109 research outputs found
Separating Regular Languages with First-Order Logic
Given two languages, a separator is a third language that contains the first
one and is disjoint from the second one. We investigate the following decision
problem: given two regular input languages of finite words, decide whether
there exists a first-order definable separator. We prove that in order to
answer this question, sufficient information can be extracted from semigroups
recognizing the input languages, using a fixpoint computation. This yields an
EXPTIME algorithm for checking first-order separability. Moreover, the
correctness proof of this algorithm yields a stronger result, namely a
description of a possible separator. Finally, we generalize this technique to
answer the same question for regular languages of infinite words
The Covering Problem
An important endeavor in computer science is to understand the expressive
power of logical formalisms over discrete structures, such as words. Naturally,
"understanding" is not a mathematical notion. This investigation requires
therefore a concrete objective to capture this understanding. In the
literature, the standard choice for this objective is the membership problem,
whose aim is to find a procedure deciding whether an input regular language can
be defined in the logic under investigation. This approach was cemented as the
right one by the seminal work of Sch\"utzenberger, McNaughton and Papert on
first-order logic and has been in use since then. However, membership questions
are hard: for several important fragments, researchers have failed in this
endeavor despite decades of investigation. In view of recent results on one of
the most famous open questions, namely the quantifier alternation hierarchy of
first-order logic, an explanation may be that membership is too restrictive as
a setting. These new results were indeed obtained by considering more general
problems than membership, taking advantage of the increased flexibility of the
enriched mathematical setting. This opens a promising research avenue and
efforts have been devoted at identifying and solving such problems for natural
fragments. Until now however, these problems have been ad hoc, most fragments
relying on a specific one. A unique new problem replacing membership as the
right one is still missing. The main contribution of this paper is a suitable
candidate to play this role: the Covering Problem. We motivate this problem
with 3 arguments. First, it admits an elementary set theoretic formulation,
similar to membership. Second, we are able to reexplain or generalize all known
results with this problem. Third, we develop a mathematical framework and a
methodology tailored to the investigation of this problem
Separation for dot-depth two
The dot-depth hierarchy of Brzozowski and Cohen classifies the star-free
languages of finite words. By a theorem of McNaughton and Papert, these are
also the first-order definable languages. The dot-depth rose to prominence
following the work of Thomas, who proved an exact correspondence with the
quantifier alternation hierarchy of first-order logic: each level in the
dot-depth hierarchy consists of all languages that can be defined with a
prescribed number of quantifier blocks. One of the most famous open problems in
automata theory is to settle whether the membership problem is decidable for
each level: is it possible to decide whether an input regular language belongs
to this level?
Despite a significant research effort, membership by itself has only been
solved for low levels. A recent breakthrough was achieved by replacing
membership with a more general problem: separation. Given two input languages,
one has to decide whether there exists a third language in the investigated
level containing the first language and disjoint from the second. The
motivation is that: (1) while more difficult, separation is more rewarding (2)
it provides a more convenient framework (3) all recent membership algorithms
are reductions to separation for lower levels.
We present a separation algorithm for dot-depth two. While this is our most
prominent application, our result is more general. We consider a family of
hierarchies that includes the dot-depth: concatenation hierarchies. They are
built via a generic construction process. One first chooses an initial class,
the basis, which is the lowest level in the hierarchy. Further levels are built
by applying generic operations. Our main theorem states that for any
concatenation hierarchy whose basis is finite, separation is decidable for
level one. In the special case of the dot-depth, this can be lifted to level
two using previously known results
The Covering Problem: A Unified Approach for Investigating the Expressive Power of Logics
An important endeavor in computer science is to precisely understand the expressive power of logical formalisms over discrete structures, such as words. Naturally, "understanding" is not a mathematical notion. Therefore, this investigation requires a concrete objective to capture such a notion. In the literature, the standard choice for this objective is the membership problem, whose aim is to find a procedure deciding whether an input regular language can be defined in the logic under study. This approach was cemented as the "right" one by the seminal work of Schuetzenberger, McNaughton and Papert on first-order logic and has been in use since then.
