Combination of Uniform Interpolants via Beth Definability

This paper is devoted to combination results concerning uniform interpolants. In this introduction, we summarize the two main (quite independant indeed) research lines that investigated uniform interpolants in the last three decades. We first recall what uniform interpolants are; we fix a logic or a theory T and a suitable fragment (propositional, first-order quantifier-free, etc.) of its language L. Given an L-formula \(\phi ({\underline{x}}, {\underline{y}})\) (here \({\underline{x}},{\underline{y}}\) are the variables occurring in \(\phi \)), a uniform interpolant of \(\phi \) (w.r.t. \({\underline{y}}\)) is an L-formula \(\phi '({\underline{x}})\) where only the \({\underline{x}}\) occur, and that satisfies the following two properties: (i) \(\phi ({\underline{x}}, {\underline{y}})\vdash _T \phi '({\underline{x}})\); (ii) for any further L-formula \(\psi ({\underline{x}}, {\underline{z}})\) such that \(\phi ({\underline{x}}, {\underline{y}}) \vdash _T \psi ({\underline{x}}, {\underline{z}})\), we have \(\phi '({\underline{x}}) \vdash _T \psi ({\underline{x}}, {\underline{z}})\). Whenever uniform interpolants exist, one can compute an interpolant for an entailment like \(\phi ({\underline{x}}, {\underline{y}}) \vdash _T \psi ({\underline{x}}, {\underline{z}})\) in a way that is independent of \(\psi \).

Uniform interpolants were originally studied in the context of non-classical logics, starting from the pioneering work by Pitts [40]. Uniform interpolants have in such non-classical logics context a ‘local’ and a ’global’ version, depending on how the entailment \(\vdash _T\) is interpreted: in the local version it is interpreted as ‘provability of implication’, whereas in the global version is interpreted as ‘provability under assumption’ (the two versions coincide for intuitionistic logic, but not for modal logics). The local version of uniform interpolation allows an (albeit not faithful) interpretation of the second order propositional calculus into plain propositional calculus, whereas the global version can be used in the axiomatization of model completions for the corresponding classes of algebras (see below). Uniform interpolants can be sematically connected to some appropriate notion of bisimulation at the level of Kripke models [13].

The existence of uniform interpolants is an exceptional phenomenon, which is however not so infrequent, as witnessed by a large literature in non-classical logics (a non-exhaustive list includes [1, 16, 22, 23, 25, 34, 37, 42, 45, 46]). The main results from the above papers are that uniform interpolants exist for intuitionistic logic and for some modal systems (like the Gödel-Löb system and the S4.Grz system); they do not exist for instance in S4 and K4, whereas for the basic modal system K they exist for the local version but not for the global version (the opposite situation is also well-possible, already in the locally finite case, as a consequence of Maksimova’s results on amalgamation and super-amalgamation [35, 36]). The connection between (global) uniform interpolants and model completions (for equational theories axiomatizing the varieties corresponding to propositional logics) was first stated in [24] and further developed in [25, 34, 37, 45].

In the last decade, also the automated reasoning community developed an increasing interest in uniform interpolants, with particular focus on quantifier-free fragments of first-order theories. This is witnessed by various talks and drafts by D. Kapur presented in many conferences and workshops (FloC 2010, ISCAS 2013-14, SCS 2017 [33]), as well as by the paper [32] by Gulwani and Musuvathi in ESOP 2008. In this last paper uniform interpolants were renamed as covers, a terminology we shall frequently adopt in this paper too. In these contributions, examples of cover computations were supplied and also some algorithms were sketched. The first formal proof about existence of covers in \(\mathcal {EUF}\) was however published by the present authors only in [6]; such a proof was equipped with powerful semantic tools (the Cover-by-Extensions Lemma 1 below) coming from the connection to model-completeness, as well as with an algorithm relying on a constrained variant of the Superposition Calculus (two simpler algorithms are studied in [27], the related completeness proofs are available in [26, 30]). The usefulness of covers in model checking was already stressed in [32] and further motivated by our recent line of research on the verification of data-aware processes [4, 5, 7, 9]. Notably, it is also operationally mirrored in the MCMT [21] implementation since version 2.8. Covers (via quantifier elimination in model completions and hierarchical reasoning) play an important role in symbol elimination problems in theory extensions, as witnesssed in the comprehensive paper [43] and in related papers [39] studying invariant synthesis in model checking applications.

An important question suggested by the applications is the cover transfer problem for combined theories: for instance, when modeling and verifying data-aware processes, it is natural to consider the combination of different theories, such as the theories accounting for the read-write and read-only data storage of the process as well as those for the elements stored therein [5‐7, 10]. Formally, the cover transfer problem can be stated as follows: by supposing that covers exist in theories \(T_1, T_2\), under which conditions do they exist also in the combined theory \(T_1\cup T_2\)? In this paper we show that the answer is affirmative in the disjoint signatures convex case, using the same hypothesis (that is, the equality interpolating condition) under which quantifier-free interpolation transfers. Thus, for convex theories we essentially obtain a necessary and sufficient condition, in the precise sense captured by Theorem 6 below. We also prove that if convexity does not hold (as it happens, e.g., for integer difference logic \(\mathcal {IDL}\) or for linear integer arithmetics \(\mathcal {LIA} \)), the non-convex equality interpolating property [2] may not be sufficient to ensure the cover transfer property. As a witness for this, we show that \(\mathcal {EUF}\) combined with integer difference logic or with linear integer arithmetics constitutes a counterexample.

