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Variant-Based Satisfiability in Initial Algebras

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Formal Techniques for Safety-Critical Systems (FTSCS 2015)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 596))

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Abstract

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 illustrated with examples.

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Notes

  1. 1.

    Roughly, u is an E, B-variant of a term t if u is the E, B-canonical form of a substitution instance, \(t \theta \), of t (for a more careful definition see Definition 5). Therefore, the variants of t are intuitively the “irreducible patterns” to which t can be symbolically evaluated by the rules E modulo B. \(E \uplus B\) has the finite variant property if there is a finite set of most general variants, which are computed by folding variant narrowing.

  2. 2.

    For a discussion of similar but not exactly equivalent versions of the variant notion see [23]. Here I follow the formulation in [42].

  3. 3.

    Such decidable QF satisfiability is of course equivalent to the decidability of whether a sentence in the existential closure of such QF formulas belongs to the theory of \(T_{\varSigma /E'}\), which is how the decidability property is actually stated in [31].

  4. 4.

    By the assumption that \(\varSigma \)’s poset of sorts \((S,\leqslant )\) is locally finite, up to variable renaming the specializations of a finite set of variables form always a finite set. When \(\phi ' \gamma \not \in T_{\varOmega ^{\wedge }}(X)\) we may still have \(\phi ' \gamma \rho \in T_{\varOmega ^{\wedge }}(X)\) for some variable specialization \(\rho \) because a constructor symbol \(f:w \rightarrow s\) may have a subsort-overloaded typing \(f:w' \rightarrow s'\) that is not a constructor but a defined symbol (see Footnote 5 below for an example).

  5. 5.

    The following example illustrates all the issues involved. In the FVP decomposition \(\mathcal {Z}_{+}\) of the integers with addition of Example 10 in Sect. 6.2, the signature \(\varOmega \) of constructors contains two typings for \(+\), namely, \(\_+\_ : Nat \; Nat \rightarrow Nat \) and \(\_+\_ : NzNat \; NzNat \rightarrow NzNat \), with \( NzNat \) the subsort of non-zero naturals, and both operations associative-commutative, and having 0 as unit element (\( ACU \)). Instead, the typing \(\_+\_ : Int \; Int \rightarrow Int \) (also ACU) is not a constructor, but a function defined by equations. Let \(\phi \) be the equation \(x+y=x'+y'\), where all variables have sort \( Int \). It has the variant \((x+y=x'+y', id )\), and \(\gamma =\{x \mapsto x',y \mapsto y'\}\) is one of the ACU-unifiers of \(x+y=x'+y'\). Case (1) fails because \(x'+y'\) is not an \(\varOmega \)-term. However, the variable specialization \(\rho =\{x' \mapsto x''\!:\! Nat ,y' \mapsto y''\!:\! Nat \}\) yields the constructor unifier \( id \gamma \rho = \{x \mapsto x''\!:\! Nat ,y \mapsto y''\!:\! Nat , x' \mapsto x''\!:\! Nat ,y' \mapsto y''\!:\! Nat \}\) because now \( x''\!:\! Nat +y''\!:\! Nat \) is an \(\varOmega \)-term (property (ii) holds) and property (iii) also holds. Furthermore, \(\rho \) is maximal with properties (ii) and (iii). For example, \(\rho > \tau \) for \(\tau =\{x' \mapsto x'''\!:\! NzNat ,y' \mapsto y'''\!:\! NzNat \}\), so that the less general unifier \( id \gamma \tau \) is unnecessary.

  6. 6.

    Using a lazy \( DPLL (T)\) solver (see, e.g., [13]) we do not have to assume that \(\varphi \) is in DNF: the \( DPLL (T)\) solver will efficiently extract from \(\varphi \) the appropriate conjunctions of T-literals to check for satisfiability.

  7. 7.

