Lights and Darks of the StarFree Star


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1 Lights and Darks of the StarFree Star Edward Ochmański & Krystyna Stawikowska Nicolaus Copernicus University, Toruń, Poland
2 Introduction: star may destroy recognizability In (finitely generated) trace monoids we have: singletons are in REC REC is closed under union REC is closed under catenation REC is included in RAT Corollary: If a rational language is not recognizable, recognizability was broken by a star operation.
3 History: questions on recognizable star Rational expressions are often more convenient (e.g. for inductive proofs) than the automataoriented or algebraic descriptions of recognizable languages Definition: Recognizable star is the classical star, working only if its result is recognizable (and undefined otherwise) Questions: Does recognizable star produce (jointly with union and catenation) the whole class REC?
4 History: recognizable stars in free and trace monoids Questions: Does recognizable star produce (jointly with union and catenation) the whole class REC? Answer 1 (Kleene): In finitely generated free monoid YES. Any star keeps recognizability! Answer 2: In arbitrary trace monoid YES. The connected star keeps recognizability and builds the whole REC.
5 Monoids, languages, free monoids Monoid (M, ) a set M with an associative operation (called product) and a neutral element ε (called unit). Language in (M, ) any subset of M. Atomic languages (atoms) empty language and singletons {m}, for m M. Free Monoid (A*, ) set of all finite words (including ε) over an alphabet A, with concatenation.
6 Operations on languages Settheoretical operations: union X Y intersection X Y difference X \ Y complement X = M \ X Algebraic operations: product: X Y = { xy x X, y Y} power: X 0 = {ε}, X n+1 = XX n star: X* = U{X n n=0,1,...}
7 Rational and starfree languages Rational languages: languages built up from atoms with operations of union, product and star. Rational expressions: expressions describing such construction. RAT(M): the class of rational subsets of M. Starfree languages: languages built up from atoms with operations of union, product and complement. Starfree expressions: expressions describing such construction. SF(M): the class of starfree subsets of M.
8 Syntactic monoid, recognizable and aperiodic languages syntactic congruence L M M of a language L M: u L v iff ( x,y M) xuy L xvy L syntactic monoid of a language L M: A language L M is: quotient monoid M L =M/ L recognizable (REC) iff its syntactic monoid M L is finite; aperiodic (AP) iff its syntactic monoid M L is finite and. aperiodic, i.e. ( n)( x M L ) x n =x n+1.
9 StarFree = Aperiodic (in A*) Theorem (Schützenberger 1965): In finitely generated free monoids SF(A*) = AP(A*) i.e. a language is starfree iff it is aperiodic Example: The language (aa)* has syntactic monoid: ε a ε ε a It is not aperiodic, a a ε as a n a n+1 for any n. By Schützenberger Theorem, (aa)* is not starfree.
10 Example: M={a,b}*, L=(ab)* The language (ab)* is starfree (ab)* = (b a aa bb ) Syntactic monoid of (ab)* ε a b ab ba 0 ε ε a b ab ba 0 a a 0 ab 0 a 0 b b ba 0 b 0 0 ab ab a 0 ab 0 0 ba ba 0 b 0 ba a 2 = a 3 = 0 b 2 = b 3 = 0 (ab) 2 = (ab) 3 = ab (ba) 2 = (ba) 3 = ba ε 2 = ε 3 = ε 0 2 = 0 3 = 0 n = 2 Hence (ab)* is aperiodic
11 Operation of STARFREE STAR Starfree star operation: L = L* if L* is starfree undefined otherwise SFSexpression: a rational expression, built from atoms with symbols of union, product and starfree star SFSlanguage: a language built up from atoms with operations of union, product and starfree star
12 Question: SFS = SF? Inclusion holds in any monoid, since union, product and starfree star preserve starfreeness. Inclusion is not general; there are monoids in which it does not hold. Example: (Z,+) integers with addition SF = { X Z X or X is finite } SFS = { X Z X=Z or X is finite }
13 Starfree star in free monoids Theorem (O/S 2005): In finitely generated free monoids SFS = SF Inclusion holds in any monoid. The proof of is based on McNaughton/Yamada (1960) construction of regular expressions for automata and uses the Schützenberger Theorem.
