131 research outputs found
Upper and lower bounds for first order expressibility
AbstractWe study first order expressibility as a measure of complexity. We introduce the new class Var&Sz[v(n),z(n)] of languages expressible by a uniform sequence of sentences with v(n) variables and size O[z(n)]. When v(n) is constant our uniformity condition is syntactical and thus the following characterizations of P and PSPACE come entirely from logic. NSPACE|log n|⊆⋃k=1,2,…Var&Sz|k, log(n)|⊆DSPACE|log2(n)|,P=⋃k=1,2,…Var&Sz|k, nk|,PSPACE=⋃k=1,2,…Var&Sz|k, 2nk|. The above means, for example, that the properties expressible with constantly many variables in polynomial size sentences are just the polynomial time recognizable properties. These results hold for languages with an ordering relation, e.g., for graphs the vertices are numbered. We introduce an “alternating pebbling game” to prove lower bounds on the number of variables and size needed to express properties without the ordering. We show, for example, that k variables are needed to express Clique(k), suggesting that this problem requires DTIME[nk]
On tractability and congruence distributivity
Constraint languages that arise from finite algebras have recently been the
object of study, especially in connection with the Dichotomy Conjecture of
Feder and Vardi. An important class of algebras are those that generate
congruence distributive varieties and included among this class are lattices,
and more generally, those algebras that have near-unanimity term operations. An
algebra will generate a congruence distributive variety if and only if it has a
sequence of ternary term operations, called Jonsson terms, that satisfy certain
equations.
We prove that constraint languages consisting of relations that are invariant
under a short sequence of Jonsson terms are tractable by showing that such
languages have bounded relational width
An Algebraic Preservation Theorem for Aleph-Zero Categorical Quantified Constraint Satisfaction
We prove an algebraic preservation theorem for positive Horn definability in
aleph-zero categorical structures. In particular, we define and study a
construction which we call the periodic power of a structure, and define a
periomorphism of a structure to be a homomorphism from the periodic power of
the structure to the structure itself. Our preservation theorem states that,
over an aleph-zero categorical structure, a relation is positive Horn definable
if and only if it is preserved by all periomorphisms of the structure. We give
applications of this theorem, including a new proof of the known complexity
classification of quantified constraint satisfaction on equality templates
A note on the expressive power of linear orders
This article shows that there exist two particular linear orders such that
first-order logic with these two linear orders has the same expressive power as
first-order logic with the Bit-predicate FO(Bit). As a corollary we obtain that
there also exists a built-in permutation such that first-order logic with a
linear order and this permutation is as expressive as FO(Bit)
Structure Theorem and Strict Alternation Hierarchy for FO^2 on Words
It is well-known that every first-order property on words is expressible
using at most three variables. The subclass of properties expressible with only
two variables is also quite interesting and well-studied. We prove precise
structure theorems that characterize the exact expressive power of first-order
logic with two variables on words. Our results apply to both the case with and
without a successor relation. For both languages, our structure theorems show
exactly what is expressible using a given quantifier depth, n, and using m
blocks of alternating quantifiers, for any m \leq n. Using these
characterizations, we prove, among other results, that there is a strict
hierarchy of alternating quantifiers for both languages. The question whether
there was such a hierarchy had been completely open. As another consequence of
our structural results, we show that satisfiability for first-order logic with
two variables without successor, which is NEXP-complete in general, becomes
NP-complete once we only consider alphabets of a bounded size
Lower Bounds for Existential Pebble Games and k-Consistency Tests
The existential k-pebble game characterizes the expressive power of the
existential-positive k-variable fragment of first-order logic on finite
structures. The winner of the existential k-pebble game on two given finite
structures can be determined in time O(n2k) by dynamic programming on the graph
of game configurations. We show that there is no O(n(k-3)/12)-time algorithm
that decides which player can win the existential k-pebble game on two given
structures. This lower bound is unconditional and does not rely on any
complexity-theoretic assumptions. Establishing strong k-consistency is a
well-known heuristic for solving the constraint satisfaction problem (CSP). By
the game characterization of Kolaitis and Vardi our result implies that there
is no O(n(k-3)/12)-time algorithm that decides if strong k-consistency can be
established for a given CSP-instance
Structure Theorem and Strict Alternation Hierarchy for FO² on Words
It is well-known that every first-order property on words is
expressible using at most three variables. The subclass of properties
expressible with only two variables is also quite interesting and
well-studied. We prove precise structure
theorems that characterize the exact expressive power of first-order
logic with two variables on words. Our results apply to
FO and FO, the latter of which includes the
binary successor relation in addition to the linear ordering on
string positions.
For both languages, our structure theorems show exactly what is
expressible using a given quantifier depth, , and using blocks
of alternating quantifiers, for any . Using these
characterizations, we prove, among other results, that there is a
strict hierarchy of alternating quantifiers for both languages. The
question whether there was such a hierarchy had been completely open
since it was asked in [Etessami, Vardi, and Wilke 1997]
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