The main tool employed in our combination result is the Beth definability theorem for primitive formulae (this theorem has been shown to be equivalent to the equality interpolating condition in [2]). In order to design a combined cover algorithm, we exploit the equivalence between implicit and explicit definability that is supplied by the Beth theorem. Implicit definability is reformulated, via covers for input theories, at the quantifier-free level. Thus, the combined cover algorithm guesses the implicitly definable variables, then eliminates them via explicit definability, and finally uses the component-wise input cover algorithms to eliminate the remaining (non implicitly definable) variables. The identification and the elimination of the implicitly defined variables via explicitly defining terms is an essential step towards the correctness of the combined cover algorithm: when computing a cover of a formula \(\phi ({\underline{x}}, {\underline{y}})\) (w.r.t. \({\underline{y}}\)), the variables \({\underline{x}}\) are (non-eliminable) parameters, and those variables among the \({\underline{y}}\) that are implicitly definable need to be discovered and treated in the same way as the parameters \({\underline{x}}\). Only after this preliminary step (Lemma 6 below), the input cover algorithms can be suitably exploited (Proposition 2 below).

The combination result we obtain is quite strong, as it is a typical ‘black box’ combination result: it applies not only to theories used in verification (like the combination of real arithmetics with \(\mathcal {EUF}\)), but also in other contexts. For instance, since the theory \({\mathcal {B}}\) of Boolean algebras satisfies our hypotheses (being model completable and strongly amalgamable [19]), we get that uniform interpolants exist in the combination of \({\mathcal {B}}\) with \(\mathcal {EUF}\). The latter is the equational theory algebraizing the basic non-normal classical modal logic system \(\mathbf{E}\) from [41] (extended to n-ary modalities). Notice that this result must be contrasted with the case of many systems of Boolean algebras with operators where existence of uniform interpolation fails [34] (recall that operators on a Boolean algebra are not just arbitrary functions, but are required to be monotonic and also to preserve either joins or meets in each coordinate).

As a last important comment on related work, it is worth mentioning that Gulwani and Musuvathi in [32] also have a combined cover algorithm for some convex, signature disjoint theories. Their algorithm looks quite different from ours; apart from the fact that a full correctness and completeness proof for such an algorithm has never been published, we underline that our algorithm is rooted on different hypotheses. In fact, we only need the equality interpolating condition and we show that this hypothesis is not only sufficient, but also necessary for cover transfer in convex theories; consequently, our result is formally stronger. The equality interpolating condition was known to the authors of [32] (but not even mentioned in their paper [32]): in fact, it was introduced by one of them some years before [47]. The equality interpolating condition was then extended to the non convex case in [2], where it was also semantically characterized via the strong amalgamation property.

The paper is organized as follows: after some preliminaries in Section 2, the crucial Covers-by-Extensions Lemma and the relationship between covers and model completions from [6] are recalled in Sect. 3. In Sect. 4, we present some preliminary results from the literature on interpolation, amalgamation, strong amalgamation and Beth definability that are instrumental to our machinery. After some useful facts about convex theories in Sect. 5, we introduce the combined cover algorithms for the convex case and we prove its correctness in Sect. 6; we also present a detailed example of application of the combined algorithm in case of the combination of \(\mathcal {EUF}\) with linear real arithmetic, and we show that the equality interpolating condition is, in a natural sense, necessary for combining covers. In Sect. 7 we exhibit a counterexample to the existence of combined covers in the non-convex case. Finally, in Sect. 8 we prove that for the ‘tame’ multi-sorted theory combinations used in our applications to data-aware processes verification, covers existence transfers to the combined theory under only the stable infiniteness requirement for the shared sorts. Section 9 is devoted to the conclusions and discussion of future work. The current paper is the extended version of [8]; in addition to supplying full self-contained proofs of the results of [8], it contains the entirely new Sect. 8 dedicated to the positive results for the non-convex tame case.

We adopt the usual first-order syntactic notions of signature, term, atom, (ground) formula, and so on; our signatures are always finite or countable and include equality. To avoid considering limit cases, we assume that signatures always contain at least an individual constant. We compactly represent a tuple \(\langle x_1,\ldots ,x_n\rangle \) of variables as \({\underline{x}}\); by abuse of notation, we sometimes use \(\langle x_1,\ldots ,x_n\rangle \) to denote also sets of variables (not just tuples). The notation \(t({\underline{x}}), \phi ({\underline{x}})\) means that the term t, the formula \(\phi \) has free variables included in the tuple \({\underline{x}}\). This tuple is assumed to be formed by distinct variables, thus we underline that when we write e.g. \(\phi ({\underline{x}}, {\underline{y}})\), we mean that the tuples \({\underline{x}}, {\underline{y}}\) are made of distinct variables that are also disjoint from each other.

A formula is said to be universal (resp., existential) if it has the form \(\forall {\underline{x}}(\phi ({\underline{x}}))\) (resp., \(\exists {\underline{x}}(\phi ({\underline{x}}))\)), where \(\phi \) is quantifier-free. Formulae with no free variables are called sentences. On the semantic side, we use the standard notion of \(\varSigma \)-structure \(\mathcal {M}\) and of truth of a formula in a \(\varSigma \)-structure under a free variables assignment. The support of \(\mathcal {M}\) is denoted as \(\vert \mathcal {M}\vert \). The interpretation of a (function, predicate) symbol \(\sigma \) in \(\mathcal {M}\) is denoted \(\sigma ^{\mathcal {M}}\).

