Decision problems for classes of semigroups and rational languages Marc Zeitoun LaBRI, Univ. Bordeaux 1, UMR CNRS 5800 Joint work with J. Almeida, J. C. Costa LaBRI, 2005-12-06 1/31 Marc Zeitoun Decision problems for classes of rational languages
Outline Motivations and context 1 Pseudovarieties of semigroups: decidability issues Equation systems and reducibility The case of R: proof outline 2/31 Marc Zeitoun Decision problems for classes of rational languages
Framework: classification of regular languages ◮ From logical definability/combinatorial properties to algebraic properties. High-level description Logics, e = ab ∗ , ϕ = a ∧ XG b of rational language L combinatorics. b a a Minimal automaton A min ( L ) 0 1 2 b a, b 0 1 2 { a, b, a 2 } Syntactic semigroup M ( L ) a 1 2 2 Algebra ab = a, ba = a 2 , (finite, [K56]) b 2 1 2 b 2 = b ◮ Is it possible to recover some properties of L from S ( L ) ? 3/31 Marc Zeitoun Decision problems for classes of rational languages
Semigroups ◮ Semigroup: ( S, · ) where · is associative. ◮ Examples: A + (free semigroup), A ∗ , square matrices,. . . ◮ Idempotent e ∈ S : e 2 = e . ◮ In a finite semigroup, s ω = unique idempotent of { s, s 2 , s 3 , . . . } . · · · · · · s n s 2 s k +1 s k s 4/31 Marc Zeitoun Decision problems for classes of rational languages
High-level vs. algebraic properties Logical Language L A min ( L ) M ( L ) definability Star free Counter free A FO ( < ) , LTL [Sch65,McN-P71,K68] � non ambiguous � FO 2 ( < ) , 2-way part. ord. DA f UTL , Σ 2 ∩ Π 2 [Sch76,SchThV01,ThW98,EVW97,PW97] Loc. threshold testable Forbidden patterns ACom ∗ LI FO ( ⋖ ) [ S85,ThW85,BP89] � left det. � Very weak R f [Ei74] Piecewise testable Very weak + . . . J Bool (Σ 1 ) [S75] 5/31 Marc Zeitoun Decision problems for classes of rational languages
Some hierarchies Membership Hierarchy of languages Hierarchy of semigroups decidability Straubing-Th´ erien V n . ? Brzozowski V n ∗ LI. ? (Product nesting/quantifier alternation) [PW95,P98] ( A ∗ G ) n ∗ A. Krohn-Rhodes complexity ? ( R ∗ MD 1 n ∗ D ) ρ . Until depth Yes [ThW96] DA ✷ MNB n ✷ LI Since-Until depth Yes [ThW02] 6/31 Marc Zeitoun Decision problems for classes of rational languages
Outline Motivations and context Pseudovarieties of semigroups: decidability issues 2 Equation systems and reducibility The case of R: proof outline 7/31 Marc Zeitoun Decision problems for classes of rational languages
Pseudovarieties ◮ In previous examples, properties can be tested on the syntactical semigroup. ◮ All corresponding classes of semigroups are closed by ◮ quotient, ◮ sub-semigroup, ◮ direct finite product. ◮ These closures properties define a pseudovariety. ◮ Eilenberg ’74: Varieties of languages ↔ pseudovarieties of semigroups. Summary ◮ Deciding combinatorial properties/logical definability of a rational language is frequently equivalent deciding the membership problem for a pseudovariety... ◮ ... obtained by combining smaller pseudovarieties using operators. ◮ Goal: to obtain tools to decide membership of such pseudovarieties. 8/31 Marc Zeitoun Decision problems for classes of rational languages
Pseudovarieties: some examples ◮ All (finite) semigroups: S. ◮ Commutative semigroups: Com = � xy = yx � . B = � x 2 = x � . ◮ Bands (idempotent semigroups): Sl = Com ∩ B = � x 2 = x, xy = yx � . ◮ Semilattices: A = � x ω = x ω +1 � . ◮ Groups-free semigroups: G = � x ω y = yx ω = y � . ◮ Groups: ◮ R -trivial semigroups: R = � ( xy ) ω x = ( xy ) ω � . J = � ( xy ) ω x = ( yx ) ω = y ( xy ) ω � . ◮ J -trivial semigroups: ◮ A pseudovariety is decidable if it has a decidable membership problem. ◮ Above pseudovarieties are trivially decidable. ◮ What about combinations through operators ( e.g. , join)? 9/31 Marc Zeitoun Decision problems for classes of rational languages
Decidability through operators ◮ Pseudovarieties arising from natural properties are usually decidable. ◮ Common operators do not preserve decidability [ABR92,R99]. Com ∨ V. ◮ Idea: strengthen decidability property to gain preservation through operators. ◮ Several successive attempts during the last decade. ◮ Tameness [AS98,A02] ◮ a property of “uniform resolution” of equation systems. ◮ a word problem. 10/31 Marc Zeitoun Decision problems for classes of rational languages
Decidability through operators ◮ Pseudovarieties arising from natural properties are usually decidable. ◮ Common operators do not preserve decidability [ABR92,R99]. Com ∨ V. ◮ Idea: strengthen decidability property to gain preservation through operators. ◮ Several successive attempts during the last decade. ◮ Tameness [AS98,A02] ◮ a property of “uniform resolution” of equation systems. ◮ a word problem. 10/31 Marc Zeitoun Decision problems for classes of rational languages
