Unification in the Datastructure Multisets

In a forthcoming paper A. Herold and J. Siekmann generalize "pure" AC-unification ([ST 75], [LS 76]) to terms containing additional function symbols (see also [ST 81], [FA 84]). Generalized AC-unification thus attains practical relevance for a broad range of applications. Pure AC-unification is used as a basic mechanism and it is this key role that has motivated our research. We have improved upon earlier approaches by basing (pure) AC-unification on a firm theoretical basis and presenting algorithms which fully exploit the properties of the underlying mathematical structure. In particular, the high degree of parallelism for AC-unification will become apparent. Our algorithms have been designed for parallel hardware but still yield significant improvements over earlier algorithms when used in sequential mode

Satisfiability Modulo Free Data Structures Combined with Bridging Functions

International audienceFree Data Structures are finite semantic trees modulo equational axioms that are useful to represent classical data structures such as lists, multisets and sets. We study the satisfiability problem when free data structures are combined with bridging functions. We discuss the possibility to get a combination methodàmethod`methodà la Nelson-Oppen for these particular non-disjoint unions of theories. In order to handle satisfiability problems with disequalities, we investigate a form of sufficient surjectivity for the bridging functions

Opening the AC-Unification Race

This note reports about the implementation of AC-unification algorithms, based on the variable-abstraction method of Stickel and on the constant-abstraction method of Livesey, Siekmann, and Herold. We give a set of 105 benchmark examples and compare execution times for implementations of the two approaches. This documents for other researchers what we consider to be the state-of-the-art performance for elementary AC-unification problems

Variant-Based Decidable Satisfiability in Initial Algebras with Predicates

[EN] Decision procedures can be either theory-specific, e.g., Presburger arithmetic, or theory-generic, applying to an infinite number of user-definable theories. Variant satisfiability is a theory-generic procedure for quantifier-free satisfiability in the initial algebra of an order-sorted equational theory (¿,E¿B) under two conditions: (i) E¿B has the finite variant property and B has a finitary unification algorithm; and (ii) (¿,E¿B) protects a constructor subtheory (¿,E¿¿B¿) that is OS-compact. These conditions apply to many user-definable theories, but have a main limitation: they apply well to data structures, but often do not hold for user-definable predicates on such data structures. We present a theory-generic satisfiability decision procedure, and a prototype implementation, extending variant-based satisfiability to initial algebras with user-definable predicates under fairly general conditions.Partially supported by NSF Grant CNS 14-09416, NRL under contract number N00173-17-1-G002, the EU (FEDER), Spanish MINECO project TIN2015-69175- C4-1-R and GV project PROMETEOII/2015/013. Ra´ul Guti´errez was also supported by INCIBE program “Ayudas para la excelencia de los equipos de investigaci´on avanzada en ciberseguridad”.Gutiérrez Gil, R.; Meseguer, J. (2018). Variant-Based Decidable Satisfiability in Initial Algebras with Predicates. Lecture Notes in Computer Science. 10855:306-322. https://doi.org/10.1007/978-3-319-94460-9_18S30632210855Armando, A., Bonacina, M.P., Ranise, S., Schulz, S.: New results on rewrite-based satisfiability procedures. TOCL 10(1), 4 (2009)Armando, A., Ranise, S., Rusinowitch, M.: A rewriting approach to satisfiability procedures. 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Ph.D. thesis, ENS Cachan (2011)Comon, H.: Complete axiomatizations of some quotient term algebras. TCS 118(2), 167–191 (1993)Comon-Lundh, H., Delaune, S.: The finite variant property: how to get rid of some algebraic properties. In: Giesl, J. (ed.) RTA 2005. LNCS, vol. 3467, pp. 294–307. Springer, Heidelberg (2005). https://doi.org/10.1007/978-3-540-32033-3_22Dershowitz, N., Jouannaud, J.P.: Rewrite systems. In: Handbook of Theoretical Computer Science, North-Holland, vol. B, pp. 243–320 (1990)Dovier, A., Piazza, C., Rossi, G.: A uniform approach to constraint-solving for lists, multisets, compact lists, and sets. TOCL 9(3), 15 (2008)Dross, C., Conchon, S., Kanig, J., Paskevich, A.: Adding decision procedures to SMT solvers using axioms with triggers. JAR 56(4), 387–457 (2016)Escobar, S., Sasse, R., Meseguer, J.: Folding variant narrowing and optimal variant termination. JALP 81, 898–928 (2012)Goguen, J.A., Meseguer, J.: Models and equality for logical programming. In: Ehrig, H., Kowalski, R., Levi, G., Montanari, U. (eds.) TAPSOFT 1987. LNCS, vol. 250, pp. 1–22. Springer, Heidelberg (1987). https://doi.org/10.1007/BFb0014969Goguen, J., Meseguer, J.: Order-sorted algebra I: equational deduction for multiple inheritance, overloading, exceptions and partial operations. TCS 105, 217–273 (1992)Gutiérrez, R., Meseguer, J.: Variant satisfiability in initial algebras with predicates. Technical report, CS Department, University of Illinois at Urbana-Champaign (2018). http://hdl.handle.net/2142/99039Jouannaud, J.P., Kirchner, H.: Completion of a set of rules modulo a set of equations. SICOMP 15, 1155–1194 (1986)Kroening, D., Strichman, O.: Decision Procedures - An algorithmic point of view. Texts in TCS. An EATCS Series. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-74105-3Lynch, C., Morawska, B.: Automatic decidability. In: Proceedings of LICS 2002, p. 7. IEEE Computer Society (2002)Lynch, C., Tran, D.-K.: Automatic decidability and combinability revisited. In: Pfenning, F. (ed.) CADE 2007. LNCS (LNAI), vol. 4603, pp. 328–344. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-73595-3_22Meseguer, J.: Variant-based satisfiability in initial algebras. SCP 154, 3–41 (2018)Meseguer, J.: Strict coherence of conditional rewriting modulo axioms. TCS 672, 1–35 (2017)Meseguer, J., Goguen, J.: Initiality, induction and computability. In: Algebraic Methods in Semantics, Cambridge, pp. 459–541 (1985)Meseguer, J., Goguen, J.: Order-sorted algebra solves the constructor-selector, multiple representation and coercion problems. I&C 103(1), 114–158 (1993)Nelson, G., Oppen, D.C.: Simplification by cooperating decision procedures. TOPLAS 1(2), 245–257 (1979)Shostak, R.E.: Deciding combinations of theories. J. ACM 31(1), 1–12 (1984)Skeirik, S., Meseguer, J.: Metalevel algorithms for variant satisfiability. In: Lucanu, D. (ed.) WRLA 2016. LNCS, vol. 9942, pp. 167–184. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-44802-2_10Stump, A., Barrett, C.W., Dill, D.L., Levitt, J.R.: A decision procedure for an extensional theory of arrays. In: Proceedings of LICS 2001, pp. 29–37. IEEE (2001)Tushkanova, E., Giorgetti, A., Ringeissen, C., Kouchnarenko, O.: A rule-based system for automatic decidability and combinability. SCP 99, 3–23 (2015

