing Space. En Logic-Based Program Synthesis and Transformation. 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Springer, Heidelberg (1996)Baader, F., Nipkow, T.: Term Rewriting and All That. Cambridge University Press (1998)Bae, K., Escobar, S., Meseguer, J.: Abstract Logical Model Checking of Infinite-State Systems Using Narrowing. In: Proceedings of the 24th International Conference on Rewriting Techniques and Applications. LIPIcs, vol. 21, pp. 81–96. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2013)De Schreye, D., Glück, R., Jørgensen, J., Leuschel, M., Martens, B., Sørensen, M.: Conjunctive partial deduction: foundations, control, algorihtms, and experiments. Journal of Logic Programming 41(2&3), 231–277 (1999)Escobar, S., Meadows, C., Meseguer, J.: A rewriting-based inference system for the NRL Protocol Analyzer and its meta-logical properties. Theoretical Computer Science 367(1–2), 162–202 (2006)Escobar, S., Meseguer, J.: Symbolic Model Checking of Infinite-State Systems Using Narrowing. In: Baader, F. (ed.) RTA 2007. LNCS, vol. 4533, pp. 153–168. Springer, Heidelberg (2007)Fribourg, L.: SLOG: A Logic Programming Language Interpreter Based on Clausal Superposition and Rewriting. In: Proceedings of the Symposium on Logic Programming, pp. 172–185. IEEE Press (1985)Gnaedig, I., Kirchner, H.: Proving weak properties of rewriting. Theoretical Computer Science 412(34), 4405–4438 (2011)Hanus, M.: The integration of functions into logic programming: From theory to practice. Journal of Logic Programming 19&20, 583–628 (1994)Hanus, M. (ed.): Curry: An integrated functional logic language (vers. 0.8.3) (2012). http://www.curry-language.orgHermenegildo, M., Rossi, F.: On the Correctness and Efficiency of Independent And-Parallelism in Logic Programs. In: Lusk, E., Overbeck, R. (eds.) Proceedings of the 1989 North American Conf. on Logic Programming, pp. 369–389. The MIT Press, Cambridge (1989)Hölldobler, S. (ed.): Foundations of Equational Logic Programming. LNCS, vol. 353. Springer, Heidelberg (1989)Meseguer, J., Thati, P.: Symbolic Reachability Analysis Using Narrowing and its Application to Verification of Cryptographic Protocols. Electronic Notes in Theoretical Computer Science 117, 153–182 (2005)Middeldorp, A., Okui, S.: A Deterministic Lazy Narrowing Calculus. Journal of Symbolic Computation 25(6), 733–757 (1998)Nishida, N., Sakai, M., Sakabe, T.: Generation of Inverse Computation Programs of Constructor Term Rewriting Systems. IEICE Transactions on Information and Systems J88–D–I(8), 1171–1183 (2005) (in Japanese)Nishida, N., Sakai, M., Sakabe, T.: Partial Inversion of Constructor Term Rewriting Systems. In: Giesl, J. (ed.) RTA 2005. LNCS, vol. 3467, pp. 264–278. Springer, Heidelberg (2005)Nishida, N., Vidal, G.: Program inversion for tail recursive functions. In: Schmidt-Schauß, M. (ed.) Proceedings of the 22nd International Conference on Rewriting Techniques and Applications. LIPIcs, vol. 10, pp. 283–298. 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Incremental theory reasoning methods for semantic tableaux

Theory reasoning is an important technique for increasing the efficiency of automated deduction systems. In this paper we present incremental theory reasoning, a method that improves the interaction between the foreground reasoner and the background (theory) reasoner and, thus, the efficiency of the combined system. The use of incremental theory reasoning in free variable semantic tableaux and the cost reduction that can be achieved are discussed; as an example, completion-based equality reasoning is presented, including experimental data obtained using an implementation

Adding homomorphisms to commutative/monoidal theories or : how algebra can help in equational unification

Two approaches to equational unification can be distinguished. The syntactic approach relies heavily on the syntactic structure of the identities that define the equational theory. The semantic approach exploits the structure of the algebras that satisfy the theory. With this paper we pursue the semantic approach to unification. We consider the class of theories for which solving unification problems is equivalent to solving systems of linear equations over a semiring. This class has been introduced by the authors independently of each other as commutative theories (Baader) and monoidal theories (Nutt). The class encompasses important examples like the theories of abelian monoids, idempotent abelian monoids, and abelian groups. We identify a large subclass of commutative/monoidal theories that are of unification type zero by studying equations over the corresponding semiring. As a second result, we show with methods from linear algebra that unitary and finitary commutative/monoidal theories do not change their unification type when they are augmented by a finite monoid of homomorphisms, and how algorithms for the extended theory can be obtained from algorithms for the basic theory. The two results illustrate how using algebraic machinery can lead to general results and elegant proofs in unification theory

Narrowing strategies for arbitrary canonical rewrite systems

Narrowing is a universal unification procedure for equational theories defined by a canonical term rewriting system. In its original form it is extremely inefficient. Therefore, many optimizations have been proposed during the last years. In this paper, we present the narrowing strategies for arbitrary canonical systems in a uniform framework and introduce the new narrowing strategy LSE narrowing. LSE narrowing is complete and improves all other strategies which are complete for arbitrary canonical systems. It is optimal in the sense that two different LSE narrowing derivations cannot generate the same narrowing substitution. Moreover, LSE narrowing computes only normalized narrowing substitutions

Lazy unification with inductive simplification

Unification in the presence of an equational theory is an important problem in theorem-proving and in the integration of functional and logic programming languages. This paper presents an improvement of the proposed lazy unification methods by incorporating simplification with inductive axioms into the unification process. Inductive simplification reduces the search space so that in some case infinite search spaces are reduced to finite ones. Consequently, more efficient unification algorithms can be achieved. We prove soundness and completeness of our method for equational theories represented by ground confluent and terminating rewrite systems

Basic paramodulation

We introduce a class of restrictions for the ordered paramodulation and superposition calculi (inspired by the {\em basic\/} strategy for narrowing), in which paramodulation inferences are forbidden at terms introduced by substitutions from previous inference steps. In addition we introduce restrictions based on term selection rules and redex orderings, which are general criteria for delimiting the terms which are available for inferences. These refinements are compatible with standard ordering restrictions and are complete without paramodulation into variables or using functional reflexivity axioms. We prove refutational completeness in the context of deletion rules, such as simplification by rewriting (demodulation) and subsumption, and of techniques for eliminating redundant inferences