A \(\varSigma \)-theory T is a set of \(\varSigma \)-sentences; a model of T is a \(\varSigma \)-structure \(\mathcal {M}\) where all sentences in T are true. We use the standard notation \(T\models \phi \) to say that \(\phi \) is true in all models of T for every assignment to the variables occurring free in \(\phi \). We say that \(\phi \) is T-satisfiable iff there is a model \(\mathcal {M}\) of T and an assignment to the variables occurring free in \(\phi \) making \(\phi \) true in \(\mathcal {M}\).

We now focus on the constraint satisfiability problem and quantifier elimination for a theory T. A \(\varSigma \)-formula \(\phi \) is a \(\varSigma \)-constraint (or just a constraint) iff it is a conjunction of literals. The constraint satisfiability problem for T is the following: we are given a constraint \(\phi ({\underline{x}})\) and we are asked whether there exist a model \(\mathcal {M}\) of T and an assignment \(\mathcal {I}\) to the free variables \({\underline{x}}\) such that \(\mathcal {M}, \mathcal {I}\models \phi ({\underline{x}})\). A theory T has quantifier elimination iff for every formula \(\phi ({\underline{x}})\) in the signature of T there is a quantifier-free formula \(\phi '({\underline{x}})\) such that \(T\models \phi ({\underline{x}})\leftrightarrow \phi '({\underline{x}})\). Since we are in a computational logic context, when we speak of quantifier elimination, we assume that it is effective, namely that it comes with an algorithm for computing \(\phi '\) out of \(\phi \). It is well-known that quantifier elimination holds in case we can eliminate quantifiers from primitive formulae, i.e., formulae of the kind \(\exists {\underline{y}}\,\phi ({\underline{x}}, {\underline{y}})\), with \(\phi \) a constraint.

We recall also some further basic notions. Let \(\varSigma \) be a first-order signature. The signature obtained from \(\varSigma \) by adding to it a set \({\underline{a}}\) of new constants (i.e., 0-ary function symbols) is denoted by \(\varSigma ^{{\underline{a}}}\). Analogously, given a \(\varSigma \)-structure \(\mathcal {M}\), the signature \(\varSigma \) can be expanded to a new signature \(\varSigma ^{|\mathcal {M}|}:=\varSigma \cup \{\bar{a}\ |\ a\in |\mathcal {M}| \}\) by adding a constant \(\bar{a}\) (the name for a) for each element a in the support of \(\mathcal {M}\), with the convention that two distinct elements are denoted by different “name” constants. \(\mathcal {M}\) can be expanded to a \(\varSigma ^{|\mathcal {M}|}\)-structure \({\overline{\mathcal {M}}}:=(\mathcal {M}, a)_{a\in \vert \mathcal {M}\vert }\) just interpreting the additional constants over the corresponding elements. From now on, when the meaning is clear from the context, we will freely use the notation \(\mathcal {M}\) and \({\overline{\mathcal {M}}}\) interchangeably: in particular, given a \(\varSigma \)-structure \(\mathcal {M}\) and a \(\varSigma \)-formula \(\phi ({\underline{x}})\) with free variables that are all in \({\underline{x}}\), we will write, by abuse of notation, \(\mathcal {M}\models \phi ({\underline{a}})\) instead of \({\overline{\mathcal {M}}}\models \phi ({\bar{{\underline{a}}}})\).

We say that a theory T is stably infinite iff every T-satisfiable constraint is satisfiable in an infinite model of T. Moreover, a theory T is convex iff for every constraint \(\delta \), if \(T\vdash \delta \rightarrow \bigvee _{i=1}^n x_i=y_i\) then \(T\vdash \delta \rightarrow x_i=y_i\) holds for some \(i\in \{1,\ldots , n\}\). Strictly speaking, convexity says that if, for a set of literals \(\phi \) and for a non empty disjunction of variables \(\bigvee _{i=1}^n x_i=y_i\), we have \(T\models \phi \rightarrow \bigvee _{i=1}^n x_i=y_i\), then we have also \(T\models \phi \rightarrow x_i=y_i\) for some \(i=1, \dots ,n\). If, instead of variables, we have terms, the same property nevertheless applies: if we have \(T\models \phi \rightarrow \bigvee _{i=1}^n t_i=u_i\), then for fresh variables \(x_i, y_i\) we get \(T\models \phi \wedge \bigwedge _{i=1}^n (x_i= t_i \wedge y_i=u_i)\rightarrow \bigvee _{i=1}^n x_i=y_i\), which implies, by applying the definition of convexity, the same property for terms.

A \(\varSigma \)-homomorphism (or, simply, a homomorphism) between two \(\varSigma \)-structures \(\mathcal {M}\) and \(\mathcal {N}\) is a map \(\mu : \vert \mathcal {M}\vert \longrightarrow \vert \mathcal {N}\vert \) among the support sets \(\vert \mathcal {M}\vert \) of \(\mathcal {M}\) and \(\vert \mathcal {N}\vert \) of \(\mathcal {N}\) satisfying the condition \((\mathcal {M}\models \varphi \quad \Rightarrow \quad \mathcal {N}\models \varphi )\) for all \(\varSigma ^{\vert \mathcal {M}\vert }\)-atoms \(\varphi \) (\(\mathcal {M}\) is regarded as a \(\varSigma ^{\vert \mathcal {M}\vert }\)-structure, by interpreting each additional constant \(a\in \vert \mathcal {M}\vert \) into itself and \(\mathcal {N}\) is regarded as a \(\varSigma ^{\vert \mathcal {M}\vert }\)-structure by interpreting each additional constant \(a\in \vert \mathcal {M}\vert \) into \(\mu (a)\)). In case the last condition holds for all \(\varSigma ^{|\mathcal {M}|}\)-literals, the homomorphism \(\mu \) is said to be an embedding and if it holds for all first order formulae, the embedding \(\mu \) is said to be elementary.