Pseudowords and pro- V topology = u = v ( u, v ∈ A + ) iff for all η : A + → S , η ( u ) = η ( v ) . ◮ S | ◮ u, v ∈ A + , d V ( u, v ) = 2 − r ( u,v ) where r ( u, v ) = min {| S | /S ∈ V , S �| = u = v } ◮ Fact 1 d V is a distance over F A V = A + / ∼ V . ◮ Fact 2 A + / ∼ V , d V is not complete: x n ! is a Cauchy sequence for any V. ◮ � F A V: completion of ( A + / ∼ V , d V ) is the topological semigroup of pseudowords. Examples F A N ≈ A + ∪ { 0 } . ◮ V = N = � x ω = 0 � , then � ◮ V = Sl = � xy = yx, x 2 = x � , then � F A Sl ≈ (2 A , ∪ ) . ◮ In general, � F A V noncountable (if A � = ∅ ). 11/31 Marc Zeitoun Decision problems for classes of rational languages
Definability by pseudoidentities ◮ A morphism ϕ : A + → S ∈ V has a unique continuous extension ϕ : � ˆ F A V → S . ◮ u, v ∈ � F A V. Define S | = u = v if ˆ ϕ ( u ) = ˆ ϕ ( v ) . ◮ This gives a precise definition of � Σ � . � Σ � = { S ∈ S | S | = Σ } . Theorem [Reiterman ’82] Pseudovarieties are (pseudo)-equational classes A class V of finite semigroups is a pseudovariety iff it is defined by pseudoidentities: ∃ Σ ⊆ � F A V × � F A V such that V = � Σ � . 12/31 Marc Zeitoun Decision problems for classes of rational languages
Outline Motivations and context Pseudovarieties of semigroups: decidability issues Equation systems and reducibility 3 The case of R: proof outline 13/31 Marc Zeitoun Decision problems for classes of rational languages
How systems of equations appear ◮ The Basis Theorem for semidirect products gives [ wu 2 = wu, wuv = wvu : V | Sl ∗ V = [ = wu = wv = w ] ] . ◮ To check that a finite semigroup S belongs to Sl ∗ V, it suffices to verify: u 2 � = ¯ If ¯ w, ¯ u, ¯ v ∈ S are such that ¯ w ¯ w ¯ u or ¯ w ¯ u ¯ v � = ¯ w ¯ v ¯ u , then there are no pseudowords w, u, v ∈ � F A S and evaluation of the generators A in S such that: 1. w, u, v are evaluated to ¯ w, ¯ u, ¯ v , respectively; 2. V | = wu = wv = w . Thus, we have the system of equations zx = zy = z upon whose variables x, y, z we impose constraints in the semigroup S . ◮ We want to decide whether there is a solution of the system modulo V. ◮ Equation systems also appear for other operators. Mal’cev products: [ u 2 = u, uv = vu : V | = u 2 = u = v ] Sl � m V = [ ] . 14/31 Marc Zeitoun Decision problems for classes of rational languages
Solving equations: the general problem Input A finite system of equations u i = v i ( i ∈ I ) over a finite set X of variables with constraints s x ( x ∈ X ) in a finite semigroup S . ◮ A mapping ϕ : X → � Output F A S (the solution modulo V), ◮ A continuous morphism ψ : � F A S → S such that 1. ∀ x ∈ X, ψ ( ϕ ( x )) = s x ; 2. ∀ i ∈ I, V | = ˆ ϕ ( u i ) = ˆ ϕ ( v i ) . The problem is to decide whether such a solution exists. 15/31 Marc Zeitoun Decision problems for classes of rational languages
Semi-algorithm for non-solvability ◮ If the system has a solution in � F A S modulo V then it also has a solution modulo any A -generated semigroup from V. ◮ By a compactness argument, the converse is also true. ◮ For a specific A -generated semigroup T from V, existence of solutions modulo T can be determined by checking a finite number of candidates. ◮ Semi-algorithm to enumerate non-solvable systems of equations: ◮ enumerate all ( A, X, ( u i = v i ) i ∈ I , S, ψ, ( s x ) x ∈ X , T ) , where ◮ A and X are finite sets, ◮ ( u i = v i ) i ∈ I is a system of word equations, ◮ S is an A -generated finite semigroup and ψ : A + → S ◮ s x is a constraint for variable x ∈ X , ◮ T is an A -generated finite semigroup from V. ◮ for each such tuple, test whether the system has a solution mod T . 16/31 Marc Zeitoun Decision problems for classes of rational languages
x ω and x ω − 1 ◮ By definition of d V , a sequence ( u n ) n converges in � F A V iff ∀ S ∈ V , ∃ N, p, q > N = ⇒ S | = u p = u q . F A V and for ϕ : A + → S ∈ V, ◮ For x ∈ � ϕ ( x n ! ) = ϕ ( x ) n ! = ϕ ( x ) ω for n > | S | . Hence the sequence ( x n ! ) n ∈ N converges in � F A V. ◮ The limit is the unique idempotent x ω of the closed subsemigroup � x � . ◮ Idem, x ω − 1 . ◮ Signature κ = { · , ω − 1 } ◮ F κ A : algebra of κ -terms, ◮ F κ A V: κ -semigroup induced by κ -terms. 17/31 Marc Zeitoun Decision problems for classes of rational languages
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