Star-free expressions for graphs, and an appropriate first-order logic

Speaker(s)

Mikołaj Bojańczyk

Affiliation

MIM UW

Date

May 5, 2021, 2:15 p.m.

Information about the event

online

Seminar

Seminar Automata Theory

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In the study of word languages, first-order logic is more frequently used with the order predicate x > y, and not the successor predicate x=y+1. (For MSO, there is no difference.) For example, the celebrated equivalence of star-free regular languages with first-order logic uses the first-order logic with order. With successor only, first-order logic for words is a less appealing logic. In contrast to words, the graph variant of first-order logic is usually studied with the edge relation E(x,y), which seems to be more in spirit of successor than order. It would be nice to define a variant of first-order logic for graphs, so that the standard notion of first-order logic for words becomes a special case for those graphs which are words. This is the topic of my talk. I will discuss a variant of star-free expressions for graphs, and a variant of first-order logic that is equivalent to these expressions. The variant is first-order logic with separator predicates of the form: “there is a path from vertex s to vertex t, which does not use vertices x1,…,xk”. (This is an infinite family of predicates, with one predicate of arity k+2 for every k.)

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