10,007 research outputs found
Extending Context-Sensitivity in Term Rewriting
We propose a generalized version of context-sensitivity in term rewriting
based on the notion of "forbidden patterns". The basic idea is that a rewrite
step should be forbidden if the redex to be contracted has a certain shape and
appears in a certain context. This shape and context is expressed through
forbidden patterns. In particular we analyze the relationships among this novel
approach and the commonly used notion of context-sensitivity in term rewriting,
as well as the feasibility of rewriting with forbidden patterns from a
computational point of view. The latter feasibility is characterized by
demanding that restricting a rewrite relation yields an improved termination
behaviour while still being powerful enough to compute meaningful results.
Sufficient criteria for both kinds of properties in certain classes of rewrite
systems with forbidden patterns are presented
Termination of Rewriting with and Automated Synthesis of Forbidden Patterns
We introduce a modified version of the well-known dependency pair framework
that is suitable for the termination analysis of rewriting under forbidden
pattern restrictions. By attaching contexts to dependency pairs that represent
the calling contexts of the corresponding recursive function calls, it is
possible to incorporate the forbidden pattern restrictions in the (adapted)
notion of dependency pair chains, thus yielding a sound and complete approach
to termination analysis. Building upon this contextual dependency pair
framework we introduce a dependency pair processor that simplifies problems by
analyzing the contextual information of the dependency pairs. Moreover, we show
how this processor can be used to synthesize forbidden patterns suitable for a
given term rewriting system on-the-fly during the termination analysis.Comment: In Proceedings IWS 2010, arXiv:1012.533
Termination of rewriting strategies: a generic approach
We propose a generic termination proof method for rewriting under strategies,
based on an explicit induction on the termination property. Rewriting trees on
ground terms are modeled by proof trees, generated by alternatively applying
narrowing and abstracting steps. The induction principle is applied through the
abstraction mechanism, where terms are replaced by variables representing any
of their normal forms. The induction ordering is not given a priori, but
defined with ordering constraints, incrementally set during the proof.
Abstraction constraints can be used to control the narrowing mechanism, well
known to easily diverge. The generic method is then instantiated for the
innermost, outermost and local strategies.Comment: 49 page
Soundness of Unravelings for Conditional Term Rewriting Systems via Ultra-Properties Related to Linearity
Unravelings are transformations from a conditional term rewriting system
(CTRS, for short) over an original signature into an unconditional term
rewriting systems (TRS, for short) over an extended signature. They are not
sound w.r.t. reduction for every CTRS, while they are complete w.r.t.
reduction. Here, soundness w.r.t. reduction means that every reduction sequence
of the corresponding unraveled TRS, of which the initial and end terms are over
the original signature, can be simulated by the reduction of the original CTRS.
In this paper, we show that an optimized variant of Ohlebusch's unraveling for
a deterministic CTRS is sound w.r.t. reduction if the corresponding unraveled
TRS is left-linear or both right-linear and non-erasing. We also show that
soundness of the variant implies that of Ohlebusch's unraveling. Finally, we
show that soundness of Ohlebusch's unraveling is the weakest in soundness of
the other unravelings and a transformation, proposed by Serbanuta and Rosu, for
(normal) deterministic CTRSs, i.e., soundness of them respectively implies that
of Ohlebusch's unraveling.Comment: 49 pages, 1 table, publication in Special Issue: Selected papers of
the "22nd International Conference on Rewriting Techniques and Applications
(RTA'11)
Turchin's Relation for Call-by-Name Computations: A Formal Approach
Supercompilation is a program transformation technique that was first
described by V. F. Turchin in the 1970s. In supercompilation, Turchin's
relation as a similarity relation on call-stack configurations is used both for
call-by-value and call-by-name semantics to terminate unfolding of the program
being transformed. In this paper, we give a formal grammar model of
call-by-name stack behaviour. We classify the model in terms of the Chomsky
hierarchy and then formally prove that Turchin's relation can terminate all
computations generated by the model.Comment: In Proceedings VPT 2016, arXiv:1607.0183
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
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
A Combination Framework for Complexity
In this paper we present a combination framework for polynomial complexity
analysis of term rewrite systems. The framework covers both derivational and
runtime complexity analysis. We present generalisations of powerful complexity
techniques, notably a generalisation of complexity pairs and (weak) dependency
pairs. Finally, we also present a novel technique, called dependency graph
decomposition, that in the dependency pair setting greatly increases
modularity. We employ the framework in the automated complexity tool TCT. TCT
implements a majority of the techniques found in the literature, witnessing
that our framework is general enough to capture a very brought setting
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