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Automatic Synthesis of Logical Models for Order-Sorted First-Order Theories

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Abstract

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.

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Notes

  1. “Quantification in first-order logic always involves all elements of the universe. However, it is often more natural to just quantify over those elements of the universe which satisfy a certain condition” [19, 52].

  2. “The unsorted theory will in general be much larger (more symbols and more axioms) than the sorted theory” [59, p. 500].

  3. “Many sorted logics can allow an increase in deductive efficiency by eliminating useless branches of the search space” [24, Abstract].

  4. Implemented as part of the mechanisms available in the programming language Clear [20] to synthesize operations out from a set of basic (and available) ones [47, 48].

  5. The idea of using domains defined by means of linear inequations in program analysis actually goes back to earlier works like [30], for instance.

  6. In the realm of program termination, see, e.g., [87] for an early reference and [25] for a more systematic approach focusing on termination of rewrite systems.

  7. As for terms, the regularity requirement guarantees the existence of a least rank w for overloaded predicates, see [49, Footnote 7].

  8. Following [64, Section 1.1], these sets can be empty.

  9. As in [64], we use ‘structure’ and reserve the word ‘model’ to refer those structures satisfying a given set of sentences (theory).

  10. Goguen and Burstall consider the notion of a derivor as a more general kind of morphism between signatures [48, Section 6]. Martí-Oliet, Meseguer, and Palomino develop a similar notion called generalized signature morphism [90, Definition 3].

  11. Second-order structures are defined as for (unsorted) FOL; the difference with second-order logic is in the treatment of second-order variables that brings a different notion of satisfaction [17, p. 280].

  12. A ring with identity is a set with a rule of addition and a rule of multiplication satisfying the commutative, associative, zero element, inverse and distributive rules [106, Section 1.3].

  13. If \(A\mathbf {x}\ge \mathbf {b}\) has no solution, i.e., S in Theorem 5 is empty, the conditional sentence (1) trivially holds. Thus, we do not need to check S for emptiness when using Farkas’ result, although the systematic use of (2) to check (1) in this case may fail, thus eventually leading to loose some positive answers for (1).

  14. We rely on the existence of inverse additive elements of the ring structure of N for this, see Remark 15; also on the antisymmetry of \(\ge \) (Remark 16).

  15. i.e., with a single domain [10, p. 116], as a particular case of heterogeneous algebras, consisting of an indexed set of domains and functions over such domains.

  16. In [34, 35], the automated generation of monotone many-sorted algebras is considered. However, only two-sorted algebras are considered for generating interpretations of function symbols, on a previously fixed interpretation of sorts, see [35, Section 5].

  17. Constrained Horn clauses are essentially Horn clauses where some atoms have a predefined structure and interpretation established by a well-known theory (e.g., linear arithmetic).

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Acknowledgements

We thank José Meseguer for his comments and suggestions after carefully reading a preliminary version of this paper. We also thank him for drawing our attention to several relevant references. Salvador Lucas is most grateful to Prof. Manfred Schmidt-Schauss for sending him a hardcopy of his PhD Thesis [110]. We also thank the anonymous referees for many remarks and suggestions that led to improve the paper.

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    Salvador Lucas & Raúl Gutiérrez

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Correspondence to Salvador Lucas.

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Partially supported by the EU (FEDER), projects TIN2015-69175-C4-1-R, and GV PROMETEOII/2015/013. R. Gutiérrez also supported by Juan de la Cierva Fellowship JCI-2012-13528.

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Lucas, S., Gutiérrez, R. Automatic Synthesis of Logical Models for Order-Sorted First-Order Theories. J Autom Reasoning 60, 465–501 (2018). https://doi.org/10.1007/s10817-017-9419-3

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  • DOI: https://doi.org/10.1007/s10817-017-9419-3

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