Lazy productivity via termination

Abstract

a b s t r a c tWe present a procedure for transforming strongly sequential constructor-based term rewriting systems (TRSs) into context-sensitive TRSs in such a way that productivity of the input system is equivalent to termination of the output system. Thereby automated termination provers become available for proving productivity. A TRS is called productive if all its finite ground terms are constructor normalizing, and all 'inductive constructor paths' through the resulting (possibly non-wellfounded) constructor normal form are finite. To our knowledge, this is the first complete transformation from productivity to termination.The transformation proceeds in two steps: (i) The strongly sequential TRS is converted into a shallow TRS, where patterns do not have nested constructors. (ii) The shallow TRS is transformed into a context-sensitive TRS, where rewriting below constructors and in arguments not 'consumed from' is disallowed.Furthermore, we show how lazy evaluation can be encoded by strong sequentiality, thus extending our transformation to, e.g., Haskell programs.Finally, we present a simple, but fruitful extension of matrix interpretations to make them applicable for proving termination of context-sensitive TRSs.

Productivity of Non-Orthogonal Term Rewrite Systems

“…Note that for this example, one could also have used µ(id) = {1}, i.e., here the removal of argument positions in the second item of Definition 11 is irrelevant. This is not the case in the next example, showing that this improvement, which was inspired by [5] and blocks more argument positions, allows to prove productivity of specifications where this would otherwise not be possible.…”

“…Intuitively, this holds because structure arguments of constructors are blocked, and if every right-hand side of R s starts with a constructor then the number of allowed redexes of sort s in a term steadily decreases. 2 Note that in [5], Endrullis and Hendriks consider orthogonal TRSs and also block arguments of symbols in Σ d which only contain variables. This however is problematic when allowing data rules that are not left-linear.…”

Termination of canonical context-sensitive rewriting and productivity of rewrite systems

“…Our result, though, does not provide a characterization of productivity, as we show by means of an example. However, our results apply to left-linear TRSs, whereas [8,30] deal with orthogonal (constructorbased) TRSs only. Actually, we also supersede the main result of [26] which applies to non-orthogonal TRSs which are still left-linear.…”

“…Consider the tree specification R in Example 3, where, according to [8,Example 6.8] Being exhaustive is a necessary condition for productivity. PROOF.…”

“…,t k ) is a redex whenever t i ∈ T ω (C ) are (possiby infinite) closed constructor terms for all i, 1 ≤ i ≤ k [8, Definition 2.9]. As in [8,Definition 2.4], we assume here a generalized notion of substitution as an S -sorted mapping σ : X → T ω (F , X ) which is also extended to a mapping σ : T ω (F , X ) → T ω (F , X ).…”