14 Manysided characterization of StarFree Trace Languages SF(A*)=SFS(A*)
15 Traces and trace languages Concurrent alphabet pair (A,I), where A is a finite alphabet, I A A is a symmetric and irreflexive independence relation; complement of I, the relation D=A A I, is called dependency. Trace monoid M=A*/ I quotient of A* by the least congruence containing the relation {ab ba aib}. Traces members of trace monoids. Trace languages subset of trace monoids. Free monoid trace monoid with I=.
16 Lemma on closing product Flattening of a trace language: UT = { w A* [w] T } Closure of a word language: L = U[L] = { w A* [w] [L] } L is closed iff L=L Lemat: If K,L A* are closed and starfree, then KL is starfree.
17 Flat characterization of starfree trace languages SF(A*)=SFS(A*) Lemma on closing product Schützenberger Theorem T SF(A*/I) iff UT SF(A*)
18 Manysided characterization of StarFree Trace Languages SF(A*)=SFS(A*) T SF(A*/I) iff UT SF(A*)
19 Aperiodic characterization of starfree trace languages Theorem (Guaiana/Restivo/Salemi 1992): In any trace monoid, T is starfree iff T is aperiodic. Proof: T AP(A*/ I ) T SF(A*/ I ) UT AP(A*) UT SF(A*)
20 Manysided characterization of StarFree Trace Languages SF(A*/I)=AP(A*/I) SF(A*)=SFS(A*) T SF(A*/I) iff UT SF(A*)
21 Logic for traces (Thomas 1989) (A,I) concurrent alphabet, w=a 1 a n A*. Model for the trace [w] A*/ I is the tracegraph V,E,λ where V={x 1 x n } vertices, E V V edges and λ:v A is s.t. ( i) λ(x i )=a i and x i Ex j iff i<j & a i Da j First order formula is build up from atomic formulas x=y, xey and λ(x)=a with logical connectives and quantifiers.
22 First order definability T(Ψ)={α A*/ I Ψ is satisfied in α} trace language defined by the sentence Ψ. Trace language T is firstorder definable iff there is a firstorder sentence Ψ s.t. T=T(Ψ). FO(A*/ I ) the class of firstorder definable languages in the trace monoid A*/ I. Theorem (McNaughton/Papert 1971): In finitely generated free monoids SF(A*)=FO(A*)
23 Lexicographic representation (A,<,I) ordered concurrent alphabet Lexicographic representative of trace α A*/ I a word Lex(α) A*, lexicographically first in α. Lex(T) = { Lex(α) A* α α T } lexicographic representation of the trace language T. LEX = Lex (A*/ I ) lexicographic representation of the trace monoid. LEX = A* U{A*b(I a )*aa* aib a<b}, where I a ={c A aic}, hence LEX is starfree.
24 Flat and Lex characterizations of firstorder trace languages Theorem (Ebinger/Muscholl 1993): For any trace language T, the following statements are equivalent: (1) T is firstorder definable in A*/ I (2) UT is firstorder definable in A* (3) Lex(T) is firstorder definable in A*
25 Manysided characterization of StarFree Trace Languages SF(A*/I)=AP(A*/I) SF(A*)=SFS(A*) T SF(A*/I) iff UT SF(A*) T FO(A*/I) ) iff UT FO(A*) iff Lex(T) FO(A*)
26 Logical characterization of starfree trace languages Theorem (Ebinger/Muschol 93, Diekert/Metivier 97): A trace language is starfree if and only if it is firstorder definable Proof. T SF(A*/ I ) iff UT SF(A*) iff UT FO(A*) iff T FO(A*/ I )
27 Manysided characterization of StarFree Trace Languages SF(A*/I)=AP(A*/I) SF(A*/I)=FO(A*/I) SF(A*)=SFS(A*) T SF(A*/I) iff UT SF(A*) T FO(A*/I) ) iff UT FO(A*) iff Lex(T) FO(A*)
28 Lexicographic characterization of starfree trace languages Theorem: A trace language is starfree in A*/ I if and only if its lexicographic representation is starfree in A* Proof. T SF(A*/ I ) iff T FO(A*/ I ) iff Lex(T) FO(A*) iff Lex(T) SF(A*)
29 Manysided characterization of StarFree Trace Languages SF(A*/I)=AP(A*/I) SF(A*/I)=FO(A*/I) SF(A*)=SFS(A*) T SF(A*/I) iff UT SF(A*) T FO(A*/I) ) iff UT FO(A*) iff Lex(T) FO(A*) T SF(A*/I) iff Lex(T) SF(A*)
30 Starfree star in trace monoids Lemma 1: If L SFS(A*) and L LEX, then [L] SFS(A*/ I ) Proof. Structural induction on SFSexpressions. Lemma 2: If UT SFS(A*), then T SFS(A*/ I ) Theorem: In trace monoids SFS = SF Proof. T SF(A*/ I ) UT SF(A*) UT SFS(A*) T SFS(A*/ I ) T SF(A*/ I )
31 Manysided characterization of StarFree Trace Languages SF(A*/I)=AP(A*/I) SF(A*/I)=FO(A*/I) SF(A*)=SFS(A*) T SF(A*/I) iff UT SF(A*) T FO(A*/I) ) iff UT FO(A*) iff Lex(T) FO(A*) SF(A*/I)=SFS(A*/I) T SF(A*/I) iff Lex(T) SF(A*)
32 Problems 13 Problem 1: To find syntactically formulated conditions for starfree star in trace monoids. Problem 2: In which monoids SFS=SF? Does it hold in concurrency monoids of Droste?
