36,194 research outputs found
Combining Spatial and Temporal Logics: Expressiveness vs. Complexity
In this paper, we construct and investigate a hierarchy of spatio-temporal
formalisms that result from various combinations of propositional spatial and
temporal logics such as the propositional temporal logic PTL, the spatial
logics RCC-8, BRCC-8, S4u and their fragments. The obtained results give a
clear picture of the trade-off between expressiveness and computational
realisability within the hierarchy. We demonstrate how different combining
principles as well as spatial and temporal primitives can produce NP-, PSPACE-,
EXPSPACE-, 2EXPSPACE-complete, and even undecidable spatio-temporal logics out
of components that are at most NP- or PSPACE-complete
Resource Bounded Unprovability of Computational Lower Bounds
This paper introduces new notions of asymptotic proofs,
PT(polynomial-time)-extensions, PTM(polynomial-time Turing
machine)-omega-consistency, etc. on formal theories of arithmetic including PA
(Peano Arithmetic). This paper shows that P not= NP (more generally, any
super-polynomial-time lower bound in PSPACE) is unprovable in a
PTM-omega-consistent theory T, where T is a consistent PT-extension of PA. This
result gives a unified view to the existing two major negative results on
proving P not= NP, Natural Proofs and relativizable proofs, through the two
manners of characterization of PTM-omega-consistency. We also show that the
PTM-omega-consistency of T cannot be proven in any PTM-omega-consistent theory
S, where S is a consistent PT-extension of T.Comment: 78 page
Propositional computability logic I
In the same sense as classical logic is a formal theory of truth, the
recently initiated approach called computability logic is a formal theory of
computability. It understands (interactive) computational problems as games
played by a machine against the environment, their computability as existence
of a machine that always wins the game, logical operators as operations on
computational problems, and validity of a logical formula as being a scheme of
"always computable" problems. The present contribution gives a detailed
exposition of a soundness and completeness proof for an axiomatization of one
of the most basic fragments of computability logic. The logical vocabulary of
this fragment contains operators for the so called parallel and choice
operations, and its atoms represent elementary problems, i.e. predicates in the
standard sense. This article is self-contained as it explains all relevant
concepts. While not technically necessary, however, familiarity with the
foundational paper "Introduction to computability logic" [Annals of Pure and
Applied Logic 123 (2003), pp.1-99] would greatly help the reader in
understanding the philosophy, underlying motivations, potential and utility of
computability logic, -- the context that determines the value of the present
results. Online introduction to the subject is available at
http://www.cis.upenn.edu/~giorgi/cl.html and
http://www.csc.villanova.edu/~japaridz/CL/gsoll.html .Comment: To appear in ACM Transactions on Computational Logi
From truth to computability I
The recently initiated approach called computability logic is a formal theory
of interactive computation. See a comprehensive online source on the subject at
http://www.cis.upenn.edu/~giorgi/cl.html . The present paper contains a
soundness and completeness proof for the deductive system CL3 which axiomatizes
the most basic first-order fragment of computability logic called the
finite-depth, elementary-base fragment. Among the potential application areas
for this result are the theory of interactive computation, constructive applied
theories, knowledgebase systems, systems for resource-bound planning and
action. This paper is self-contained as it reintroduces all relevant
definitions as well as main motivations.Comment: To appear in Theoretical Computer Scienc
Internal Diffusion-Limited Aggregation: Parallel Algorithms and Complexity
The computational complexity of internal diffusion-limited aggregation (DLA)
is examined from both a theoretical and a practical point of view. We show that
for two or more dimensions, the problem of predicting the cluster from a given
set of paths is complete for the complexity class CC, the subset of P
characterized by circuits composed of comparator gates. CC-completeness is
believed to imply that, in the worst case, growing a cluster of size n requires
polynomial time in n even on a parallel computer.
A parallel relaxation algorithm is presented that uses the fact that clusters
are nearly spherical to guess the cluster from a given set of paths, and then
corrects defects in the guessed cluster through a non-local annihilation
process. The parallel running time of the relaxation algorithm for
two-dimensional internal DLA is studied by simulating it on a serial computer.
The numerical results are compatible with a running time that is either
polylogarithmic in n or a small power of n. Thus the computational resources
needed to grow large clusters are significantly less on average than the
worst-case analysis would suggest.
For a parallel machine with k processors, we show that random clusters in d
dimensions can be generated in O((n/k + log k) n^{2/d}) steps. This is a
significant speedup over explicit sequential simulation, which takes
O(n^{1+2/d}) time on average.
Finally, we show that in one dimension internal DLA can be predicted in O(log
n) parallel time, and so is in the complexity class NC
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