30 research outputs found
On Second-Order Monadic Monoidal and Groupoidal Quantifiers
We study logics defined in terms of second-order monadic monoidal and
groupoidal quantifiers. These are generalized quantifiers defined by monoid and
groupoid word-problems, equivalently, by regular and context-free languages. We
give a computational classification of the expressive power of these logics
over strings with varying built-in predicates. In particular, we show that
ATIME(n) can be logically characterized in terms of second-order monadic
monoidal quantifiers
The descriptive complexity approach to LOGCFL
Building upon the known generalized-quantifier-based first-order
characterization of LOGCFL, we lay the groundwork for a deeper investigation.
Specifically, we examine subclasses of LOGCFL arising from varying the arity
and nesting of groupoidal quantifiers. Our work extends the elaborate theory
relating monoidal quantifiers to NC1 and its subclasses. In the absence of the
BIT predicate, we resolve the main issues: we show in particular that no single
outermost unary groupoidal quantifier with FO can capture all the context-free
languages, and we obtain the surprising result that a variant of Greibach's
``hardest context-free language'' is LOGCFL-complete under quantifier-free
BIT-free projections. We then prove that FO with unary groupoidal quantifiers
is strictly more expressive with the BIT predicate than without. Considering a
particular groupoidal quantifier, we prove that first-order logic with majority
of pairs is strictly more expressive than first-order with majority of
individuals. As a technical tool of independent interest, we define the notion
of an aperiodic nondeterministic finite automaton and prove that FO
translations are precisely the mappings computed by single-valued aperiodic
nondeterministic finite transducers.Comment: 10 pages, 1 figur
Axiomatizing proof tree concepts in Bounded Arithmetic
We construct theories of Cook-Nguyen style two-sort bounded arithmetic
whose provably total functions are exactly those in LOGCFL and LOGDCFL.
Axiomatizations of both theories are based on the proof tree size
characterizations of these classes. We also show that our theory for LOGCFL proves a certain formulation of the pumping lemma for context-free languages
Existential Second-Order Logic Over Graphs: A Complete Complexity-Theoretic Classification
Descriptive complexity theory aims at inferring a problem's computational
complexity from the syntactic complexity of its description. A cornerstone of
this theory is Fagin's Theorem, by which a graph property is expressible in
existential second-order logic (ESO logic) if, and only if, it is in NP. A
natural question, from the theory's point of view, is which syntactic fragments
of ESO logic also still characterize NP. Research on this question has
culminated in a dichotomy result by Gottlob, Kolatis, and Schwentick: for each
possible quantifier prefix of an ESO formula, the resulting prefix class either
contains an NP-complete problem or is contained in P. However, the exact
complexity of the prefix classes inside P remained elusive. In the present
paper, we clear up the picture by showing that for each prefix class of ESO
logic, its reduction closure under first-order reductions is either FO, L, NL,
or NP. For undirected, self-loop-free graphs two containment results are
especially challenging to prove: containment in L for the prefix and containment in FO for the prefix
for monadic . The complex argument by
Gottlob, Kolatis, and Schwentick concerning polynomial time needs to be
carefully reexamined and either combined with the logspace version of
Courcelle's Theorem or directly improved to first-order computations. A
different challenge is posed by formulas with the prefix : We show that they express special constraint satisfaction problems
that lie in L.Comment: Technical report version of a STACS 2015 pape
Logic Meets Algebra: the Case of Regular Languages
The study of finite automata and regular languages is a privileged meeting
point of algebra and logic. Since the work of Buchi, regular languages have
been classified according to their descriptive complexity, i.e. the type of
logical formalism required to define them. The algebraic point of view on
automata is an essential complement of this classification: by providing
alternative, algebraic characterizations for the classes, it often yields the
only opportunity for the design of algorithms that decide expressibility in
some logical fragment.
We survey the existing results relating the expressibility of regular
languages in logical fragments of MSO[S] with algebraic properties of their
minimal automata. In particular, we show that many of the best known results in
this area share the same underlying mechanics and rely on a very strong
relation between logical substitutions and block-products of pseudovarieties of
monoid. We also explain the impact of these connections on circuit complexity
theory.Comment: 37 page
Regular Representations of Uniform TC^0
The circuit complexity class DLOGTIME-uniform AC^0 is known to be a modest
subclass of DLOGTIME-uniform TC^0. The weakness of AC^0 is caused by the fact
that AC^0 is not closed under restricting AC^0-computable queries into simple
subsequences of the input. Analogously, in descriptive complexity, the logics
corresponding to DLOGTIME-uniform AC^0 do not have the relativization property
and hence they are not regular. This weakness of DLOGTIME-uniform AC^0 has been
elaborated in the line of research on the Crane Beach Conjecture. The
conjecture (which was refuted by Barrington, Immerman, Lautemann, Schweikardt
and Th{\'e}rien) was that if a language L has a neutral letter, then L can be
defined in first-order logic with the collection of all numerical built-in
relations, if and only if L can be already defined in FO with order.
In the first part of this article we consider logics in the range of AC^0 and
TC^0. First we formulate a combinatorial criterion for a cardinality quantifier
C_S implying that all languages in DLOGTIME-uniform TC^0 can be defined in
FO(C_S). For instance, this criterion is satisfied by C_S if S is the range of
some polynomial with positive integer coefficients of degree at least two. In
the second part of the paper we first adapt the key properties of abstract
logics to accommodate built-in relations. Then we define the regular interior
R-int(L) and regular closure R-cl(L), of a logic L, and show that the Crane
Beach Conjecture can be interpreted as a statement concerning the regular
interior of first-order logic with built-in relations B. We show that if B={+},
or B contains only unary relations besides the order, then R-int(FO_B)
collapses to FO with order. In contrast, our results imply that if B contains
the order and the range of a polynomial of degree at least two, then R-cl(FO_B)
includes all languages in DLOGTIME-uniform TC^0