If \(\mu : \mathcal {M}\longrightarrow \mathcal {N}\) is an embedding which is just the identity inclusion \(\vert \mathcal {M}\vert \subseteq \vert \mathcal {N}\vert \), we say that \(\mathcal {M}\) is a substructure of \(\mathcal {N}\) or that \(\mathcal {N}\) is an extension of \(\mathcal {M}\). Universal theories can be characterized as those theories T having the property that if \(\mathcal {N}\models T\) and \(\mathcal {M}\) is a substructure of \(\mathcal {N}\), then \(\mathcal {M}\models T\) (see [11]). If \(\mathcal {M}\) is a structure and \(X\subseteq \vert \mathcal {M}\vert \), then there is the smallest substructure of \(\mathcal {M}\) including X in its support; this is called the substructure generated by X. If X is the set of elements of a finite tuple \({\underline{a}}\), then the substructure generated by X has in its support precisely the \(b\in \vert \mathcal {M}\vert \) such that \(\mathcal {M}\models b=t({\underline{a}})\) for some term t.

Let \(\mathcal {M}\) be a \(\varSigma \)-structure. The diagram of \(\mathcal {M}\), written \(\varDelta _{\varSigma }(\mathcal {M})\) (or just \(\varDelta (\mathcal {M})\)), is the set of ground \(\varSigma ^{|\mathcal {M}|}\)-literals that are true in \(\mathcal {M}\). An easy but important result, called Robinson Diagram Lemma [11], says that, given any \(\varSigma \)-structure \(\mathcal {N}\), the embeddings \(\mu : \mathcal {M}\longrightarrow \mathcal {N}\) are in bijective correspondence with expansions of \(\mathcal {N}\) to \(\varSigma ^{\vert \mathcal {M}\vert }\)-structures which are models of \(\varDelta _{\varSigma }(\mathcal {M})\). The expansions and the embeddings are related in the obvious way: \({\bar{a}}\) is interpreted as \(\mu (a)\).

We report the notion of a cover taken from [32] and also the basic results proved in [6, 10]. Fix a theory T and an existential formula \(\exists {\underline{e}}\, \phi ({\underline{e}}, {\underline{y}})\); call a residue of \(\exists {\underline{e}}\, \phi ({\underline{e}}, {\underline{y}})\) any quantifier-free formula belonging to the set of quantifier-free formulae(the above two sets are trivially equal, by applying the \(\exists \)-left introduction rule). A quantifier-free formula \(\psi ({\underline{y}})\) is said to be a T-cover (or, simply, a cover) of \(\exists {\underline{e}}\, \phi ({\underline{e}},{\underline{y}})\) iff \(\psi ({\underline{y}})\in Res(\exists {\underline{e}}\, \phi )\) and \(\psi ({\underline{y}})\) implies (modulo T) all the other formulae in \(Res(\exists {\underline{e}}\, \phi )\). The following “cover-by-extensions” Lemma [6] (to be widely used throughout the paper) supplies a semantic counterpart to the notion of a cover:

We underline that, since our language is at most countable, we can assume that the models \(\mathcal {M}\), \(\mathcal {N}\) from (ii) above are at most countable too, by a Löwenheim-Skolem argument.

We say that a theory T has uniform quantifier-free interpolation iff every existential formula \(\exists {\underline{e}}\, \phi ({\underline{e}},{\underline{y}})\) (equivalently, every primitive formula \(\exists {\underline{e}}\, \phi ({\underline{e}},{\underline{y}})\)) has a T-cover. Notice that a cover is also called (quantifier-free) uniform interpolant for the following reason. Indeed, it is clear that if T has uniform quantifier-free interpolation, then it has ordinary quantifier-free interpolation [2], in the sense that if we have \(T\models \phi ({\underline{e}}, {\underline{y}})\rightarrow \phi '({\underline{y}}, {\underline{z}})\) (for quantifier-free formulae \(\phi , \phi '\)), then there is a quantifier-free formula \(\theta ({\underline{y}})\) such that \(T\models \phi ({\underline{e}}, {\underline{y}})\rightarrow \theta ({\underline{y}})\) and \(T\models \theta ({\underline{y}})\rightarrow \phi '({\underline{y}}, {\underline{z}})\). In fact, if T has uniform quantifier-free interpolation, then the interpolant \(\theta \) is independent on \(\phi '\) (the same \(\theta ({\underline{y}})\) can be used as interpolant for all entailments \(T\models \phi ({\underline{e}}, {\underline{y}})\rightarrow \phi '({\underline{y}}, {\underline{z}})\), varying \(\phi '\)). Hence, it is straightforward to see that the definition of cover is equivalent to the one of uniform interpolant given in the introduction.

We say that a universal theory T has a model completion iff there is a stronger theory \(T^*\supseteq T\) (still within the same signature \(\varSigma \) of T) such that: Other equivalent definitions are possible [11]: for instance, (i) is equivalent to the fact that T and \(T^*\) prove the same universal formulae or again to the fact that every model of T can be embedded into a model of \(T^*\). We recall that the model completion, if it exists, is unique and that its existence implies the quantifier-free interpolation property for T [11] (the latter can be seen directly or via the correspondence between quantifier-free interpolation and amalgamability, see [2]).

