20,567 research outputs found
A Reduction-Preserving Completion for Proving Confluence of Non-Terminating Term Rewriting Systems
We give a method to prove confluence of term rewriting systems that contain
non-terminating rewrite rules such as commutativity and associativity. Usually,
confluence of term rewriting systems containing such rules is proved by
treating them as equational term rewriting systems and considering E-critical
pairs and/or termination modulo E. In contrast, our method is based solely on
usual critical pairs and it also (partially) works even if the system is not
terminating modulo E. We first present confluence criteria for term rewriting
systems whose rewrite rules can be partitioned into a terminating part and a
possibly non-terminating part. We then give a reduction-preserving completion
procedure so that the applicability of the criteria is enhanced. In contrast to
the well-known Knuth-Bendix completion procedure which preserves the
equivalence relation of the system, our completion procedure preserves the
reduction relation of the system, by which confluence of the original system is
inferred from that of the completed system
Canonized Rewriting and Ground AC Completion Modulo Shostak Theories : Design and Implementation
AC-completion efficiently handles equality modulo associative and commutative
function symbols. When the input is ground, the procedure terminates and
provides a decision algorithm for the word problem. In this paper, we present a
modular extension of ground AC-completion for deciding formulas in the
combination of the theory of equality with user-defined AC symbols,
uninterpreted symbols and an arbitrary signature disjoint Shostak theory X. Our
algorithm, called AC(X), is obtained by augmenting in a modular way ground
AC-completion with the canonizer and solver present for the theory X. This
integration rests on canonized rewriting, a new relation reminiscent to
normalized rewriting, which integrates canonizers in rewriting steps. AC(X) is
proved sound, complete and terminating, and is implemented to extend the core
of the Alt-Ergo theorem prover.Comment: 30 pages, full version of the paper TACAS'11 paper "Canonized
Rewriting and Ground AC-Completion Modulo Shostak Theories" accepted for
publication by LMCS (Logical Methods in Computer Science
Smart matching
One of the most annoying aspects in the formalization of mathematics is the
need of transforming notions to match a given, existing result. This kind of
transformations, often based on a conspicuous background knowledge in the given
scientific domain (mostly expressed in the form of equalities or isomorphisms),
are usually implicit in the mathematical discourse, and it would be highly
desirable to obtain a similar behavior in interactive provers. The paper
describes the superposition-based implementation of this feature inside the
Matita interactive theorem prover, focusing in particular on the so called
smart application tactic, supporting smart matching between a goal and a given
result.Comment: To appear in The 9th International Conference on Mathematical
Knowledge Management: MKM 201
Reinterpreting Compression in Infinitary Rewriting
Departing from a computational interpretation of compression in infinitary rewriting, we view compression as a degenerate case of standardisation. The change in perspective comes about via two observations: (a) no compression property can be recovered for non-left-linear systems and (b) some standardisation procedures, as a ‘side-effect’, yield compressed reductions
Superposition as a logical glue
The typical mathematical language systematically exploits notational and
logical abuses whose resolution requires not just the knowledge of domain
specific notation and conventions, but not trivial skills in the given
mathematical discipline. A large part of this background knowledge is expressed
in form of equalities and isomorphisms, allowing mathematicians to freely move
between different incarnations of the same entity without even mentioning the
transformation. Providing ITP-systems with similar capabilities seems to be a
major way to improve their intelligence, and to ease the communication between
the user and the machine. The present paper discusses our experience of
integration of a superposition calculus within the Matita interactive prover,
providing in particular a very flexible, "smart" application tactic, and a
simple, innovative approach to automation.Comment: In Proceedings TYPES 2009, arXiv:1103.311
Termination orders for 3-dimensional rewriting
This paper studies 3-polygraphs as a framework for rewriting on
two-dimensional words. A translation of term rewriting systems into
3-polygraphs with explicit resource management is given, and the respective
computational properties of each system are studied. Finally, a convergent
3-polygraph for the (commutative) theory of Z/2Z-vector spaces is given. In
order to prove these results, it is explained how to craft a class of
termination orders for 3-polygraphs.Comment: 30 pages, 35 figure
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