676 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
Synchronizing non-deterministic finite automata
In this paper, we show that every D3-directing CNFA can be mapped uniquely to
a DFA with the same synchronizing word length. This implies that \v{C}ern\'y's
conjecture generalizes to CNFAs and that the general upper bound for the length
of a shortest D3-directing word is equal to the Pin-Frankl bound for DFAs. As a
second consequence, for several classes of CNFAs sharper bounds are
established. Finally, our results allow us to detect all critical CNFAs on at
most 6 states. It turns out that only very few critical CNFAs exist.Comment: 21 page
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 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
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
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
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