A close relationship between model completion and uniform interpolation emerged in the area of propositional logic (see the book [25]) and can be formulated roughly as follows. It is well-known that most propositional calculi, via Lindenbaum constructions, can be algebraized: the algebraic analogue of classical logic are Boolean algebras, the algebraic analogue of intuitionistic logic are Heyting algebras, the algebraic analogue of modal calculi are suitable varieties of modal algebras, etc. Under suitable hypotheses, it turns out that a propositional logic has uniform interpolation (for the global consequence relation) iff the equational theory axiomatizing the corresponding variety of algebras has a model completion [25]. In the context of first order theories, we prove an even more direct connection:

We report here some definitions and results we need concerning combined quantifier-free interpolation. Most definitions and results come from [2], but are simplified here

because we restrict them to the case of universal convex theories.

We recall that a theory T is stably infinite iff every T-satisfiable constraint is satisfiable in an infinite model of T. The following lemma comes from a compactness argument:

We also recall that theory T is convex iff for every constraint \(\delta \), if \(T\vdash \delta \rightarrow \bigvee _{i=1}^n x_i=y_i\) then \(T\vdash \delta \rightarrow x_i=y_i\) holds for some \(i\in \{1,\ldots , n\}\).

A convex theory T is ‘almost’ stably infinite in the sense that it can be shown that every constraint which is T-satisfiable in a T-model whose support has at least two elements is satisfiable also in an infinite T-model. The one-element model can be used to build counterexamples, though: e.g., the theory of Boolean algebras is convex (like any other universal Horn theory) but the constraint \(x=0 \wedge x=1\) is only satisfiable in the degenerate one-element Boolean algebra. Since we take into account these limit cases, we do not assume that convexity implies stable infiniteness.

Quantifier-free interpolation and combined quantifier-free interpolation can be semantically characterized, as we are going to show.

We underline that Theorem 2 extends also to the non convex case provided the notion of an equality interpolating theory is suitably adjusted [2].

Next two results (supplied without proof) will be used only in Sect. 6.1 to show that, in some sense, the sufficient conditions of our main combination Theorem 5 are also necessary.

The previous theorem essentially states that the equality interpolating property is a sufficient condition for the transfer of quantifier-free interpolation to theory combinations. There is a converse of the previous result, in the sense that it is possible to show that the equality interpolating property is, to some extent, necessary in order to guarantee the transfer of quantifier-free interpolation for minimal combinations with signatures adding only uninterpreted symbols. For this purpose, for a signature \(\varSigma \), we call \(\mathcal {EUF}(\varSigma )\) the pure equality theory over the signature \(\varSigma \) (this theory is equality interpolating and has the quantifier-free interpolation property).

In [2] the above definitions and results are extended to the non-convex case and a long list of universal quantifier-free interpolating and equality interpolating theories is given. The list includes \(\mathcal {EUF}(\varSigma )\), recursive data theories, as well as linear arithmetics. For linear arithmetics (and fragments of its), it is essential to make a very careful choice of the signature, see again [2] (especially Subsection 4.1) for details. All the above theories admit a model completion (which coincides with the theory itself in case the theory admits quantifier elimination).

The equality interpolating property in a theory T can be equivalently characterized using Beth definability as follows. Consider a primitive formula \(\exists {\underline{z}}\phi ({\underline{x}}, {\underline{z}},y)\) (here \(\phi \) is a conjunction of literals); we say that \(\exists {\underline{z}}\, \phi ({\underline{x}}, {\underline{z}},y)\) implicitly defines y in T iff the formulais T-valid. We say that \(\exists {\underline{z}}\phi ({\underline{x}}, {\underline{z}},y)\) explicitly defines y in T iff there is a term \(t({\underline{x}})\) such that the formulais T-valid.

For future use, we notice that, by trivial logical manipulations, the formulae (4) and (5) are logically equivalent toand torespectively (we shall use such equivalences without explicit mention).

We say that a theory T has the Beth definability property for primitive formulae iff whenever a primitive formula \(\exists {\underline{z}}\, \phi ({\underline{x}}, {\underline{z}},y)\) implicitly defines the variable y then it also explicitly defines it.

We remark that the above Proposition can be inverted (see [2]).

We now collect some useful facts concerning convex theories. We fix for this section a convex, stably infinite, equality interpolating universal theory T admitting a model completion \(T^*\). We let \(\varSigma \) be the signature of T. We fix also a \(\varSigma \)-constraint \(\phi ({\underline{x}}, {\underline{y}})\), where we assume that \({\underline{y}}=y_1, \dots , y_n\) (recall that the tuple \({\underline{x}}\) is disjoint from the tuple \({\underline{y}}\) according to our conventions from Sect. 2).

For \(i=1, \dots , n\), we let the formula \(\mathtt {ImplDef}_{\phi ,y_i}^T({\underline{x}})\) be the quantifier-free formula equivalent in \(T^*\) to the formulawhere the \({\underline{y}}'\) are renamed copies of the \({\underline{y}}\). Notice that the variables occurring free in \(\phi \) are \({\underline{x}}, {\underline{y}}\), whereas only the \({\underline{x}}\) occur free in \(\mathtt {ImplDef}_{\phi ,y_i}^T({\underline{x}})\) (the variable \(y_i\) is among the \({\underline{y}}\) and does not occur free in \(\mathtt {ImplDef}_{\phi ,y_i}^T({\underline{x}})\)): these facts coming from our notational conventions are crucial and should be kept in mind when reading this and next section. We need a first semantic technical lemma.

