238 research outputs found
Combinatorics of reductions between equivalence relations
We discuss combinatorial conditions for the existence of various types of
reductions between equivalence relations, and in particular identify necessary
and sufficient conditions for the existence of injective reductions.Comment: 7 pages, 2 figure
Tameness in least fixed-point logic and McColm's conjecture
We investigate four model-theoretic tameness properties in the context of
least fixed-point logic over a family of finite structures. We find that each
of these properties depends only on the elementary (i.e., first-order) limit
theory, and we completely determine the valid entailments among them. In
contrast to the context of first-order logic on arbitrary structures, the order
property and independence property are equivalent in this setting. McColm
conjectured that least fixed-point definability collapses to first-order
definability exactly when proficiency fails. McColm's conjecture is known to be
false in general. However, we show that McColm's conjecture is true for any
family of finite structures whose limit theory is model-theoretically tame
Laver and set theory
In this commemorative article, the work of Richard Laver is surveyed in its full range and extent.Accepted manuscrip
The prospects for mathematical logic in the twenty-first century
The four authors present their speculations about the future developments of
mathematical logic in the twenty-first century. The areas of recursion theory,
proof theory and logic for computer science, model theory, and set theory are
discussed independently.Comment: Association for Symbolic Logi
Decreasing Diagrams for Confluence and Commutation
Like termination, confluence is a central property of rewrite systems. Unlike
for termination, however, there exists no known complexity hierarchy for
confluence. In this paper we investigate whether the decreasing diagrams
technique can be used to obtain such a hierarchy. The decreasing diagrams
technique is one of the strongest and most versatile methods for proving
confluence of abstract rewrite systems. It is complete for countable systems,
and it has many well-known confluence criteria as corollaries.
So what makes decreasing diagrams so powerful? In contrast to other
confluence techniques, decreasing diagrams employ a labelling of the steps with
labels from a well-founded order in order to conclude confluence of the
underlying unlabelled relation. Hence it is natural to ask how the size of the
label set influences the strength of the technique. In particular, what class
of abstract rewrite systems can be proven confluent using decreasing diagrams
restricted to 1 label, 2 labels, 3 labels, and so on? Surprisingly, we find
that two labels suffice for proving confluence for every abstract rewrite
system having the cofinality property, thus in particular for every confluent,
countable system.
Secondly, we show that this result stands in sharp contrast to the situation
for commutation of rewrite relations, where the hierarchy does not collapse.
Thirdly, investigating the possibility of a confluence hierarchy, we
determine the first-order (non-)definability of the notion of confluence and
related properties, using techniques from finite model theory. We find that in
particular Hanf's theorem is fruitful for elegant proofs of undefinability of
properties of abstract rewrite systems
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