Stream Productivity by Outermost Termination

Streams are infinite sequences over a given data type. A stream specification is a set of equations intended to define a stream. A core property is productivity: unfolding the equations produces the intended stream in the limit. In this paper we show that productivity is equivalent to termination with respect to the balanced outermost strategy of a TRS obtained by adding an additional rule. For specifications not involving branching symbols balancedness is obtained for free, by which tools for proving outermost termination can be used to prove productivity fully automatically

Finding small counter examples for abstract rewriting properties

Rewriting notions like termination, normal forms and confluence can be described in an abstract way referring to rewriting only as a binary relation. Several theorems on rewriting, like Newman's lemma, can be proved in this abstract setting. For investigating possible generalizations of such theorems, it is fruitful to have counterexamples showing that particular generalizations do not hold. In this paper, we develop a technique to find such counterexamples fully automatically, and we describe our tool Carpa that follows this technique. The basic idea is to fix the number of objects of the abstract rewrite system, and to express the conditions and the negation of the conclusion in a satisfiability (SAT) formula, and then call a current SAT solver. In case the formula turns out to be satisfiable, the resulting satisfying assignment yields a counterexample to the encoded property. We give several examples of finite abstract rewrite systems having remarkable properties that are found in this way fully automatically.</p

Termination of term rewriting by semantic labelling

A new kind of transformation of term rewriting systems (TRS) is proposed, depending on a choice for a model for the TRS. The labelled TRS is obtained from the original one by labelling operation symbols, possibly creating extra copies of some rules. This construction has the remarkable property that the labelled TRS is terminating if and only if the original TRS is terminating. Although the labelled version has more operation symbols and may have more rules (sometimes innitely many), termination is often easier to prove for the labelled TRS than for the original one. This provides a new technique for proving termination, making classical techniques like path orders and polynomial interpretations applicable even for non-simplifying TRS's. The requirement of having a model can slightly be weakened, yielding a remarkably simple termination proof of the system SUBST of [11] describing explicit substitution in -calculus

Combining Insertion and Deletion in RNA-editing Preserves Regularity

Inspired by RNA-editing as occurs in transcriptional processes in the living cell, we introduce an abstract notion of string adjustment, called guided rewriting. This formalism allows simultaneously inserting and deleting elements. We prove that guided rewriting preserves regularity: for every regular language its closure under guided rewriting is regular too. This contrasts an earlier abstraction of RNA-editing separating insertion and deletion for which it was proved that regularity is not preserved. The particular automaton construction here relies on an auxiliary notion of slice sequence which enables to sweep from left to right through a completed rewrite sequence.Comment: In Proceedings MeCBIC 2012, arXiv:1211.347

A complete characterization of termination of 0p1q → 1r0s

We completely characterize termination of one-rule string rewriting systems of the form 0p1q → 1r0s for every choice of positive integers p, q, r, and s. For the simply terminating cases, we give a sharp estimate of the complexity of derivation lengths

Relaxation of 3-partition instances

The 3-partition problem admits a straightforward formulation as a 0-1 Integer Linear Program (ILP). We investigate problem instances for which the half-integer relaxation of the ILP is feasible, while the ILP is not. We prove that this only occurs on a set of at least 18 elements, and in case of 18 elements such an instance always contains an element of weight = 10. These bounds are sharp: we give all 14 instances consisting of 18 elements all having weight = 10. Our approach is based on analyzing an underlying graph structure