A short overview of topology and languages interactions Introduction Crash course on regular languages Genus of a regular language Genus and size The genus of regular languages and other ideas from low-dimensional topology Florian Deloup Institut de Math´ ematiques de Toulouse, France June 21, 2016 Florian Deloup The genus of regular languages and other ideas from low-dimensional
A short overview of topology and languages interactions Introduction Crash course on regular languages Genus of a regular language Genus and size Joint work with Guillaume Bonfante. Florian Deloup The genus of regular languages and other ideas from low-dimensional
A short overview of topology and languages interactions Introduction Crash course on regular languages Genus of a regular language Genus and size Joint work with Guillaume Bonfante. 1) The genus of regular languages, 2012. Math. Str. Computer Sc., 2016. 2) The decidability of language genus computation, 2016. Available on ArXiv. Florian Deloup The genus of regular languages and other ideas from low-dimensional
A short overview of topology and languages interactions Introduction Crash course on regular languages Genus of a regular language Genus and size What I won’t talk about in this talk Topology = ⇒ Languages (as tool to study topology): languages as topological invariants Florian Deloup The genus of regular languages and other ideas from low-dimensional
A short overview of topology and languages interactions Introduction Crash course on regular languages Genus of a regular language Genus and size What I won’t talk about in this talk Topology = ⇒ Languages (as tool to study topology): languages as topological invariants - Fundamental group of a topological space, languages (Poincar´ e, 1895, “Analysis situs” paper, also Riemann and Klein) Florian Deloup The genus of regular languages and other ideas from low-dimensional
A short overview of topology and languages interactions Introduction Crash course on regular languages Genus of a regular language Genus and size What I won’t talk about in this talk Topology = ⇒ Languages (as tool to study topology): languages as topological invariants - Fundamental group of a topological space, languages (Poincar´ e, 1895, “Analysis situs” paper, also Riemann and Klein) - Knots: encoding Reidemeister moves (1927) yields language(s). Particular cases: quandles, Wirtinger presentation of the fundamental group of the complement of a knot. Florian Deloup The genus of regular languages and other ideas from low-dimensional
A short overview of topology and languages interactions Introduction Crash course on regular languages Genus of a regular language Genus and size What I will talk about in this talk Languages = ⇒ Topology (as a tool to study languages): topology as a language invariant This talk: language invariants from low-dimensional topology . Florian Deloup The genus of regular languages and other ideas from low-dimensional
A short overview of topology and languages interactions Introduction Crash course on regular languages Genus of a regular language Genus and size ”Moore’s Law” Moore’s ”Law” (1960s) The number of transistors in a dense integrated circuit doubles every two years. Florian Deloup The genus of regular languages and other ideas from low-dimensional
A short overview of topology and languages interactions Introduction Crash course on regular languages Genus of a regular language Genus and size ”Moore’s Law” Moore’s ”Law” (1960s) The number of transistors in a dense integrated circuit doubles every two years. Correction to Moore’s ”Law” (2005) Moore’s Law has to end. Florian Deloup The genus of regular languages and other ideas from low-dimensional
A short overview of topology and languages interactions Introduction Crash course on regular languages Genus of a regular language Genus and size ”Moore’s Law” Moore’s ”Law” (1960s) The number of transistors in a dense integrated circuit doubles every two years. Correction to Moore’s ”Law” (2005) Moore’s Law has to end. Reason invoked: physical limit of matter processing. Florian Deloup The genus of regular languages and other ideas from low-dimensional
A short overview of topology and languages interactions Introduction Crash course on regular languages Genus of a regular language Genus and size ”Moore’s Law” Moore’s ”Law” (1960s) The number of transistors in a dense integrated circuit doubles every two years. Correction to Moore’s ”Law” (2005) Moore’s Law has to end. Reason invoked: physical limit of matter processing. Shape and space organization become central = ⇒ Low-dimensional topology = ⇒ Invariants of Languages from low-dimensional topology Florian Deloup The genus of regular languages and other ideas from low-dimensional
A short overview of topology and languages interactions Introduction Regular languages as computations Crash course on regular languages Automata Genus of a regular language Minimal automaton Genus and size Regular languages Set-up: - the class Reg A of regular languages on a finite alphabet A . - the class DFA A of deterministic finite automata on A . Florian Deloup The genus of regular languages and other ideas from low-dimensional
A short overview of topology and languages interactions Introduction Regular languages as computations Crash course on regular languages Automata Genus of a regular language Minimal automaton Genus and size Regular languages Set-up: - the class Reg A of regular languages on a finite alphabet A . - the class DFA A of deterministic finite automata on A . Working-out definition: a regular language L on alphabet A is a subset of A ∗ , starting from a subset of A and recursively computed by a finite number of the familiar 3 operations: Florian Deloup The genus of regular languages and other ideas from low-dimensional
A short overview of topology and languages interactions Introduction Regular languages as computations Crash course on regular languages Automata Genus of a regular language Minimal automaton Genus and size Regular languages Set-up: - the class Reg A of regular languages on a finite alphabet A . - the class DFA A of deterministic finite automata on A . Working-out definition: a regular language L on alphabet A is a subset of A ∗ , starting from a subset of A and recursively computed by a finite number of the familiar 3 operations: - Union of two languages: ( L , L ′ ) �→ L ∪ L ′ = { w ∈ A ∗ | w ∈ L , or w ∈ L ′ } . - Composition of two languages: ( L , L ′ ) �→ LL ′ = { ww ′ | w ∈ L , w ′ ∈ L ′ } . - Star operation: L �→ L ∗ = � n ≥ 0 L n Florian Deloup The genus of regular languages and other ideas from low-dimensional
A short overview of topology and languages interactions Introduction Regular languages as computations Crash course on regular languages Automata Genus of a regular language Minimal automaton Genus and size Automata An automaton is a decorated directed (multi)graph. Florian Deloup The genus of regular languages and other ideas from low-dimensional
A short overview of topology and languages interactions Introduction Regular languages as computations Crash course on regular languages Automata Genus of a regular language Minimal automaton Genus and size Automata An automaton is a decorated directed (multi)graph. 0 0 1 1 0 2 2 0 2 1 4 2 2 1 1 0 0 1 2 3 Florian Deloup The genus of regular languages and other ideas from low-dimensional
A short overview of topology and languages interactions Introduction Regular languages as computations Crash course on regular languages Automata Genus of a regular language Minimal automaton Genus and size Automata Decoration: - label each directed edge (transition) by a letter of the alphabet A . a Florian Deloup The genus of regular languages and other ideas from low-dimensional
A short overview of topology and languages interactions Introduction Regular languages as computations Crash course on regular languages Automata Genus of a regular language Minimal automaton Genus and size Automata Decoration: - label each directed edge (transition) by a letter of the alphabet A . a - distinguish special states: one initial state, one subset of final states. Pictorial convention for initial and final states: , initial final final Florian Deloup The genus of regular languages and other ideas from low-dimensional
A short overview of topology and languages interactions Introduction Regular languages as computations Crash course on regular languages Automata Genus of a regular language Minimal automaton Genus and size Deterministic automaton The automaton is deterministic if there is at most one transition labelled by a given letter. a a Florian Deloup The genus of regular languages and other ideas from low-dimensional
A short overview of topology and languages interactions Introduction Regular languages as computations Crash course on regular languages Automata Genus of a regular language Minimal automaton Genus and size Deterministic automaton The automaton is deterministic if there is at most one transition labelled by a given letter. a a Florian Deloup The genus of regular languages and other ideas from low-dimensional
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