29 research outputs found
Projections for infinitary rewriting
Proof terms in term rewriting are a representation means for reduction
sequences, and more in general for contraction activity, allowing to
distinguish e.g simultaneous from sequential reduction. Proof terms for
finitary, first-order, left-linear term rewriting are described in the Terese
book, chapter 8. In a previous work, we defined an extension of the finitary
proof-term formalism, that allows to describe contractions in infinitary
first-order term rewriting, and gave a characterisation of permutation
equivalence.
In this work, we discuss how projections of possibly infinite rewrite
sequences can be modeled using proof terms. Again, the foundation is a
characterisation of projections for finitary rewriting described in Terese,
Section 8.7. We extend this characterisation to infinitary rewriting and also
refine it, by describing precisely the role that structural equivalence plays
in the development of the notion of projection. The characterisation we propose
yields a definite expression, i.e. a proof term, that describes the projection
of an infinitary reduction over another.
To illustrate the working of projections, we show how a common reduct of a
(possibly infinite) reduction and a single step that makes part of it can be
obtained via their respective projections. We show, by means of several
examples, that the proposed definition yields the expected behavior also in
cases beyond those covered by this result. Finally, we discuss how the notion
of limit is used in our definition of projection for infinite reduction
A Calculus of Lambda Calculus Contexts
The calculus c serves as a general framework for representing contexts. Essential features are control over variable capturing and the freedom to manipulate contexts before or after hole lling, by a mechanism of delayed substitution. The context calculus c is given in the form of an extension of the lambda calculus. Many notions of context can be represented within the framework; a particular variation can be obtained by the choice of a pretyping, which we illustrate by three examples. 1
Four Equivalent Equivalences of Reductions
Two co-initial reductions in a term rewriting system are said to be equivalent if they perform the same steps, albeit maybe in a di#erent order. We present four characterisations of such a notion of equivalence, based on permutation, standardisation, labelling and projection, respectively. We prove that the characterisations all yield the same notion of equivalence, for the class of first-order left-linear term rewriting systems. A crucial role in our development is played by the notion of a proof term.
Projections for Infinitary Rewriting
Proof terms in term rewriting are a representation means for reduction sequences, and more in general for contraction activity, allowing to distinguish e.g. simultaneous from sequential reduction. Proof terms for finitary, first-order, left-linear term rewriting are described in [Terese. Term Rewriting Systems, volume 55 of Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, Cambridge, UK, 2003], ch. 8. In a previous work [C. Lombardi, A. RÃos, and R. de Vrijer. Proof terms for infinitary rewriting. In G. Dowek, editor, RTA-TLCA'14, volume 8560 of Lecture Notes in Computer Science, pages 303–318. Springer, 2014] we defined an extension of the finitary proof-term formalism, that allows to describe contractions in infinitary first-order term rewriting, and gave a characterisation of permutation equivalence. In this work, we discuss how projections of possibly infinite rewrite sequences can be modeled using proof terms. Again, the foundation is a characterisation of projections for finitary rewriting described in [Terese. Term Rewriting Systems, volume 55 of Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, Cambridge, UK, 2003], Sec. 8.7. We extend this characterisation to infinitary rewriting and also refine it, by describing precisely the role that structural equivalence plays in the development of the notion of projection. The characterisation we propose yields a definite expression, i.e. a proof term, that describes the projection of an infinitary reduction over another. To illustrate the working of projections, we show how a common reduct of a (possibly infinite) reduction and a single step that makes part of it can be obtained via their respective projections. We show, by means of several examples, that the proposed definition yields the expected behavior also in cases beyond those covered by this result. Finally, we discuss how the notion of limit is used in our definition of projection for infinite reduction