Classifying Non-periodic Sequences by Permutation Transducers

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Data-Oblivious Stream Productivity

We are concerned with demonstrating productivity of specifications of infinite streams of data, based on orthogonal rewrite rules. In general, this property is undecidable, but for restricted formats computable sufficient conditions can be obtained. The usual analysis disregards the identity of data, thus leading to approaches that we call data-oblivious. We present a method that is provably optimal among all such data-oblivious approaches. This means that in order to improve on the algorithm in this paper one has to proceed in a data-aware fashion

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

Termination and Productivity

Klop, J.W. [Promotor]Vrijer, R.C. de [Copromotor

Local Termination: theory and practice

The characterisation of termination using well-founded monotone algebras has been a milestone on the way to automated termination techniques, of which we have seen an extensive development over the past years. Both the semantic characterisation and most known termination methods are concerned with global termination, uniformly of all the terms of a term rewriting system (TRS). In this paper we consider local termination, of specific sets of terms within a given TRS. The principal goal of this paper is generalising the semantic characterisation of global termination to local termination. This is made possible by admitting the well-founded monotone algebras to be partial. We also extend our approach to local relative termination. The interest in local termination naturally arises in program verification, where one is probably interested only in sensible inputs, or just wants to characterise the set of inputs for which a program terminates. Local termination will be also be of interest when dealing with a specific class of terms within a TRS that is known to be non-terminating, such as combinatory logic (CL) or a TRS encoding recursive program schemes or Turing machines. We show how some of the well-known techniques for proving global termination, such as stepwise removal of rewrite rules and semantic labelling, can be adapted to the local case. We also describe transformations reducing local to global termination problems. The resulting techniques for proving local termination have in some cases already been automated. One of our applications concerns the characterisation of the terminating S-terms in CL as regular language. Previously this language had already been found via a tedious analysis of the reduction behaviour of S-terms. These findings have now been vindicated by a fully automated and verified proof

Braids via term rewriting

We present a brief introduction to braids, in particular simple positive braids, with a double emphasis: first, we focus on term rewriting techniques, in particular, reduction diagrams and decreasing diagrams. The second focus is our employment of the colored braid notation next to the more familiar Artin notation. Whereas the latter is a relative, position dependent, notation, the former is an absolute notation that seems more suitable for term rewriting techniques such as symbol tracing. Artin's equations translate in this notation to simple word inversions. With these points of departure we treat several basic properties of positive braids, in particular related to the word problem, confluence property, projection equivalence, and the congruence property. In our introduction the beautiful diamond known as the permutohedron plays a decisive role

Transducer degrees: atoms, infima and suprema

Although finite state transducers are very natural and simple devices, surprisingly little is known about the transducibility relation they induce on streams (infinite words). We collect some intriguing problems that have been unsolved since several years. The transducibility relation arising from finite state transduction induces a partial order of stream degrees, which we call Transducer degrees, analogous to the well-known Turing degrees or degrees of unsolvability. We show that there are pairs of degrees without supremum and without infimum. The former result is somewhat surprising since every finite set of degrees has a supremum if we strengthen the machine model to Turing machines, but also if we weaken it to Mealy machines

Decreasing diagrams for confluence and commutation

Like termination, confluence is a central property of rewrite systems. Unlike for termination, however, there exists no known complexity hierarchy for confluence. In this paper we investigate whether the decreasing diagrams technique can be used to obtain such a hierarchy. The decreasing diagrams technique is one of the strongest and most versatile methods for proving confluence of abstract rewrite systems. It is complete for countable systems, and it has many well-known confluence criteria as corollaries. So what makes decreasing diagrams so powerful? In contrast to other confluence techniques, decreasing diagrams employ a labelling of the steps with labels from a wellfounded order in order to conclude confluence of the underlying unlabelled relation. Hence it is natural to ask how the size of the label set influences the strength of the technique. In particular, what class of abstract rewrite systems can be proven confluent using decreasing diagrams restricted to 1 label, 2 labels, 3 labels, and so on? Surprisingly, we find that two labels suffice for proving confluence for every abstract rewrite system having the cofinality property, thus in particular for every confluent, countable system. Secondly, we show that this result stands in sharp contrast to the situation for commutation of rewrite relations, where the hierarchy does not collapse. Thirdly, investigating the possibility of a confluence hierarchy, we determine the first-order (non-)definability of the notion of confluence and related properties, using techniques from finite model theory. We find that in particular Hanf ’s theorem is fruitful for elegant proofs of undefinability of properties of abstract rewrite systems