However, membership questions are hard: for several important fragments, researchers have failed in this endeavor despite decades of investigation. In view of recent results on one of the most famous open questions, namely the quantifier alternation hierarchy of first-order logic, an explanation may be that membership is too restrictive as a setting. These new results were indeed obtained by considering more general problems than membership, taking advantage of the increased flexibility of the enriched mathematical setting. This opens a promising avenue of research and efforts have been devoted at identifying and solving such problems for natural fragments. However, until now, these problems have been ad hoc, most fragments relying on a specific one. A unique new problem replacing membership as the right one is still missing.
The main contribution of this paper is a suitable candidate to play this role: the Covering Problem. We motivate this problem with three arguments. First, it admits an elementary set theoretic formulation, similar to membership. Second, we are able to reexplain or generalize all known results with this problem. Third, we develop a mathematical framework as well as a methodology tailored to the investigation of this problem
Separation and the Successor Relation
We investigate two problems for a class C of regular word languages. The
C-membership problem asks for an algorithm to decide whether an input language belongs to C. The C-separation problem asks for an algorithm that, given as input two regular languages, decides whether there exists a third language in C containing the first language, while being disjoint from the second. These problems are considered as means to obtain a deep understanding of the class C.
It is usual for such classes to be defined by logical formalisms. Logics are often built on top of each other, by adding new predicates. A natural construction is to enrich a logic with the successor relation. In this paper, we obtain new and simple proofs of two transfer results: we show that for suitable logically defined classes, the membership, resp. the separation problem for a class enriched with the successor relation reduces to the same problem for the original class.
Our reductions work both for languages of finite words and infinite words. The proofs are mostly self-contained, and only require a basic background on regular languages. This paper therefore gives simple proofs of results that were considered as difficult, such as the decidability of the membership problem for the levels 1, 3/2, 2 and 5/2 of the dot-depth hierarchy
A generic characterization of generalized unary temporal logic and two-variable first-order logic
We investigate an operator on classes of languages. For each class , it
outputs a new class associated with a variant of two-variable
first-order logic equipped with a signature built from . For , we get the variant equipped with the linear
order. For , we get the variant
, which also includes the successor. If consists of all Boolean
combinations of languages where is a letter, we get the variant
, which also includes ``between relations''. We prove a generic
algebraic characterization of the classes . It smoothly and
elegantly generalizes the known ones for all aforementioned cases. Moreover, it
implies that if has decidable separation (plus mild properties), then
has a decidable membership problem.
We actually work with an equivalent definition of \fodc in terms of unary
temporal logic. For each class , we consider a variant of unary
temporal logic whose future/past modalities depend on and such that . Finally, we also characterize and , the
pure-future and pure-past restrictions of . These characterizations as
well imply that if \Cs is a class with decidable separation, then and
have decidable membership
A Generic Polynomial Time Approach to Separation by First-Order Logic Without Quantifier Alternation
We look at classes of languages associated to fragments of first-order logic B??, in which quantifier alternations are disallowed. Each such fragment is fully determined by choosing the set of predicates on positions that may be used. Equipping first-order logic with the linear ordering and possibly the successor relation as predicates yields two natural fragments, which were investigated by Simon and Knast, who proved that these two variants have decidable membership: "does an input regular language belong to the class ?". We extend their results in two orthogonal directions.
- First, instead of membership, we explore the more general separation problem: decide if two regular languages can be separated by a language from the class under study.
- Second, we use more general inputs: classes ? of group languages (i.e., recognized by a DFA in which each letter induces a permutation of the states) and extensions thereof, written ?^+. We rely on a characterization of B?? by the operator BPol: given an input class ?, it outputs a class BPol(?) that corresponds to a variant of B?? equipped with special predicates associated to ?. The classes BPol(?) and BPol(?^+) capture many natural variants of B?? which use predicates such as the linear ordering, the successor, the modular predicates or the alphabetic modular predicates.
We show that separation is decidable for BPol(?) and BPol(?^+) when this is the case for ?. This was already known for BPol(?) and for two particular classes of the form BPol(?^+). Yet, the algorithms were indirect and relied on involved frameworks, yielding poor upper complexity bounds. In contrast, our approach is direct. We work only with elementary concepts (mainly, finite automata). Our main contribution consists in polynomial time Turing reductions from both BPol(?)- and BPol(?^+)-separation to ?-separation. This yields polynomial time algorithms for several key variants of B??, including those equipped with the linear ordering and possibly the successor and/or the modular predicates
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