137 research outputs found
On the strictness of the quantifier structure hierarchy in first-order logic
We study a natural hierarchy in first-order logic, namely the quantifier
structure hierarchy, which gives a systematic classification of first-order
formulas based on structural quantifier resource. We define a variant of
Ehrenfeucht-Fraisse games that characterizes quantifier classes and use it to
prove that this hierarchy is strict over finite structures, using strategy
compositions. Moreover, we prove that this hierarchy is strict even over
ordered finite structures, which is interesting in the context of descriptive
complexity.Comment: 38 pages, 8 figure
State Complexity of Reversals of Deterministic Finite Automata with Output
We investigate the worst-case state complexity of reversals of deterministic
finite automata with output (DFAOs). In these automata, each state is assigned
some output value, rather than simply being labelled final or non-final. This
directly generalizes the well-studied problem of determining the worst-case
state complexity of reversals of ordinary deterministic finite automata. If a
DFAO has states and possible output values, there is a known upper
bound of for the state complexity of reversal. We show this bound can be
reached with a ternary input alphabet. We conjecture it cannot be reached with
a binary input alphabet except when , and give a lower bound for the
case . We prove that the state complexity of reversal depends
solely on the transition monoid of the DFAO and the mapping that assigns output
values to states.Comment: 18 pages, 3 tables. Added missing affiliation/funding informatio
On the isomorphism conjecture for 2DFA reductions
The degree structure of complete sets under 2DFA reductions is investigated. It is shown that, for any class C that is closed under log-lin reductions: All complete sets for the class C under 2DFA reductions are also complete under one-one, length-increasing 2DFA reductions and are first-order isomorphic. The 2DFA-isomorphism conjecture is false, i.e., the complete sets under 2DFA reductions are not isomorphic to each other via 2DFA reductions
Verifying proofs in constant depth
In this paper we initiate the study of proof systems where verification of proofs proceeds by NC circuits. We investigate the question which languages admit proof systems in this very restricted model. Formulated alternatively, we ask which languages can be enumerated by NC functions. Our results show that the answer to this problem is not determined by the complexity of the language. On the one hand, we construct NC proof systems for a variety of languages ranging from regular to NP-complete. On the other hand, we show by combinatorial methods that even easy regular languages such as Exact-OR do not admit NC proof systems. We also present a general construction of proof systems for regular languages with strongly connected NFA's
For completeness, sublogarithmic space is no space
It is shown that for any class C closed under linear-time reductions, the complete sets for C under sublogarithmic reductions are also complete under 2DFA reductions, and thus are isomorphic under first-order reductions
A SURVEY OF LIMITED NONDETERMINISM IN COMPUTATIONAL COMPLEXITY THEORY
Nondeterminism is typically used as an inherent part of the computational models used incomputational complexity. However, much work has been done looking at nondeterminism asa separate resource added to deterministic machines. This survey examines several differentapproaches to limiting the amount of nondeterminism, including Kintala and Fischer\u27s βhierarchy, and Cai and Chen\u27s guess-and-check model
Regularization of Toda lattices by Hamiltonian reduction
The Toda lattice defined by the Hamiltonian with , which
exhibits singular (blowing up) solutions if some of the , can be
viewed as the reduced system following from a symmetry reduction of a subsystem
of the free particle moving on the group G=SL(n,\Real ). The subsystem is
, where consists of the determinant one matrices with
positive principal minors, and the reduction is based on the maximal nilpotent
group . Using the Bruhat decomposition we show that the full
reduced system obtained from , which is perfectly regular, contains
Toda lattices. More precisely, if is odd the reduced system
contains all the possible Toda lattices having different signs for the .
If is even, there exist two non-isomorphic reduced systems with different
constituent Toda lattices. The Toda lattices occupy non-intersecting open
submanifolds in the reduced phase space, wherein they are regularized by being
glued together. We find a model of the reduced phase space as a hypersurface in
{\Real}^{2n-1}. If for all , we prove for that the
Toda phase space associated with is a connected component of this
hypersurface. The generalization of the construction for the other simple Lie
groups is also presented.Comment: 42 pages, plain TeX, one reference added, to appear in J. Geom. Phy
05301 Abstracts Collection -- Exact Algorithms and Fixed-Parameter Tractability
From 24.07.05 to 29.07.05, the Dagstuhl Seminar 05301 ``Exact Algorithms and Fixed-Parameter Tractability\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl.
This is a collection of abstracts of the presentations given during the seminar
Homogeneity and Homogenizability: Hard Problems for the Logic SNP
We show that the question whether a given SNP sentence defines a
homogenizable class of finite structures is undecidable, even if the sentence
comes from the connected Datalog fragment and uses at most binary relation
symbols. As a byproduct of our proof, we also get the undecidability of some
other properties for Datalog programs, e.g., whether they can be rewritten in
MMSNP, whether they solve some finite-domain CSP, or whether they define the
age of a reduct of a homogeneous Ramsey structure in a finite relational
signature. We subsequently show that the closely related problem of testing the
amalgamation property for finitely bounded classes is EXPSPACE-hard or
PSPACE-hard, depending on whether the input is specified by a universal
sentence or a set of forbidden substructures.Comment: 34 pages, 3 figure
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