Automata, Semigroups and Groups: 60 Years of Synergy
Automata, semigroups and groups: 60 years of synergy Jean-Eric´ Pin1 1LIAFA, CNRS and University Paris Diderot 60th birthday of Stuart W. Margolis June 2013, Bar Ilan Don’t forget to turn your mobile phone back on AFTER this lecture LIAFA, CNRS and University Paris Diderot The precursors A, A, ⊲ A, A, ⊲ a,♯,⊲ ♯,♯,⊲ 0 1 5 Turing (1936) a,♯,⊲ a, a, ⊲ a, A, ⊲ Turing Machine 4 2 3 ♯,♯,⊳ a, a, ⊲ a, a, ⊳ A, A, ⊲ A, A, ⊲ A, A, ⊳ McCulloch and Pitts (1943) Neural networks Shannon (1948) LIAFA, CNRS and University Paris Diderot The founders Kleene (1951, 1956) Equivalence between automata and regular expressions. Sch¨utzenberger (1956): Ordered syntactic monoid Codes, unambiguous expressions. Chomsky (1956): Chomsky hierarchy LIAFA, CNRS and University Paris Diderot 1956: Sch¨utzenberger’s paper • Birth of the theory of variable length codes • Free submonoids of the free monoid • Link with probabilities • Definition of the syntactic preorder LIAFA, CNRS and University Paris Diderot Theory of codes: lead to the following books 1985 2011 Berstel, Perrin Berstel, Perrin, Reutenauer LIAFA, CNRS and University Paris Diderot Definition of the syntactic preorder The syntactic preorder of a language K of A∗ is the ∗ relation 6K defined on A by u 6K v iff, for every x, y ∈ A∗, xuy ∈ K ⇒ xvy ∈ K. The syntactic congruence ∼K is the associated equivalence relation: u ∼K v iff u 6K v and v 6K u. LIAFA, CNRS and University Paris Diderot Codes and syntactic monoids ∗ ∗ a3 a a2ba aba a2 aba2 abab a2b a2bab ab2 aba2b ab ∗ ∗ baba ba2 ba3 ba ba2ba baba2 b3 bab ba2b b b2 bab2 LIAFA, CNRS and University Paris Diderot Early results • Automata theory: Medvedev (1956), Myhill (1957), Nerode (1958), Rabin and Scott (1958), Brzozowski (1964).
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