5 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
Automatic Synthesis of Logical Models for Order-Sorted First-Order Theories
[EN] In program analysis, the synthesis of models of logical theories representing the program semantics is often useful to prove program properties. We use order-sorted first- order logic as an appropriate framework to describe the semantics and properties of programs as given theories. Then we investigate the automatic synthesis of models for such theories. We use convex polytopic domains as a flexible approach to associate different domains to different sorts. We introduce a framework for the piecewise definition of functions and predicates. We develop its use with linear expressions (in a wide sense, including linear transformations represented as matrices) and inequalities to specify functions and predicates. In this way, algorithms and tools from linear algebra and arithmetic constraint solving (e.g., SMT) can be used as a backend for an efficient implementation.Partially supported by the EU (FEDER), projects TIN2015-69175-C4-1-R, and GV PROMETEOII/2015/ 013. R. GutiĂ©rrez also supported by Juan de la Cierva Fellowship JCI-2012-13528.Lucas Alba, S.; GutiĂ©rrez Gil, R. (2018). Automatic Synthesis of Logical Models for Order-Sorted First-Order Theories. Journal of Automated Reasoning. 60(4):465-501. https://doi.org/10.1007/s10817-017-9419-3S465501604AlarcĂłn, B., GutiĂ©rrez, R., Lucas, S., Navarro-Marset, R.: Proving termination properties with MU-TERM. In: Proceedings of AMASTâ10. LNCS, vol. 6486, pp. 201â208 (2011)AlarcĂłn, B., Lucas, S., Navarro-Marset, R.: Using matrix interpretations over the reals in proofs of termination. In: Proceedings of PROLEâ09, pp. 255â264 (2009)Albert, E., Genaim, S., GutiĂ©rrez, R.: A Transformational Approach to Resource Analysis with Typed-Norms. Revised Selected Papers from LOPSTRâ13. LNCS, vol. 8901, pp 38â53 (2013)de Angelis, E., Fioravante, F., Pettorossi, A., Proietti, M.: Proving correctness of imperative programs by linearizing constrained Horn clauses. Theory Pract. Log. 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Matching Logic
This paper presents matching logic, a first-order logic (FOL) variant for
specifying and reasoning about structure by means of patterns and pattern
matching. Its sentences, the patterns, are constructed using variables,
symbols, connectives and quantifiers, but no difference is made between
function and predicate symbols. In models, a pattern evaluates into a power-set
domain (the set of values that match it), in contrast to FOL where functions
and predicates map into a regular domain. Matching logic uniformly generalizes
several logical frameworks important for program analysis, such as:
propositional logic, algebraic specification, FOL with equality, modal logic,
and separation logic. Patterns can specify separation requirements at any level
in any program configuration, not only in the heaps or stores, without any
special logical constructs for that: the very nature of pattern matching is
that if two structures are matched as part of a pattern, then they can only be
spatially separated. Like FOL, matching logic can also be translated into pure
predicate logic with equality, at the same time admitting its own sound and
complete proof system. A practical aspect of matching logic is that FOL
reasoning with equality remains sound, so off-the-shelf provers and SMT solvers
can be used for matching logic reasoning. Matching logic is particularly
well-suited for reasoning about programs in programming languages that have an
operational semantics, but it is not limited to this
Programming and symbolic computation in Maude
[EN] Rewriting logic is both a flexible semantic framework within which widely different concurrent systems can be naturally specified and a logical framework in which widely different logics can be specified. Maude programs are exactly rewrite theories. Maude has also a formal environment of verification tools. Symbolic computation is a powerful technique for reasoning about the correctness of concurrent systems and for increasing the power of formal tools. We present several new symbolic features of Maude that enhance formal reasoning about Maude programs and the effectiveness of formal tools. They include: (i) very general unification modulo user-definable equational theories, and (ii) symbolic reachability analysis of concurrent systems using narrowing. The paper does not focus just on symbolic features: it also describes several other new Maude features, including: (iii) Maude's strategy language for controlling rewriting, and (iv) external objects that allow flexible interaction of Maude object-based concurrent systems with the external world. In particular, meta-interpreters are external objects encapsulating Maude interpreters that can interact with many other objects. To make the paper self-contained and give a reasonably complete language overview, we also review the basic Maude features for equational rewriting and rewriting with rules, Maude programming of concurrent object systems, and reflection. Furthermore, we include many examples illustrating all the Maude notions and features described in the paper.Duran has been partially supported by MINECO/FEDER project TIN2014-52034-R. Escobar has been partially supported by the EU (FEDER) and the MCIU under grant RTI2018-094403-B-C32, by the Spanish Generalitat Valenciana under grant PROMETE0/2019/098, and by the US Air Force Office of Scientific Research under award number FA9550-17-1-0286. MartiOliet and Rubio have been partially supported by MCIU Spanish project TRACES (TIN2015-67522-C3-3-R). Rubio has also been partially supported by a MCIU grant FPU17/02319. Meseguer and Talcott have been partially supported by NRL Grant N00173 -17-1-G002. Talcott has also been partially supported by ONR Grant N00014-15-1-2202.DurĂĄn, F.; Eker, S.; Escobar RomĂĄn, S.; NARCISO MARTĂ OLIET; JosĂ© Meseguer; RubĂ©n Rubio; Talcott, C. (2020). Programming and symbolic computation in Maude. Journal of Logical and Algebraic Methods in Programming. 110:1-58. https://doi.org/10.1016/j.jlamp.2019.100497S158110Alpuente, M., Escobar, S., Espert, J., & Meseguer, J. (2014). A modular order-sorted equational generalization algorithm. 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