The following lemma supplies terms which will be used as ingredients in our combined covers algorithm:

In all our concrete examples, the theory T has a decidable quantifier-free fragment (namely it is decidable whether a quantifier-free formula is a logical consequence of T or not), thus the terms \(t_{ij}\) mentioned in Lemma 4 can be computed just by enumerating all possible \(\varSigma ({\underline{x}})\)-terms: the computation terminates, because the above proof shows that the appropriate terms always exist. However, this is terribly inefficient and, from a practical point of view, one needs to have at disposal dedicated algorithms to find the required equality interpolating terms. For some common theories (\(\mathcal {EUF}\), Lisp-structures, linear real arithmetic), such algorithms are designed in [47]; in [2] [Lemma 4.3 and Theorem 4.4], the algorithms for computing equality interpolating terms are connected to quantifier elimination algorithms in the case of universal theories admitting quantifier elimination.

The following lemma will be useful in the next section:

Let us now fix two theories \(T_1, T_2\) over disjoint signatures \(\varSigma _1, \varSigma _2\).

We assume that both of them satisfy the assumptions from the previous section, meaning that they are convex, stably infinite, equality interpolating, universal and admit model completions \(T^*_1, T^*_2\) respectively. We will prove in this section (Theorem 5) that \(T_1\cup T_2\) admits a model completion too. We achieve this by supplying a combined algorithm, called \(\mathsf {ConvexCombCover}\), for computing \(T_1\cup T_2\)-covers: in order to construct the \(T_1\cup T_2\)-cover, this combined algorithm exploits the cover algorithms of the component theories \(T_i\) (\(i=1,2\)).

We need to compute a cover for \(\exists {\underline{e}}\,\phi ({\underline{x}}, {\underline{e}})\), where \(\phi \) is a conjunction of \(\varSigma _1\cup \varSigma _2\)-literals. By applying rewriting purification steps like(where d is a fresh variable and t is a pure term, i.e. it is either a \(\varSigma _1\)- or a \(\varSigma _2\)-term), we can assume that our formula \(\phi \) is of the kind \(\phi _1\wedge \phi _2\), where \(\phi _1\) is a \(\varSigma _1\)-formula and \(\phi _2\) is a \(\varSigma _2\)-formula. Thus we need to compute a cover for a formula of the kindwhere \(\phi _i\) is a conjunction of \(\varSigma _i\)-literals (\(i=1,2\)). By guessing a partition of the \({\underline{e}}\) and by replacing each variable e in \({\underline{e}}\) with the representative element of its equivalence class, we also assume that both \(\phi _1\) and \(\phi _2\) contain the literals \(e_i\ne e_j\) (for \(i\ne j\)) as a conjunct.

To manipulate formulae, our algorithm employs acyclic explicit definitions as follows. When we write \(\mathtt {ExplDef}({\underline{z}}, {\underline{x}})\) (where \({\underline{z}}, {\underline{x}}\) are tuples of distinct variables), we mean any formula of the kind (let \({\underline{z}}:=z_1 \dots , z_m\))where the term \(t_i\) is pure (i.e. it is a \(\varSigma _i\)-term) and only the variables \(z_1, \dots , z_{i-1}, {\underline{x}}\) can occur in it. We notice that an existential formula like \(\exists {\underline{z}}\;(\mathtt {ExplDef}({\underline{z}}, {\underline{x}})\wedge \psi ({\underline{z}}, {\underline{x}}))\) can be equivalently converted into a quantifier-free formula: indeed, since the ’explicit definitions’ \(z_i =t_i\) are in fact arranged acyclically, the existentially quantified variables \({\underline{z}}\) can be recursively eliminated by substituting them with terms containing eventually only the parameters \({\underline{x}}\).

A working formula is a formula of the kindwhere \(\psi _1\) is a conjunction of \(\varSigma _1\)-literals and \(\psi _2\) is a conjunction of \(\varSigma _2\)-literals. The variables \({\underline{x}}\) are called parameters, the variables \({\underline{z}}\) are called defined variables and the variables \({\underline{e}}\) (truly) existential variables. The parameters do not change during the execution of the algorithm. We assume that \(\psi _1, \psi _2\) in a working formula (14) always contain the literals \(e_i\ne e_j\) (for distinct \(e_i, e_j\) from \({\underline{e}}\)) as a conjunct.

In our starting formula (13), there are no defined variables. However, if via some syntactic check it happens that some of the existential variables can be recognized as defined, then it is useful to display them as such (this observation may avoid redundant cases - leading to inconsistent disjuncts - in the computations below).

A working formula like (14) is said to be terminal iff for every existential variable \(e_i\in {\underline{e}}\) we have thatRoughly speaking, we can say that in a terminal working formula, all variables which are not parameters are either explicitly definable or recognized as not implicitly definable by both theories; of course, a working formula with no existential variables is terminal.

Thus we are left to the problem of computing a cover of a terminal working formula; this problem is solved in the following proposition:

From Lemma 6, Proposition 2 and Theorem 1, we immediately get

We recall from Theorem 3 that the equality interpolating property transfers to combination of theories too, when it holds in the component theories.

We now summarize the steps of the combined cover algorithm \(\mathsf {ConvexCombCover}\) that takes as input the primitive formula \(\exists {\underline{e}}\,\phi ({\underline{x}}, {\underline{e}})\), where \(\phi \) is a conjunction of \(\varSigma _1\cup \varSigma _2\)-literals:

Notice that the input cover algorithms in the above combined cover computation algorithm are used not only in the final step described in Proposition 2, but also every time we need to compute a formula \(\mathtt {ImplDef}_{\psi _h,e_i}^{T_h}({\underline{x}},{\underline{z}})\): according to its definition, this formula is obtained by eliminating quantifiers in \(T_i^*\) from (8) (this is done via a cover computation, reading \(\forall \) as \(\lnot \exists \lnot \)). In practice, implicit definability is not very frequent, so that in many concrete cases \(\mathtt {ImplDef}_{\psi _h,e_i}^{T_h}({\underline{x}},{\underline{z}})\) is trivially equivalent to \(\bot \) (in such cases, Step (2.i) above can obviously be disregarded).

