Abstract
In this survey, we report our recent work concerning combination results for interpolation and uniform interpolation in the context of quantifier-free fragments of first-order theories. We stress model-theoretic and algebraic aspects connecting this topic with amalgamation, strong amalgamation, and model-completeness. We give sufficient (and, in relevant situations, also necessary) conditions for the transfer of the quantifier-free interpolation property to combined first-order theories; we also investigate the non-disjoint signature case under the assumption that the shared theory is universal Horn. For convex, strong-amalgamating, stably infinite theories over disjoint signatures, we also provide a modular transfer result for the existence of uniform interpolants. Model completions play a key role in the whole paper: They enter into transfer results in the non-disjoint signature case and also represent a semantic counterpart of uniform interpolants.
Highlights
Suppose that the system to be verified is specified via a triple h x, ι( x ), τ ( x, x 0 )i given by a tuple of variables x, a formula ι( x ) describing initial states, and a formula τ ( x, x 0 ) describing state evolutions; suppose that we are given a further formula υ( x ) describing undesired ‘error’ states
The following extension of the above definition is considered more natural: Definition 2. [General quantifier-free interpolation] Let T be a theory in a signature Σ; we say that T has the general quantifier-free interpolation property iff, for every signature Σ0 and for every pair of ground Σ ∪ Σ0 -formulæ φ, ψ such that ψ T φ, there exists a ground formula θ, such that: (i) ψ T θ, (ii) θ T φ, and (iii) all predicates, constants, and function symbols from Σ0 occurring in θ occur both in φ and ψ
We show how compute the cover of a primitive formula ∃e φ(e, z), where we freely assume that the literals in φ are all flat: if we let Σ1 to be the signature of
Summary
Craig’s interpolation theorem [1] is a classical well-known result in first-order logic; it says that whenever an implication φ→ψ (1). We recall here what uniform interpolants are in the context of the quantifier-free fragment of a first-order theory T We use notations such as ψ( x ) to say that at most the variables from the tuple x occur freely in ψ. The usefulness of uniform interpolants in model checking was already stressed in [30] and further motivated by our recent line of research on the verification of data-aware processes [31,33,34,35]
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