E-Generalization Using Grammars

We extend the notion of anti-unification to cover equational theories and present a method based on regular tree grammars to compute a finite representation of E-generalization sets. We present a framework to combine Inductive Logic Programming and E-generalization that includes an extension of Plotkin's lgg theorem to the equational case. We demonstrate the potential power of E-generalization by three example applications: computation of suggestions for auxiliary lemmas in equational inductive proofs, computation of construction laws for given term sequences, and learning of screen editor command sequences.Comment: 49 pages, 16 figures, author address given in header is meanwhile outdated, full version of an article in the "Artificial Intelligence Journal", appeared as technical report in 2003. An open-source C implementation and some examples are found at the Ancillary file

Smart matching

One of the most annoying aspects in the formalization of mathematics is the need of transforming notions to match a given, existing result. This kind of transformations, often based on a conspicuous background knowledge in the given scientific domain (mostly expressed in the form of equalities or isomorphisms), are usually implicit in the mathematical discourse, and it would be highly desirable to obtain a similar behavior in interactive provers. The paper describes the superposition-based implementation of this feature inside the Matita interactive theorem prover, focusing in particular on the so called smart application tactic, supporting smart matching between a goal and a given result.Comment: To appear in The 9th International Conference on Mathematical Knowledge Management: MKM 201

A resolution principle for clauses with constraints

We introduce a general scheme for handling clauses whose variables are constrained by an underlying constraint theory. In general, constraints can be seen as quantifier restrictions as they filter out the values that can be assigned to the variables of a clause (or an arbitrary formulae with restricted universal or existential quantifier) in any of the models of the constraint theory. We present a resolution principle for clauses with constraints, where unification is replaced by testing constraints for satisfiability over the constraint theory. We show that this constrained resolution is sound and complete in that a set of clauses with constraints is unsatisfiable over the constraint theory if we can deduce a constrained empty clause for each model of the constraint theory, such that the empty clauses constraint is satisfiable in that model. We show also that we cannot require a better result in general, but we discuss certain tractable cases, where we need at most finitely many such empty clauses or even better only one of them as it is known in classical resolution, sorted resolution or resolution with theory unification

Refactoring pattern matching

Defining functions by pattern matching over the arguments is advantageous for understanding and reasoning, but it tends to expose the implementation of a datatype. Significant effort has been invested in tackling this loss of modularity; however, decoupling patterns from concrete representations while maintaining soundness of reasoning has been a challenge. Inspired by the development of invertible programming, we propose an approach to program refactoring based on a right-invertible language rinv—every function has a right (or pre-) inverse. We show how this new design is able to permit a smooth incremental transition from programs with algebraic datatypes and pattern matching, to ones with proper encapsulation, while maintaining simple and sound reasoning

Datalog-Based program analysis with BES and RWL

This paper describes two techniques for Datalog query evaluation and their application to object-oriented program analysis. The first technique transforms Datalog programs into an implicit Boolean Equation System (Bes) that can then be solved by using linear-time complexity algorithms that are available in existing, general purpose verification toolboxes such as Cadp. In order to improve scalability and to enable analyses involving advanced meta-programming features, we develop a second methodology that transforms Datalog programs into rewriting logic (Rwl) theories. This method takes advantage of the preeminent features and facilities that are available within the high-performance system Maude, which provides a very efficient implementation of Rwl. We provide evidence of the practicality of both approaches by reporting on some experiments with a number of real-world Datalog-based analyses. © 2011 Springer-Verlag.This work has been partially supported by the eu(feder), the Spanish mec/micinn under grants tin2007-68093-C02 and tin2010-21062-C02-02, and the Generalitat Valenciana under grant Emergentes gv/2009/024. M.A.Feliu was partially supported by the Spanish mec fpu grant AP2008-00608.Alpuente Frasnedo, M.; Feliú Gabaldón, MA.; Joubert, C.; Villanueva García, A. (2011). Datalog-Based program analysis with BES and RWL. En Datalog Reloaded. Springer Verlag (Germany). 6702:1-20. https://doi.org/10.1007/978-3-642-24206-9_1S1206702Afrati, F.N., Ullman, J.D.: Optimizing joins in a map-reduce environment. In: Manolescu, I., Spaccapietra, S., Teubner, J., Kitsuregawa, M., Léger, A., Naumann, F., Ailamaki, A., Özcan, F. (eds.) EDBT. ACM International Conference Proceeding Series, vol. 426, pp. 99–110. 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