28 research outputs found
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
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