Unification in Abelian Semigroups

Unification in equational theories, i.e. solving of equations in varieties, is a basic operation in Computational Logic, in Artificial Intelligence (AI) and in many applications of Computer Science. In particular the unification of terms in the presence of an associative and commutative f unction, i.e. solving of equations in Abelian Semigroups, turned out to be of practical relevance for Term Rewriting Systems, Automated Theorem Provers and many AI-programming languages. The observation that unification under associativity and commutativity reduces to the solution of certain linear diophantine equations is the basis for a complete and minimal unification algorithm. The set of most general unifiers is closely related to the notion of a basis for the linear solution space of these equations. These results are extended to unification in free term algebras combined with Abelian Semigroups

Relational extensions to feature logic: applications to constraint based grammars

This thesis investigates the logical and computational foundations of unification-based or more appropriately constraint based grammars. The thesis explores extensions to feature logics (which provide the basic knowledge representation services to constraint based grammars) with multi-valued or relational features. These extensions are useful for knowledge representation tasks that cannot be expressed within current feature logics.The approach bridges the gap between concept languages (such as KL-ONE), which are the mainstay of knowledge representation languages in AI, and feature logics. Va¬ rious constraints on relational attributes are considered such as existential membership, universal membership, set descriptions, transitive relations and linear precedence con¬ straints.The specific contributions of this thesis can be summarised as follows: 1. Development of an integrated feature/concept logic 2. Development of a constraint logic for so called partial set descriptions 3. Development of a constraint logic for expressing linear precedence constraints 4. The design of a constraint language CL-ONE that incorporates the central ideas provided by the above study 5. A study of the application of CL-ONE for constraint based grammarsThe thesis takes into account current insights in the areas of constraint logic programming, object-oriented languages, computational linguistics and knowledge representation

Type indexing in OCaml for function search in a large ecosystem

International audienceWhen it comes to finding functions in libraries, in whichever language, programmers struggle with the need for efficient tools able to retrieve all relevant functions in a reasonable time. For the OCaml language, dowsindex was developed to offer a solution to this problem, via the notion of isomorphisms of types. It relies on the construction of an Index to precompute information on the libraries and thus facilitate the search algorithm. We aim to improve its performances by proposing a new data structure representing the matching relation between types, along with a new heuristic to optimize the type search

Declarative programming for agent applications

This paper introduces the execution model of a declarative programming language intended for agent applications. Features supported by the language include functional and logic programming idioms, higher-order functions, modal computation, probabilistic computation, and some theorem-proving capabilities. The need for these features is motivated and examples are given to illustrate the central ideas

Extending a Brainiac Prover to Lambda-Free Higher-Order Logic

International audienceDecades of work have gone into developing efficient proof calculi, data structures, algorithms, and heuristics for first-order automatic theorem proving. Higher-order provers lag behind in terms of efficiency. Instead of developing a new higher-order prover from the ground up, we propose to start with the state-of-the-art superposition prover E and gradually enrich it with higher-order features. We explain how to extend the prover’s data structures, algorithms, and heuristics to $$\lambda$$ λ -free higher-order logic, a formalism that supports partial application and applied variables. Our extension outperforms the traditional encoding and appears promising as a stepping stone toward full higher-order logic