191,275 research outputs found
Expansion Trees with Cut
Herbrand's theorem is one of the most fundamental insights in logic. From the
syntactic point of view it suggests a compact representation of proofs in
classical first- and higher-order logic by recording the information which
instances have been chosen for which quantifiers, known in the literature as
expansion trees.
Such a representation is inherently analytic and hence corresponds to a
cut-free sequent calculus proof. Recently several extensions of such proof
representations to proofs with cut have been proposed. These extensions are
based on graphical formalisms similar to proof nets and are limited to prenex
formulas.
In this paper we present a new approach that directly extends expansion trees
by cuts and covers also non-prenex formulas. We describe a cut-elimination
procedure for our expansion trees with cut that is based on the natural
reduction steps. We prove that it is weakly normalizing using methods from the
epsilon-calculus
Space Complexity of Perfect Matching in Bounded Genus Bipartite Graphs
We investigate the space complexity of certain perfect matching problems over
bipartite graphs embedded on surfaces of constant genus (orientable or
non-orientable). We show that the problems of deciding whether such graphs have
(1) a perfect matching or not and (2) a unique perfect matching or not, are in
the logspace complexity class \SPL. Since \SPL\ is contained in the logspace
counting classes \oplus\L (in fact in \modk\ for all ), \CeqL, and
\PL, our upper bound places the above-mentioned matching problems in these
counting classes as well. We also show that the search version, computing a
perfect matching, for this class of graphs is in \FL^{\SPL}. Our results
extend the same upper bounds for these problems over bipartite planar graphs
known earlier. As our main technical result, we design a logspace computable
and polynomially bounded weight function which isolates a minimum weight
perfect matching in bipartite graphs embedded on surfaces of constant genus. We
use results from algebraic topology for proving the correctness of the weight
function.Comment: 23 pages, 13 figure
On the relative proof complexity of deep inference via atomic flows
We consider the proof complexity of the minimal complete fragment, KS, of
standard deep inference systems for propositional logic. To examine the size of
proofs we employ atomic flows, diagrams that trace structural changes through a
proof but ignore logical information. As results we obtain a polynomial
simulation of versions of Resolution, along with some extensions. We also show
that these systems, as well as bounded-depth Frege systems, cannot polynomially
simulate KS, by giving polynomial-size proofs of certain variants of the
propositional pigeonhole principle in KS.Comment: 27 pages, 2 figures, full version of conference pape
Proof Theory of Finite-valued Logics
The proof theory of many-valued systems has not been investigated to an extent comparable to the work done on axiomatizatbility of many-valued logics. Proof theory requires appropriate formalisms, such as sequent calculus, natural deduction, and tableaux for classical (and intuitionistic) logic. One particular method for systematically obtaining calculi for all finite-valued logics was invented independently by several researchers, with slight variations in design and presentation. The main aim of this report is to develop the proof theory of finite-valued first order logics in a general way, and to present some of the more important results in this area. In Systems covered are the resolution calculus, sequent calculus, tableaux, and natural deduction. This report is actually a template, from which all results can be specialized to particular logics
Criteria for virtual fibering
We prove that an irreducible 3-manifold whose fundamental group satisfies a
certain group-theoretic property called RFRS is virtually fibered. As a
corollary, we show that 3-dimensional reflection orbifolds and arithmetic
hyperbolic orbifolds defined by a quadratic form virtually fiber. These include
the Seifert Weber dodecahedral space and the Bianchi orbifolds. Moreover, we
show that a taut sutured compression body has a finite-sheeted cover with a
depth one taut-oriented foliation.Comment: 17 pages, 3 figures; new theorem 7.2; to appear in Journal of
Topolog
A new Lenstra-type Algorithm for Quasiconvex Polynomial Integer Minimization with Complexity 2^O(n log n)
We study the integer minimization of a quasiconvex polynomial with
quasiconvex polynomial constraints. We propose a new algorithm that is an
improvement upon the best known algorithm due to Heinz (Journal of Complexity,
2005). This improvement is achieved by applying a new modern Lenstra-type
algorithm, finding optimal ellipsoid roundings, and considering sparse
encodings of polynomials. For the bounded case, our algorithm attains a
time-complexity of s (r l M d)^{O(1)} 2^{2n log_2(n) + O(n)} when M is a bound
on the number of monomials in each polynomial and r is the binary encoding
length of a bound on the feasible region. In the general case, s l^{O(1)}
d^{O(n)} 2^{2n log_2(n) +O(n)}. In each we assume d>= 2 is a bound on the total
degree of the polynomials and l bounds the maximum binary encoding size of the
input.Comment: 28 pages, 10 figure
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