A Prolog Implementation of Kem

In this paper, we describe a Prolog implementation of a new theorem prover for (normal propositional) modal and multi–modal logics. The theorem prover, which is called KEM , arises from the combination of a classical refutation system which incorporates a restricted (“analytic”) version of the cut rule with a label formalism which allows for a specialised, logic–dependent unification algorithm. An essential feature of KEM is that it yields a rather simple and efficient proof search procedure which offers many computational advantages over the usual tableau–based proof search methods. This is due partly to the use of linear 2–premise β rules in place of the branching β rules of the standard tableau method, and partly to the crucial role played by the analytic cut (the only branching rule) in eliminating redundancy from the search space. It turns out that KEM method of proof search is not only computationally more efficient but also intuitively more natural than other (e.g. resolution–based) methods leading to simple and easily implementable procedures (two KEM Theorem Prover–like systems have been implemented: an LPA interpreter on Macintosh, and a Quintus compiler on Sun-Sparcstation) which make it well suited for efficient automated proof search in modal logics. 1 An overview of KEM KEM [AG94, Gov95] is a tableau–like modal proof system based on D’Agostino and Mondadori’s [DM94] classical refutation system KE. The basic feature of KEM is that it uses KE rules in combination with a label unification scheme constituted of (1) a label formalism, and (2) a specialised, logic–dependent unification algorithm. The label formalism arises from two (non empty) sets ΦC = {w1, w2, · · ·} andΦV = {W1,W2, · · ·} respectively of constant and variable world– simbols through the following definition: a world–label is either (i) an element of the set ΦC , or (ii) an element of the set ΦV , or (iii) a path term (k′, k) where (iiia) k′ ∈ ΦC ∪ ΦV and (iiib) k ∈ ΦC or k = (m′,m) where (m′,m) is a label. Intuitively, we may think of a label i ∈ ΦC as denoting a (given) world, and a label i ∈ ΦV as denoting a set or worlds (any world) in some Kripke model. A label i = (k′, k) may be viewed as representing a path from k to a (set of) world(s) k′ accessible from k (according to the appropriate accessibility relation. For any label i = (k′, k) we shall call k′ the head of i, k the body of i, and denote them by h(i) and b(i) respectively. Notice that these notions are recursive: if b(i) denotes the body of i, then b(b(i)) will denote the body of b(i), b(b(b(i))) will denote the body of b(b(i)); and so on. We shall call each of b(i),b(b(i)), etc., a segment of i. Let s(i) denote any segment of i (obviously, by definition every segment s(i) of a label i is a label); then h(s(i)) will denote the head of s(i). For any label i, we shall define the length of i, l(i), as the number of world–symbols in i (obviously l(s(i)) will denote the lenght of s(i)). We shall call a label i restricted if h(i) ∈ ΦC , otherwise we shall call it unrestricted. KEM ’s label unification scheme involves two kinds of unifications, respectively “high” and “low” unifications. “High” unifications are meant to mirror specific accessibility constraints. They are used to build “low” unifications which account for the full range of conditions governing the appropriate accessibility relation. Let = denote the set of labels. A substitution is defined in the usual way as a function ΦV −→ =− where =− = = − ΦV . For two labels i, k and a substitution σ, if σ is a unifier of i and k, then we shall say that i and k are σ-unifiable. We shall (somewhat unconventionally) use (i, k)σ to denote both that i and k are σ-unifiable and the result of their unification. On this basis we can define several specialised, logic–dependent notions of σ “high”(or σ-) unification. In particular, the notion of two labels i, k being σ-, σ-, and σ -unifiable is defined in the following way: (i, k)σK = (i, k)σ ⇐⇒ (i) at least one of i and k is restricted, and (ii) for every s(i), s(k), l(s(i)) = l(s(k)), (s(i), s(k))σK