13 research outputs found
Non-termination using Regular Languages
We describe a method for proving non-looping non-termination, that is, of
term rewriting systems that do not admit looping reductions. As certificates of
non-termination, we employ regular (tree) automata.Comment: Published at International Workshop on Termination 201
Proving Looping and Non-Looping Non-Termination by Finite Automata
A new technique is presented to prove non-termination of term rewriting. The
basic idea is to find a non-empty regular language of terms that is closed
under rewriting and does not contain normal forms. It is automated by
representing the language by a tree automaton with a fixed number of states,
and expressing the mentioned requirements in a SAT formula. Satisfiability of
this formula implies non-termination. Our approach succeeds for many examples
where all earlier techniques fail, for instance for the S-rule from combinatory
logic
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
Star Games and Hydras
The recursive path ordering is an established and crucial tool in term
rewriting to prove termination. We revisit its presentation by means of some
simple rules on trees (or corresponding terms) equipped with a 'star' as
control symbol, signifying a command to make that tree (or term) smaller in the
order being defined. This leads to star games that are very convenient for
proving termination of many rewriting tasks. For instance, using already the
simplest star game on finite unlabeled trees, we obtain a very direct proof of
termination of the famous Hydra battle, direct in the sense that there is not
the usual mention of ordinals. We also include an alternative road to setting
up the star games, using a proof method of Buchholz, adapted by van Oostrom,
resulting in a quantitative version of the star as control symbol. We conclude
with a number of questions and future research directions
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
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