Obtaining the syntactic monoid via duality

Transcription

1 Radboud University Nijmegen MLNL Groningen May 19th, 2011

2 Formal languages An alphabet is a non-empty finite set of symbols. If Σ is an alphabet, then Σ denotes the set of all words over Σ. The set Σ forms a free monoid. Operation: concatenation. Identity element: the empty word ε. A language over Σ is a subset of Σ.

3 Regular languages The class of regular languages over an alphabet Σ is defined inductively as follows. The languages, {ε}, and {a} for a Σ are regular. If K and L are regular languages, then so are the union K L; the concatenation K L = {uv u K, v L}; the Kleene star L = n 0 Ln. Examples of regular languages: {a m (ba) n m, n N} = {a} ({b}{a}) is regular. All finite languages are regular.

4 Finite automata An example of a deterministic finite automaton: a a,b b b b a a

5 Finite automata An example of a deterministic finite automaton: a a,b b b b a a The language recognized by this automaton is {a m (ba) n m, n N}.

6 Finite automata An example of a deterministic finite automaton: a a,b b b b a a The language recognized by this automaton is {a m (ba) n m, n N}. Theorem (Kleene) A language is regular if and only if it is recognized by a finite automaton.

7 Syntactic monoid Let Σ be an alphabet, and L Σ a language. Define an equivalence relation L on Σ by u L v if and only if x, y Σ (xuy L xvy L) for u, v Σ. The syntactic monoid of L is Σ / L.

8 Applications of the syntactic monoid Theorem The syntactic monoid of a regular language is effectively computable.

9 Applications of the syntactic monoid Theorem The syntactic monoid of a regular language is effectively computable. The syntactic monoid can be used to characterize properties of languages. Theorem (Myhill, Nerode) A language is regular if and only if its syntactic monoid is finite.

10 Applications of the syntactic monoid Theorem The syntactic monoid of a regular language is effectively computable. The syntactic monoid can be used to characterize properties of languages. Theorem (Myhill, Nerode) A language is regular if and only if its syntactic monoid is finite. Theorem (Schützenberger) A regular language is star-free if and only if its syntactic monoid is acyclic.

11 Overview Goal: find an algorithm to compute the syntactic monoid of a regular language using duality for residuated Boolean algebras. Duality theory for finite Boolean algebras Duality theory for residuated Boolean algebras Application to Boolean algebras associated to regular languages

12 Duality for finite Boolean algebras There is a duality between finite Boolean algebras and finite sets. Objects: Morphisms: The map dual to B 1 The map dual to X 1 B Atoms(B) P(X ) X f B 2 is x {b B 2 x f (b)} : X 2 X 1. g X 2 is g 1 : B 2 B 1.

13 Residuated Boolean algebras Definition A residuated Boolean algebra is a Boolean algebra B with three binary operations, \, / and a constant e B such that (B,, e) is a monoid For all a, b, c B: a b c b a\c a c/b

14 Residuated Boolean algebras Definition A residuated Boolean algebra is a Boolean algebra B with three binary operations, \, / and a constant e B such that (B,, e) is a monoid For all a, b, c B: a b c b a\c For the set P(Σ ) of languages over Σ: e = {ε} a c/b K L = {uv u K, v L} K\L = {u Σ v K(vu L)} K/L = {u Σ v L(uv K)}

15 Residuation ideals Definition Let B be a residuated Boolean algebra. A Boolean subalgebra A B is called a residuation ideal if, for each a A and b B, we have b\a A and a/b A. Under duality, the residuation operations \ and / on a residuation ideal A correspond to a ternary relation on the set of atoms of A.

16 Applying the duality to regular languages Let L be a regular language. Let B(L) be the Boolean algebra generated by {x\l/y x, y Σ }. Then B(L) is a residuation ideal in P(Σ ). Theorem (Gehrke, Grigorieff, Pin) The relation dual to the residuation operations on B(L) is functional. Denote the corresponding function by. The function is a monoid operation on Atoms(B(L)). The set Atoms(B(L)) together with the operation is the syntactic monoid of L.

17 Summary Each formal language has an associated syntactic monoid. Properties of the syntactic monoid can be used to characterize properties of classes of languages. If L is a regular language, then the dual of B(L) is the syntactic monoid of L.