175 research outputs found
The pro--solvable topology on a free group
We prove that, given a finitely generated subgroup of a free group ,
the following questions are decidable: is closed (dense) in for the
pro-(met)abelian topology? is the closure of in for the
pro-(met)abelian topology finitely generated? We show also that if the latter
question has a positive answer, then we can effectively construct a basis for
the closure, and the closure has decidable membership problem in any case.
Moreover, it is decidable whether is closed for the pro- topology
when is an equational pseudovariety of finite groups, such as the
pseudovariety of all finite solvable groups with derived length
. We also connect the pro-abelian topology with the topologies defined
by abelian groups of bounded exponent
Term rewriting systems from Church-Rosser to Knuth-Bendix and beyond
Term rewriting systems are important for computability theory of abstract data types, for automatic theorem proving, and for the foundations of functional programming. In this short survey we present, starting from first principles, several of the basic notions and facts in the area of term rewriting. Our treatment, which often will be informal, covers abstract rewriting, Combinatory Logic, orthogonal systems, strategies, critical pair completion, and some extended rewriting formats
Decision problems concerning sets of equations
This thesis is about "decision problems concerning properties of sets of equations". If L is a first-order language with equality and if P is a property of sets of L-equations, then "the decision problem of P in L" is the problem of the existence or not of an algorithm, which enables us to decide whether, given a set Sigma of L-equations, Sigma has the property P or not. If such an algorithm exists, P is decidable in L. Otherwise, it is undecidable in L. After surveying the work that has been done in the field, we present a new method for proving the undecidability of a property P, for finite sets of L-equations. As an application, we establish the undecidability of some basic model-theoretical properties, for finite sets of equations of non-trivial languages. Then, we prove the non-existence of an algorithm for deciding whether a field is finite and, as a corollary, we derive the undecidability of certain properties, for recursive sets of equations of infinite non-trivial languages. Finally, we consider trivial languages, and we prove that a number of properties, undecidable in languages with higher complexity, are decidable in them.<p
A Calculus and Algebra Derived from Directed Graph Algebras
Shallon invented a means of deriving algebras from graphs, yielding numerous
examples of so-called graph algebras with interesting equational properties. Here we study directed graph algebras, derived from directed graphs in the same way that Shallon’s undirected graph algebras are derived from graphs
E-Generalization Using Grammars
We extend the notion of anti-unification to cover equational theories and
present a method based on regular tree grammars to compute a finite
representation of E-generalization sets. We present a framework to combine
Inductive Logic Programming and E-generalization that includes an extension of
Plotkin's lgg theorem to the equational case. We demonstrate the potential
power of E-generalization by three example applications: computation of
suggestions for auxiliary lemmas in equational inductive proofs, computation of
construction laws for given term sequences, and learning of screen editor
command sequences.Comment: 49 pages, 16 figures, author address given in header is meanwhile
outdated, full version of an article in the "Artificial Intelligence
Journal", appeared as technical report in 2003. An open-source C
implementation and some examples are found at the Ancillary file
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