In this section, we show by giving a suitable counterexample that the convexity hypothesis cannot be dropped from Theorems 5, 6. We make use of basic facts about ultrapowers (see [11] for the essential information we need). We take as \(T_1\) integer difference logic \(\mathcal {IDL}\), i.e. the theory of integer numbers under the unary operations of successor and predecessor, the constant 0 and the strict order relation <. This is stably infinite, universal and has quantifier elimination (thus it coincides with its own model completion). It is not convex, but it satisfies the equality interpolating condition, once the latter is suitably adjusted to non-convex theories, see [2] for the related definition and all the above mentioned facts.

As \(T_2\), we take \(\mathcal {EUF}(\varSigma _{f})\), where \(\varSigma _f\) has just one unary free function symbol f (this f is supposed not to belong to the signature of \(T_1\)).

The counterexample still applies when replacing integer difference logic with linear integer arithmetics (the proof is literally the same).

So far, we only analyzed the mono-sorted case. However, many interesting examples arising in model-checking verification are multi-sorted: this is the case of array-based systems [20] and in particular of the array-based system used in data-aware processes verification [5, 9]. The above examples suggest restrictions on the theories to be combined other than convexity, in particular they suggest restrictions that make sense in a multi-sorted context.

Most definitions we gave in Sect. 2 have straightforward natural extensions to the multi-sorted case (we leave the reader to formulate them). A little care is needed however for the disjoint signatures requirement. Let \(T_1, T_2\) be multisorted theories in the signatures \(\varSigma _1, \varSigma _2\); the disjointness requirement for \(\varSigma _1\) and \(\varSigma _2\) can be formulated in this context by saying that the only function or relation symbols in \(\varSigma _1\cap \varSigma _2\) are the equality predicates over the common sorts in \(\varSigma _1\cap \varSigma _2\). We want to strengthen this requirement: we say that the combination \(T_1\cup T_2\) is tame iff the sorts in \(\varSigma _1\cap \varSigma _2\) cannot be a domain sort of a symbol from \(\varSigma _1\) other than an equality predicate. In other words, if a relation or a function symbol has as among its domain sorts a sort from \(\varSigma _1\cap \varSigma _2\), then this symbol is from \(\varSigma _2\) (and not from \(\varSigma _1\), unless it is the equality predicate).

Tame combinations arise in infinite-state model-checking (in fact, the definition is suggested by this application domain), where signatures can be split into a signature \(\varSigma _2\) used to represent ‘datatypes’ like integers and a signature \(\varSigma _1\) for representing elements contained in a database: this is customary in the literature on data-aware processes verification [5, 9].

Notice that the notion of a tame combination is not symmetric in \(T_1\) and \(T_2\): to see this, notice that if the sorts of \(\varSigma _1\) are included in the sorts of \(\varSigma _2\), then \(T_1\) must be a pure equality theory (but this is not the case if we swap \(T_1\) with \(T_2\)). The combination of \(\mathcal {IDL}\) and \(\mathcal {EUF}(\varSigma )\) used in the counterexample of Sect. 7 is not tame: even if we formulate \(\mathcal {EUF}(\varSigma )\) as a two-sorted theory, the unique sort of \(\mathcal {IDL}\) must be a sort of \(\mathcal {EUF}(\varSigma )\) too, as witnessed by the impure atom \(f(e)=0\) in the formula (25). Because of this, for the combination to be tame, \(\mathcal {IDL}\) should play the role of \(T_2\) (the arithmetic operation symbols are defined on a shared sort); however, the unary function symbol \(f\in \varSigma \) has a shared sort as domain sort, so the combination is not tame anyway.

In a tame combination, an atomic formula A can only be of two kinds: (1) we say that A is of the first kind iff the sorts of its root predicate are from \(\varSigma _1\setminus \varSigma _2\); (2) we say that A is of the second kind iff the sorts of its root predicate are from \(\varSigma _2\). We use the roman letters \(e, x,\dots \) for variables ranging over sorts in \(\varSigma _1\setminus \varSigma _2\) and the greek letters \(\eta , \xi , \dots \) for variables ranging over sorts in \(\varSigma _2\). Thus, if we want to display free variables, atoms of the first kind can be represented as \(A(e,x, \dots )\), whereas atoms of the second kind can be represented as \(A(\eta , \xi , \dots , t(e, x, \dots ), \dots )\), where the t are \(\varSigma _1\)-terms. In the following, given two tuples of \(\varSigma _i\)-terms \(\underline{\alpha }:=\langle \alpha _1,\ldots ,\alpha _n\rangle \) and \(\underline{\beta }:=\langle \beta _1,\ldots ,\beta _n\rangle \) (for some \(i=1,2\)), we use the notation \(\underline{\alpha }=\underline{\beta }\) for denoting the conjunction of equalities \(\bigwedge _j \alpha _j = \beta _j\).

