automatic presentation

Let S be a relational structure (for example, a graph).

S has an automatic presentation if (for some alphabet Σ) there is a languagePlanetmathPlanetmath LΣ* and a surjective map f from L onto the ) of S such that

  • L can be checked by a finite automaton ( (Membership)

  • The language of all convolutions of x,yL where f(x)=f(y) can be checked by a (Equality)

  • For all n-ary relationsMathworldPlanetmathPlanetmathPlanetmath ( Ri in S, the language of all convolutions of x1,x2,,xnL where Ri(f(x1),f(x2),,f(xn)) can be checked by a ()

The class of languages accepted by finite automata, i.e. regular languages, is closed underPlanetmathPlanetmath operations like union, intersectionMathworldPlanetmathPlanetmath, complementation etc, and it is decidable whether or not a finite accepts the empty language. This allows any first order sentenceMathworldPlanetmath over the structureMathworldPlanetmath to be decided - using union for ’and’, complementation for ’not’ etc., and emptiness for dealing with ’there exists’. As such, the first order theory of any structure with an automatic presentation is decidable.

Note that wrt group this is inspired by, but not to, the definition of automatic groupsMathworldPlanetmath.

Title automatic presentation
Canonical name AutomaticPresentation
Date of creation 2013-03-22 14:16:57
Last modified on 2013-03-22 14:16:57
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 8
Author mathcam (2727)
Entry type Definition
Classification msc 03D05
Classification msc 03C57
Synonym automatic structure
Synonym FA presentation
Related topic AutomaticGroup