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

Proving Termination of Graph Transformation Systems using Weighted Type Graphs over Semirings

We introduce techniques for proving uniform termination of graph transformation systems, based on matrix interpretations for string rewriting. We generalize this technique by adapting it to graph rewriting instead of string rewriting and by generalizing to ordered semirings. In this way we obtain a framework which includes the tropical and arctic type graphs introduced in a previous paper and a new variant of arithmetic type graphs. These type graphs can be used to assign weights to graphs and to show that these weights decrease in every rewriting step in order to prove termination. We present an example involving counters and discuss the implementation in the tool Grez

Resolution and binary decision diagrams cannot simulate each other polynomially

There are many different ways of proving formulas in proposition logic. Many of these can easily be characterized as forms of resolution. Others use so-called binary decision diagrams (BDDs). Experimental evidence suggests that BDDs and resolution based techniques are fundamentally different, in the sense that their performance can differ very much on benchmarks. In this paper we confirm these findings by mathematical proof. We provide examples that are easy for BDDS and exponentially hard for any form of resolution, and vice versa, examples that ar easy for resolution and exponentially hard for BDDs

A rewriting approach to binary decision diagrams

AbstractBinary decision diagrams (BDDs) provide an established technique for propositional formula manipulation. In this paper, we present the basic BDD theory by means of standard rewriting techniques. Since a BDD is a DAG instead of a tree we need a notion of shared rewriting and develop appropriate theory. A rewriting system is presented by which canonical reduced ordered BDDs (ROBDDs) can be obtained and for which uniqueness of ROBDD representation is proved. Next, an alternative rewriting system is presented, suitable for actually computing ROBDDs from formulas. For this rewriting system a layerwise strategy is defined, and it is proved that when replacing the classical apply-algorithm by layerwise rewriting, roughly the same complexity bound is reached as in the classical algorithm. Moreover, a layerwise innermost strategy is defined and it is proved that the full classical algorithm for computing ROBDDs can be replaced by layerwise innermost rewriting without essentially affecting the complexity. Finally a lazy strategy is proposed sometimes performing much better than the traditional algorithm

Well-definedness of Streams by Transformation and Termination

Streams are infinite sequences over a given data type. A stream specification is a set of equations intended to define a stream. We propose a transformation from such a stream specification to a term rewriting system (TRS) in such a way that termination of the resulting TRS implies that the stream specification is well-defined, that is, admits a unique solution. As a consequence, proving well-definedness of several interesting stream specifications can be done fully automatically using present powerful tools for proving TRS termination. In order to increase the power of this approach, we investigate transformations that preserve semantics and well-definedness. We give examples for which the above mentioned technique applies for the ransformed specification while it fails for the original one

Book review: Automatic Sequences. Theory, Applications, Generalizations

No abstrac

Counting symbol switches in synchronizing automata

Instead of looking at the lengths of synchronizing words as in Černý's conjecture, we look at the switch count of such words, that is, we only count the switches from one letter to another. Where the synchronizing words of the Černý automata C n have switch count linear in n , we wonder whether synchronizing automata exist for which every synchronizing word has quadratic switch count. The answer is positive: we prove that switch count has the same complexity as synchronizing word length. We give some series of synchronizing automata yielding quadratic switch count, the best one reaching 23 n 2 +O(n) as switch count. \u3cbr/\u3eWe investigate all binary automata on at most 9 states and determine the maximal possible switch count. For all 3≤n≤9 , a strictly higher switch count can be reached by allowing more symbols. This behaviour differs from length, where for every n , no automata are known with higher synchronization length than C n , which has only two symbols. It is not clear if this pattern extends to larger n . For n≥12 , our best construction only has two symbol