Suppose that \(T_1\cup T_2\) is a tame combination and that \(T_1, T_2\) are universal theories admitting model completions \(T_1^*, T_2^*\). We propose the following algorithm, called \(\mathsf {TameCombCover}\), to compute the cover of a primitive formula; this formula must be of the kindwhere \(\phi \) is a \(\varSigma _1\)-conjunction of literals, \(\psi \) is a conjunction of \(\varSigma _2\)-literals and the \({\underline{t}}\) are \(\varSigma _1\)-terms.

The \(\mathsf {TameCombCover}\) algorithm has three steps: We now show that the above algorithm is correct under very mild hypotheses. We need some technical facts about stably infinite theories in a multi-sorted context. We say that a multi-sorted theory T is stably infinite with respect to a set of sorts \(\mathcal {S}\) from its signature iff every T-satisfiable constraint is satisfiable in a model \(\mathcal {M}\) where, for every \(S\in \mathcal {S}\), the set \(S^\mathcal {M}\) (namely the interpretation of the sort S in \(\mathcal {M}\)) is infinite. The next Lemma is a light generalization of Lemma 2 and is proved in the same way:

We are now ready for the main result of this section:

We conclude this subsection discussing the applications that inspired tame combinations. In the context of data-aware processes verification [4, 5, 9], where relational databases can be extended with arithmetical values such as integers and reals, tame combinations become particularly interesting. Consider the combination \(T_{DB}\cup T_{int}\), where: It can be trivially seen that this combination is tame.

As explained in [9], \((\varSigma , T_{DB})\) can be thought of as a DB schema, i.e. as the formalization of a classical relational database with primary and foreign keys: for instance, from unary functions \(f_{R,1}\) and \(f_{R,2}\), one can reconstruct the corresponding database relation \(R(A_1,A_2,A_3)\), where each attribute \(A_i\) has type \(\mathcal {S}_i\) (for \(i=1,\ldots ,3\)) and \(A_1\) is the primary key of R. The interested reader is referred to [9, 31] for details on this. In addition, \(\mathcal {S}_3\), which is interpreted into a model of \(T_{int}\), can be used to formalize a value domain (using again the nomenclature of [9]), i.e., an infinite arithmetic domain whose elements are constrained by \(T_{int}\): in this sense, these elements can be thought of as (possibly infinitely many and fresh) values that can be injected into the database, e.g., by an external user (they are essential for applications in data-aware process verification). For details on this and its use in formal verification, see [31].

In this paper we showed that covers (aka uniform interpolants) exist in the combination of two convex universal theories over disjoint signatures in case they exist in the component theories and in case the component theories also satisfy the equality interpolating condition. Notice that the last condition is needed to transfer to combinations the existence of (ordinary) quantifier-free interpolants. In order to prove our result on combined covers, Beth definability property for primitive fragments turned out to be the crucial ingredient to extensively employ. In case convexity fails, we showed by a counterexample that covers might not exist in the combined theory. The last result raises the following research problem: even if in general covers do not exist for the combination of non-convex theories, under which conditions can one decide whether covers exist and, if so, how can one compute them?

Another interesting research question concerns complexity of the convex combined algorithm. It generates a tree whose depth is linear, hence the number of created nodes are in the worst case exponential. In order to generate new nodes, the algorithm makes use of the cover algorithms for the component theories and of the algorithms for generating the equality interpolating terms: these algorithms are given as input to our algorithm. Taking into consideration also the fact that these algorithms are used recursively, it is not immediate to give a significant upper bound to the overall complexity in the general case: instead, notice that this problem strongly depends on the component theories considered, hence it should be tackled separately for each involved theory and in view of the specific, concrete applications that the users have in mind. For these reasons, we leave an exhaustive investigation of this to future work, since it would require genuinely novel research and a thorough analysis of different examples of theories.

Applications suggested a different line of investigations, which led us to consider so-called ‘tame combinations’. In data-aware processes verification [4, 5, 9] one uses tame combinations \(T_1\cup T_2\), where \(T_1\) is a multi-sorted version of \(\mathcal {EUF}(\varSigma )\) in a signature \(\varSigma \) containing only unary function symbols and relation symbols of any arity, and where \(T_2\) is typically some fragment of linear arithmetics (\(T_2\)-sorts are called value sorts in the terminology of [4, 5, 9]). In this context, quantifier elimination in \(T_1^*\) for primitive formulae is quadratic in complexity. Model-checkers like MCMT represent sets of reachable states by using conjunctions of literals and during preimage computations quantifier elimination needs to be applied to primitive formulae. Now, if all relation symbols are at most binary, such a quantifier elimination in \(T_1^*\) produces conjunctions of literals out of primitive formulae. Thus, step (ii) in the algorithm from Sect. 8 becomes deterministic and the only reason why such an algorithm may become expensive (i.e., non polynomial) lies in the final quantifier elimination step for \(T_2^*\). This step might be extremely expensive if substantial arithmetic is involved, but it might still be efficiently handled in practical cases where only very limited arithmetic is used (e.g., difference bound constraints like \(x-y\le n\) or \(x\le n\), where n is a constant). Our algorithm for covers in tame combinations has been implemented in version 3.0 of MCMT.

We also feel that this algorithm can be really useful in various model-checking applications. More specifically, such a model checking framework can be applied along the recent line of research concerning analysis of data-aware processes, in which data representation and manipulation capabilities can be extended with arithmetic. Like that, one could adapt the results of this paper to the existing formalism for data-aware extensions of the de-facto standard for business process modeling [4] or to data-aware classes of Petri nets [14, 15, 28, 29]. We leave it for future work.

A final future research line could consider cover transfer properties to non-disjoint signatures combinations, analogously to similar results obtained in [18, 19] for the transfer of quantifier-free interpolation.