8 research outputs found
Modulo Counting on Words and Trees
We consider the satisfiability problem for the two-variable fragment of the first-order logic extended with modulo counting quantifiers and interpreted over finite words or trees. We prove a small-model property of this logic, which gives a technique for deciding the satisfiability problem. In the case of words this gives a new proof of EXPSPACE upper bound, and in the case of trees it gives a 2EXPTIME algorithm. This algorithm is optimal: we prove a matching lower bound by a generic reduction from alternating Turing machines working in exponential space; the reduction involves a development of a new version of tiling games
Algebraic recognizability of regular tree languages
We propose a new algebraic framework to discuss and classify recognizable
tree languages, and to characterize interesting classes of such languages. Our
algebraic tool, called preclones, encompasses the classical notion of syntactic
Sigma-algebra or minimal tree automaton, but adds new expressivity to it. The
main result in this paper is a variety theorem \`{a} la Eilenberg, but we also
discuss important examples of logically defined classes of recognizable tree
languages, whose characterization and decidability was established in recent
papers (by Benedikt and S\'{e}goufin, and by Bojanczyk and Walukiewicz) and can
be naturally formulated in terms of pseudovarieties of preclones. Finally, this
paper constitutes the foundation for another paper by the same authors, where
first-order definable tree languages receive an algebraic characterization
Deciding definability in FO2(<h,<v) on trees
We provide a decidable characterization of regular forest languages definable
in FO2(<h,<v). By FO2(<h,<v) we refer to the two variable fragment of first
order logic built from the descendant relation and the following sibling
relation. In terms of expressive power it corresponds to a fragment of the
navigational core of XPath that contains modalities for going up to some
ancestor, down to some descendant, left to some preceding sibling, and right to
some following sibling. We also show that our techniques can be applied to
other two variable first-order logics having exactly the same vertical
modalities as FO2(<h,<v) but having different horizontal modalities
Proving that a Tree Language is not First-Order Definable
We explore from an algebraic viewpoint the properties of the tree languages
definable with a first-order formula involving the ancestor predicate, using
the description of these languages as those recognized by iterated block
products of forest algebras defined from finite counter monoids. Proofs of
nondefinability are infinite sequences of sets of forests, one for each level
of the hierarchy of quantification levels that defines the corresponding
variety of languages. The forests at a given level are built recursively by
inserting forests from previous level at the ports of a suitable set of
multicontexts. We show that a recursive proof exists for the syntactic algebra
of every non-definable language. We also investigate certain types of uniform
recursive proofs. For this purpose, we define from a forest algebra an algebra
of mappings and an extended algebra, which we also use to redefine the notion
of aperiodicity in a way that generalizes the existing ones
APERIODICITY, STAR-FREENESS, AND FIRST-ORDER LOGIC DEFINABILITY OF OPERATOR PRECEDENCE LANGUAGES
A classic result in formal language theory is the equivalence among non-counting, or aperiodic, regular languages, and languages defined through star-free regular expressions, or first-order logic. Past attempts to extend this result beyond the realm of regular languages have met with difficulties: for instance it is known that star-free tree languages may violate the non-counting property and there are aperiodic tree languages that cannot be defined through first-order logic. We extend such classic equivalence results to a significant family of deterministic context-free languages, the operator-precedence languages (OPL), which strictly includes the widely investigated visibly pushdown, alias input-driven, family and other structured context-free languages. The OP model originated in the ’60s for defining programming languages and is still used by high performance compilers; its rich algebraic properties have been investigated initially in connection with grammar learning and recently completed with further closure properties and with monadic second order logic definition. We introduce an extension of regular expressions, the OP-expressions (OPE) which define the OPLs and, under the star-free hypothesis, define first-order definable and non-counting OPLs. Then, we prove, through a fairly articulated grammar transformation, that aperiodic OPLs are first-order definable. Thus, the classic equivalence of star-freeness, aperiodicity, and first-order definability is established for the large and powerful class of OPLs. We argue that the same approach can be exploited to obtain analogous results for visibly pushdown languages too
Aperiodicity, Star-freeness, and First-order Logic Definability of Operator Precedence Languages
A classic result in formal language theory is the equivalence among
non-counting, or aperiodic, regular languages, and languages defined through
star-free regular expressions, or first-order logic. Past attempts to extend
this result beyond the realm of regular languages have met with difficulties:
for instance it is known that star-free tree languages may violate the
non-counting property and there are aperiodic tree languages that cannot be
defined through first-order logic. We extend such classic equivalence results
to a significant family of deterministic context-free languages, the
operator-precedence languages (OPL), which strictly includes the widely
investigated visibly pushdown, alias input-driven, family and other structured
context-free languages. The OP model originated in the '60s for defining
programming languages and is still used by high performance compilers; its rich
algebraic properties have been investigated initially in connection with
grammar learning and recently completed with further closure properties and
with monadic second order logic definition. We introduce an extension of
regular expressions, the OP-expressions (OPE) which define the OPLs and, under
the star-free hypothesis, define first-order definable and non-counting OPLs.
Then, we prove, through a fairly articulated grammar transformation, that
aperiodic OPLs are first-order definable. Thus, the classic equivalence of
star-freeness, aperiodicity, and first-order definability is established for
the large and powerful class of OPLs. We argue that the same approach can be
exploited to obtain analogous results for visibly pushdown languages too