33 Decision Problems Theorem (Muscholl/Petersen 1996): The problem "Is T starfree?" is decidable for rational trace languages only in trace monoids with transitive independency. Problem 3 (star problem for sf trace languages): Is the question Is T* starfree? decidable for: starfree trace languages? finite trace languages?
34 A part without logic SF(A*/I)=AP(A*/I) SF(A*/I)=FO(A*/I) SF(A*)=SFS(A*) T SF(A*/I) iff UT SF(A*) T FO(A*/I) ) iff UT FO(A*) iff Lex(T) FO(A*) SF(A*/I)=SFS(A*/I) T SF(A*/I) iff Lex(T) SF(A*)
35 A part without logic SF(A*/I)=AP(A*/I) SF(A*/I)=FO(A*/I) SF(A*)=SFS(A*) T SF(A*/I) iff UT SF(A*) T FO(A*/I) ) iff UT FO(A*) iff Lex(T) FO(A*) SF(A*/I)=SFS(A*/I) T SF(A*/I) iff Lex(T) SF(A*)
36 Problem 4 avoiding logic Problem 4: To prove the nonlogic part of To prove the nonlogic part of the characterization without logic
37 LSF languages lexicographic product: X Y = XY if XY LEX else undefined lexicographic complement: X" = LEX X LSF = the class of word languages that are built from atoms with union, lexicographic product and lexicographic complement
38 Question: LSF = SF inside LEX? Lemma: If L LSF, then [L] SF(A*/ I ) Proof: By lemma on closing product. By definition: If L LSF, then L SF and L LEX Question: Does the converse hold?
39 Transitively oriented alphabets An ordered alphabet (A,<,I ) is transitively oriented iff < I is transitive. Lemma: If L SF and L LEX, then L LSF Proposition: LSF = {L A* L SF and L LEX} if (A,<,I ) is transitively oriented Problem 4a: Does the proposition hold for any ordered alphabet?
40 Example D: a c b LEX = (a*b*c)*a*b* = a*b*(ca*b*)* a* = LEX (LEX c LEX LEX b) b* = LEX (LEX c LEX a LEX) c* = LEX ((LEX LEX b) a LEX LEX b (LEX a LEX)) (a c)* = LEX (LEX b LEX bc LEX) a*b* = LEX LEX c LEX (ac)* = LEX (c LEX (LEX LEX b) a LEX cc LEX (LEX LEX b) aa LEX)
41 How is LSF in the Pentagon? Pentagon cannot be transitively oriented D: What about LSF in the pentagon? Does a* belong to LSF?
42 Publications Edward Ochmański, Krystyna Stawikowska: On Closures of Lexicographic StarFree Languages, Proceedings of AFL 2005, pp , University of Szeged, Edward Ochmański, Krystyna Stawikowska: StarFree Star and Trace Languages, Fundamenta Informaticae 72, pp , IOS Press Krystyna Stawikowska, Edward Ochmański: On StarFree Trace Languages and their Lexicographic Representations, Proceedings of LATA 2007, pp , Universitat Rovira i Virgili, Tarragona, Spain, Edward Ochmański, Krystyna Stawikowska: A Star Operation for StarFree Trace Languages, Proceedings of DLT 2007, LNCS 4588, pp , Springer Krystyna Stawikowska: Word Languages Inducing Recognizable and StarFree Trace Languages, PhD Thesis (in Polish), ToruńWarszawa, 2007.
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