321 research outputs found
Equational Formulas and Pattern Operations in Initial Order-Sorted Algebras
A pattern, i.e., a term possibly with variables, denotes the set
(language) of all its ground instances. In an untyped setting,
symbolic operations on finite sets of patterns can represent Boolean
operations on languages. But for the more expressive patterns needed
in declarative languages supporting rich type disciplines such as
subtype polymorphism untyped pattern operations and algorithms break
down. We show how they can be properly defined by means of a
signature transformation that enriches the types of the original
signature. We also show that this transformation allows a systematic
reduction of the first-order logic properties of an initial
order-sorted algebra supporting subtype-polymorphic functions to
equivalent properties of an initial many-sorted (i.e., simply typed)
algebra. This yields a new, simple proof of the known decidability of
the first-order theory of an initial order-sorted algebra.Partially supported by NSF Grant CNS 13-19109.Ope
Formalization of Universal Algebra in Agda
In this work we present a novel formalization of universal algebra in Agda. We show that heterogeneous signatures can be elegantly modelled in type-theory using sets indexed by arities to represent operations. We prove elementary results of heterogeneous algebras, including the proof that the term algebra is initial and the proofs of the three isomorphism theorems. We further formalize equational theory and prove soundness and completeness. At the end, we define (derived) signature morphisms, from which we get the contravariant functor between algebras; moreover, we also proved that, under some restrictions, the translation of a theory induces a contra-variant functor between models.Fil: Gunther, Emmanuel. Universidad Nacional de CĂłrdoba. Facultad de Matemática, AstronomĂa y FĂsica; Argentina. Consejo Nacional de Investigaciones CientĂficas y TĂ©cnicas; ArgentinaFil: Gadea, Alejandro Emilio. Universidad Nacional de CĂłrdoba. Facultad de Matemática, AstronomĂa y FĂsica; Argentina. Consejo Nacional de Investigaciones CientĂficas y TĂ©cnicas; ArgentinaFil: Pagano, Miguel Maria. Consejo Nacional de Investigaciones CientĂficas y TĂ©cnicas; Argentina. Universidad Nacional de CĂłrdoba. Facultad de Matemática, AstronomĂa y FĂsica; Argentin
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. I&C 183(2), 140–164 (2003)Barrett, C., Shikanian, I., Tinelli, C.: An abstract decision procedure for satisfiability in the theory of inductive data types. JSAT 3, 21–46 (2007)Bouchard, C., Gero, K.A., Lynch, C., Narendran, P.: On forward closure and the finite variant property. In: Fontaine, P., Ringeissen, C., Schmidt, R.A. (eds.) FroCoS 2013. LNCS (LNAI), vol. 8152, pp. 327–342. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-40885-4_23Bradley, A.R., Manna, Z.: The Calculus of Computation - Decision Procedures with Applications to Verification. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-74113-8Cholewa, A., Meseguer, J., Escobar, S.: Variants of variants and the finite variant property. Technical report, CS Dept. University of Illinois at Urbana-Champaign (2014). http://hdl.handle.net/2142/47117Ciobaca., S.: Verification of composition of security protocols with applications to electronic voting. 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
Variant-Based Satisfiability
Although different satisfiability decision procedures
can be combined by algorithms such as those of Nelson-Oppen or
Shostak, current tools typically can only support a finite number of
theories to use in such combinations. To make SMT solving more
widely applicable, generic satisfiability algorithms that can
allow a potentially infinite number of decidable theories to be
user-definable, instead of needing to be built in by the
implementers, are highly desirable. This work studies how
folding variant narrowing, a generic
unification algorithm that offers
good extensibility in unification theory, can be extended to
a generic variant-based satisfiability algorithm for the initial
algebras of its user-specified input theories when such theories
satisfy Comon-Delaune's finite variant property (FVP) and some
extra conditions. Several, increasingly larger infinite classes of
theories whose initial algebras enjoy decidable variant-based satisfiability
are identified, and a method based on descent maps to bring other theories
into these classes and to improve the generic
algorithm's efficiency is proposed and illustrated with examples.Partially supported by NSF Grant CNS 13-19109.Ope
Actors, actions, and initiative in normative system specification
The logic of norms, called deontic logic, has been used to specify normative constraints for information systems. For example, one can specify in deontic logic the constraints that a book borrowed from a library should be returned within three weeks, and that if it is not returned, the library should send a reminder. Thus, the notion of obligation to perform an action arises naturally in system specification. Intuitively, deontic logic presupposes the concept of anactor who undertakes actions and is responsible for fulfilling obligations. However, the concept of an actor has not been formalized until now in deontic logic. We present a formalization in dynamic logic, which allows us to express the actor who initiates actions or choices. This is then combined with a formalization, presented earlier, of deontic logic in dynamic logic, which allows us to specify obligations, permissions, and prohibitions to perform an action. The addition of actors allows us to expresswho has the responsibility to perform an action. In addition to the application of the concept of an actor in deontic logic, we discuss two other applications of actors. First, we show how to generalize an approach taken up by De Nicola and Hennessy, who eliminate from CCS in favor of internal and external choice. We show that our generalization allows a more accurate specification of system behavior than is possible without it. Second, we show that actors can be used to resolve a long-standing paradox of deontic logic, called the paradox of free-choice permission. Towards the end of the paper, we discuss whether the concept of an actor can be combined with that of an object to formalize the concept of active objects
Rewriting Modulo SMT
Combining symbolic techniques such as: (i) SMT solving, (ii) rewriting modulo theories, and (iii) model checking can enable the analysis of infinite-state systems outside the scope of each such technique. This paper proposes rewriting modulo SMT as a new technique combining the powers of (i)-(iii) and ideally suited to model and analyze infinite-state open systems; that is, systems that interact with a non-deterministic environment. Such systems exhibit both internal non-determinism due to the system, and external non-determinism due to the environment. They are not amenable to finite-state model checking analysis because they typically are infinite-state. By being reducible to standard rewriting using reflective techniques, rewriting modulo SMT can both naturally model and analyze open systems without requiring any changes to rewriting-based reachability analysis techniques for closed systems. This is illustrated by the analysis of a real-time system beyond the scope of timed automata methods
Maude: specification and programming in rewriting logic
Maude is a high-level language and a high-performance system supporting executable specification and declarative programming in rewriting logic. Since rewriting logic contains equational logic, Maude also supports equational specification and programming in its sublanguage of functional modules and theories. The underlying equational logic chosen for Maude is membership equational logic, that has sorts, subsorts, operator overloading, and partiality definable by membership and equality conditions. Rewriting logic is reflective, in the sense of being able to express its own metalevel at the object level. Reflection is systematically exploited in Maude endowing the language with powerful metaprogramming capabilities, including both user-definable module operations and declarative strategies to guide the deduction process. This paper explains and illustrates with examples the main concepts of Maude's language design, including its underlying logic, functional, system and object-oriented modules, as well as parameterized modules, theories, and views. We also explain how Maude supports reflection, metaprogramming and internal strategies. The paper outlines the principles underlying the Maude system implementation, including its semicompilation techniques. We conclude with some remarks about applications, work on a formal environment for Maude, and a mobile language extension of Maude
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