94 research outputs found
The Expressive Power of CSP-Quantifiers
A generalized quantifier QK is called a CSP-quantifier if its defining class K consists of all structures that can be homomorphically mapped to a fixed finite template structure. For all positive integers n ≥ 2 and k, we define a pebble game that characterizes equivalence of structures with respect to the logic Lk∞ω(CSP+n ), where CSP+n is the union of the class Q1 of all unary quantifiers and the class CSPn of all CSP-quantifiers with template structures that have at most n elements. Using these games we prove that for every n ≥ 2 there exists a CSP-quantifier with template of size n + 1 which is not definable in Lω∞ω(CSP+n ). The proof of this result is based on a new variation of the well-known Cai-Fürer-Immerman construction.publishedVersionPeer reviewe
Beyond Hypertree Width: Decomposition Methods Without Decompositions
The general intractability of the constraint satisfaction problem has
motivated the study of restrictions on this problem that permit polynomial-time
solvability. One major line of work has focused on structural restrictions,
which arise from restricting the interaction among constraint scopes. In this
paper, we engage in a mathematical investigation of generalized hypertree
width, a structural measure that has up to recently eluded study. We obtain a
number of computational results, including a simple proof of the tractability
of CSP instances having bounded generalized hypertree width
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]
Dependence Logic with Generalized Quantifiers: Axiomatizations
We prove two completeness results, one for the extension of dependence logic
by a monotone generalized quantifier Q with weak interpretation, weak in the
meaning that the interpretation of Q varies with the structures. The second
result considers the extension of dependence logic where Q is interpreted as
"there exists uncountable many." Both of the axiomatizations are shown to be
sound and complete for FO(Q) consequences.Comment: 17 page
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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