MTT: The Maude Termination Tool (System Description)

Despite the remarkable development of the theory of termination of rewriting, its application to high-level programming languages is far from being optimal. This is due to the need for features such as conditional equations and rules, types and subtypes, (possibly programmable) strategies for controlling the execution, matching modulo axioms, and so on, that are used in many programs and tend to place such programs outside the scope of current termination tools. The operational meaning of such features is often formalized in a proof-theoretic manner by means of an inference system (see, e.g., [2, 3, 17]) rather than just by a rewriting relation. In particular, Generalized Rewrite Theories (GRT) [3] are a recent generalization of rewrite theories at the heart of the most recent formulation of Maude [4]. The corresponding termination notions can also differ from the standard ones, see [14]. For these reasons, although there is a good number of tools which can be used to automatically prove termination of rewriting (e.g., [9], [13], . . . ) they cannot directly prove termination of, e.g., Elan [1], Maude [4] or CafeOBJ [8] programs. As an illustrative example, consider the Maude module MARKS-LISTS in the MTT snapshot in Figure 1. Given a list representation of a multiset of natural numbers it (nondeterministically) computes its submultisets of size 2. A mark ‘#’ is introduced into a given List of numbers (of sort Nat) to yield a marked list of sort MList (supersort of List). The matching condition < N1 ; N2 ; N3 ; L’ > := < L > in the conditional rule ensures that ‘#’ is introduced into lists of at least three elements. Symbol # is intended to mark a number to be removed by using the third rule (thus producing a sublist of the original one). The mark is propagated inside the structure of the list until it is finally removed (together with its companion number) to produce a list of sort List on which we can restart the process. Objects from both List and MList can be built by using a single overloaded constructor _;_. Modeling MARKS-LISTS as a Conditional Term Rewriting System (CTRS, see [18]) by translating the matching condition into a rewriting condition as: