From algebra to logic: there and back again -- the story of a hierarchy
aa r X i v : . [ c s . F L ] J un From algebra to logic: There and back againThe story of a hierarchy ⋆ Pascal Weil , CNRS, LaBRI, UMR 5800, F-33400 Talence, France, [email protected] Univ. Bordeaux, LaBRI, UMR 5800, F-33400 Talence, France.
Formal language theory historically arose from the definition of models ofcomputation (automata, grammars, etc) and relied for its first step on com-binatorial reasoning, especially combinatorics on words. Very quickly however,algebra and logic were identified as powerful tools for the classification of ra-tional languages, e.g. with the definition of the syntactic monoid of a languageand B¨uchi’s theorem on monadic second-order logic. It didn’t take much timeafter that to observe that, conversely, formal language theory is itself a tool foralgebra and logic.The results which we will present are an illustration of this back-and-forthmovement between languages, algebra and logic. They deal with a hierarchy ofclasses of rational languages which arises in different contexts and turned outto solve a problem in logic, namely the decidability of the quantifier alternationhierarchy within the two-variable fragment of first-order logic FO [ < ].The full picture uses a collection of results in logic, combinatorics on wordsand algebra which were obtained independently of the quantifier alternationhierarchy by various authors over several decades.Let R be the class of piecewise testable languages, which is natural from acombinatorial and automata-theoretic point of view, and corresponds to the firstlevel of the quantifier alternation hierarchy within FO [ < ] (and within FO [ < ])as well). This class is rather simple and reasonably well understood, see [10,1].We first consider the hierarchies of classes of languages obtained from R byalternatingly closing it under deterministic and co-deterministic closure: we let L = R , R k +1 (resp. L k +1 ) be the deterministic (resp. co-deterministic) closureof L k (resp. R k ).Results from the 1970s and 1980s [12,9] show that the classes R k and L k are varieties (whether a language L belongs to one of these classes dependsonly on its syntactic monoid) and describe the corresponding varieties of finitemonoids R k and L k . Results from the 1960s [5] (see also [17,11,6] shows thattheir membership problems are decidable and they form an infinite hierarchy.A first view of the structure of the lattice formed by these varieties canbe obtained by using purely algebraic results from the 1970s on a seeminglydifferent hierarchy, that of varieties of idempotent monoids [2]. The theory ofthe latter varieties is particularly well understood, and one can exhibit for eachof them structurally elegant identities and solutions of the word problem (of thecorresponding relatively free object) [3]. ⋆ This work was partially supported by the ANR through ANR-2010-BLAN-0204 o completely elucidate the structure of the lattice generated by the R k and L k , Kufleitner and Weil introduced the notion of condensed rankers [8].These are a rather natural extension of the algorithm to solve the word problemin the relatively free idempotent monoids and have natural connections withdeterministic and codeterministic products. But they are also – and foremost– a variant of the rankers introduced by Weiss and Immerman [18] (followingthe turtle programs of Schwentick, Th´erien and Vollmer [13]) to characterize thelevels of the quantifier alternation hierarchy of FO [ < ]. As a result one can showthat the k -th level of this hierarchy coincides with the intersection R k +1 ∩ L k +1 ,thus proving the decidability of each level of the hierarchy [7].The story does not end there: using algebraic methods similar to those de-scribed in his book [15], Straubing showed [16] that the k -th level of the quan-tifier alternation hierarchy of FO [ < ] is the variety of languages whose syntacticmonoid is in the k -th term of the sequence given by V = J and V n +1 = V n ⊓⊔ J .Here J is the class of J -trivial monoids, which characterizes piecewiste testablelanguages by Simon’s theorem [14] and ⊓⊔ denotes the two-sided block product,the bilateral version of the more classical wreath product. Then Straubing andKrebs showed that every class of finite monoids is decidable [4], thus providingan alternate proof of the decidability of the quantifier alternation hierarchy, butalso giving an alternative characterization of the classes V k : a finite monoid M is in V k if and only if it sits in both R k +1 and L k +1 .The coincidence of these two very differently defined hierarchies raises an in-triguing question: what connects the block product with the alternate operationof deterministic and co-deterministic closure?. . . References
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