    A complete set of constructor variants for a term t is obtained by inspecting each \((u,\theta ) \in [\![t ]\!]_{R,B}\) and either: (1) choosing \((u,\theta )\) when \(u \in T_{\varOmega }(X)\), or otherwise (2) choosing those \((u\rho ,\theta \rho )\) such that \(\rho \) is a variable specialization and: (i) \(u \rho \in T_{\varOmega }(X)\), (ii) \((u\rho ,\theta \rho )\) is a variant of t, and (iii) \(\rho \) is maximal with properties (i)–(ii).

  8. 8.

    An order-sorted version \(\mathcal {N}_{+}\) of \(\mathcal {N}^{u}_{+}\) is obtained by adding a subsort inclusion \( NzNat < Nat \), where \( NzNat \) denotes the non-zero naturals, typing 1 with sort \( NzNat \), and adding the typing \(\_+\_ : NzNat \; NzNat \rightarrow NzNat \). \(\mathcal {N}_{+}\) is also OS-compact for the exact same reasons. A reduction of satisfiability in the initial agebra of \(\mathcal {N}_{+}\) to satisfiability in the initial algebra of \(\mathcal {N}^{u}_{+}\) is discussed in [69]. \(\mathcal {N}_{+}\) makes the language more expressive: instead of stating \(x \not = 0\) we can just type x as having sort \( NzNat \).

  9. 9.

    See [69] for a version \(\mathcal {N}_{+,>}\) of natural Pressburger arithmetic in which \(>\) is only explictly defined in the positive case.

  10. 10.

    Note the interesting phenomenon, impossible in a many-sorted setting, that a subsort-polymorphic symbol like s or p can be a constructor for some typings and a defined symbol for other typings.

  11. 11.

    See [69] for an even simpler version \(\mathcal {Z}_{+,>}\) of integer Presburger arithmetic in which \(>\) is only explicitly defined in the positive case.

  12. 12.

    This violates the general assumption that sorts are non-empty; however, parameter sorts instantiated to target theories with non-empty sorts become non-empty.

  13. 13.

    There is no real loss of generality because we can make it so by renaming its sorts and operations. In fact, disjointness must in any case be enforced by the “pushout construction” for parameter instantiation, implicitly described in what follows for this simple class of uni-parametric parameterized theories.

  14. 14.

    For more details about sufficient completeness of parameterized OS theories and methods for checking it see [67].

  15. 15.

    For combining variant-based decision procedures with other decision procedures, the order-sorted NO combination method in [82] will be particulary useful.

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Acknowledgements

I thank the organizers of FTSCS 2015 for inviting me to present these ideas in Paris, and the FTSCS participants for their interest and very helpful comments. I thank Andrew Cholewa, Steven Eker, Santiago Escobar, Ralf Sasse, and Carolyn Talcott for their contributions to the development of the theory and Maude implementation of folding variant narrowing. I have learned much about satisfiability from Maria-Paola Bonacina, Vijay Ganesh and Cesare Tinelli along many conversations; I am most grateful to them for their kind enlightenment. I also thank the following persons for their very helpful comments on earlier drafts: Maria-Paola Bonacina, Santiago Escobar, Dorel Lucau, Peter Ölveczky, Vlad Rusu, Ralf Sasse, Natarajan Shankar, and Cesare Tinelli. The pioneering work of Hubert Comon-Lundh about compact theories [29], and that of him with Stephanie Delaune about the finite variant property [31], have both been important sources of inspiration for the ideas presented here. This work has been partially supported by NSF Grant CNS 13-19109.

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  1. Department of Computer Science, University of Illinois at Urbana-Champaign, Urbana, Illinois, USA

    José Meseguer

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  1. José Meseguer
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  1. AIST Ikeda, Ikeda,Osaka, Japan

    Cyrille Artho

  2. University of Oslo, Oslo, Norway

    Peter Csaba Ölveczky

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Meseguer, J. (2016). Variant-Based Satisfiability in Initial Algebras. In: Artho, C., Ölveczky, P. (eds) Formal Techniques for Safety-Critical Systems. FTSCS 2015. Communications in Computer and Information Science, vol 596. Springer, Cham. https://doi.org/10.1007/978-3-319-29510-7_1

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