523 research outputs found
Phase Transition in Matched Formulas and a Heuristic for Biclique Satisfiability
A matched formula is a CNF formula whose incidence graph admits a matching
which matches a distinct variable to every clause. We study phase transition in
a context of matched formulas and their generalization of biclique satisfiable
formulas. We have performed experiments to find a phase transition of property
"being matched" with respect to the ratio where is the number of
clauses and is the number of variables of the input formula . We
compare the results of experiments to a theoretical lower bound which was shown
by Franco and Gelder (2003). Any matched formula is satisfiable, moreover, it
remains satisfiable even if we change polarities of any literal occurrences.
Szeider (2005) generalized matched formulas into two classes having the same
property -- var-satisfiable and biclique satisfiable formulas. A formula is
biclique satisfiable if its incidence graph admits covering by pairwise
disjoint bounded bicliques. Recognizing if a formula is biclique satisfiable is
NP-complete. In this paper we describe a heuristic algorithm for recognizing
whether a formula is biclique satisfiable and we evaluate it by experiments on
random formulas. We also describe an encoding of the problem of checking
whether a formula is biclique satisfiable into SAT and we use it to evaluate
the performance of our heuristicComment: Conference version submitted to SOFSEM 2018
(https://beda.dcs.fmph.uniba.sk/sofsem2019/) 18 pages(17 without refernces),
3 figures, 8 tables, an algorithm pseudocod
Fully Dynamic Matching in Bipartite Graphs
Maximum cardinality matching in bipartite graphs is an important and
well-studied problem. The fully dynamic version, in which edges are inserted
and deleted over time has also been the subject of much attention. Existing
algorithms for dynamic matching (in general graphs) seem to fall into two
groups: there are fast (mostly randomized) algorithms that do not achieve a
better than 2-approximation, and there slow algorithms with \O(\sqrt{m})
update time that achieve a better-than-2 approximation. Thus the obvious
question is whether we can design an algorithm -- deterministic or randomized
-- that achieves a tradeoff between these two: a approximation
and a better-than-2 approximation simultaneously. We answer this question in
the affirmative for bipartite graphs.
Our main result is a fully dynamic algorithm that maintains a 3/2 + \eps
approximation in worst-case update time O(m^{1/4}\eps^{/2.5}). We also give
stronger results for graphs whose arboricity is at most \al, achieving a (1+
\eps) approximation in worst-case time O(\al (\al + \log n)) for constant
\eps. When the arboricity is constant, this bound is and when the
arboricity is polylogarithmic the update time is also polylogarithmic.
The most important technical developement is the use of an intermediate graph
we call an edge degree constrained subgraph (EDCS). This graph places
constraints on the sum of the degrees of the endpoints of each edge: upper
bounds for matched edges and lower bounds for unmatched edges. The main
technical content of our paper involves showing both how to maintain an EDCS
dynamically and that and EDCS always contains a sufficiently large matching. We
also make use of graph orientations to help bound the amount of work done
during each update.Comment: Longer version of paper that appears in ICALP 201
Maximal Sharing in the Lambda Calculus with letrec
Increasing sharing in programs is desirable to compactify the code, and to
avoid duplication of reduction work at run-time, thereby speeding up execution.
We show how a maximal degree of sharing can be obtained for programs expressed
as terms in the lambda calculus with letrec. We introduce a notion of `maximal
compactness' for lambda-letrec-terms among all terms with the same infinite
unfolding. Instead of defined purely syntactically, this notion is based on a
graph semantics. lambda-letrec-terms are interpreted as first-order term graphs
so that unfolding equivalence between terms is preserved and reflected through
bisimilarity of the term graph interpretations. Compactness of the term graphs
can then be compared via functional bisimulation.
We describe practical and efficient methods for the following two problems:
transforming a lambda-letrec-term into a maximally compact form; and deciding
whether two lambda-letrec-terms are unfolding-equivalent. The transformation of
a lambda-letrec-term into maximally compact form proceeds in three
steps:
(i) translate L into its term graph ; (ii) compute the maximally
shared form of as its bisimulation collapse ; (iii) read back a
lambda-letrec-term from the term graph with the property . This guarantees that and have the same unfolding, and that
exhibits maximal sharing.
The procedure for deciding whether two given lambda-letrec-terms and
are unfolding-equivalent computes their term graph interpretations and , and checks whether these term graphs are bisimilar.
For illustration, we also provide a readily usable implementation.Comment: 18 pages, plus 19 pages appendi
Applications of Automata and Graphs: Labeling-Operators in Hilbert Space I
We show that certain representations of graphs by operators on Hilbert space
have uses in signal processing and in symbolic dynamics. Our main result is
that graphs built on automata have fractal characteristics. We make this
precise with the use of Representation Theory and of Spectral Theory of a
certain family of Hecke operators. Let G be a directed graph. We begin by
building the graph groupoid G induced by G, and representations of G. Our main
application is to the groupoids defined from automata. By assigning weights to
the edges of a fixed graph G, we give conditions for G to acquire fractal-like
properties, and hence we can have fractaloids or G-fractals. Our standing
assumption on G is that it is locally finite and connected, and our labeling of
G is determined by the "out-degrees of vertices". From our labeling, we arrive
at a family of Hecke-type operators whose spectrum is computed. As
applications, we are able to build representations by operators on Hilbert
spaces (including the Hecke operators); and we further show that automata built
on a finite alphabet generate fractaloids. Our Hecke-type operators, or
labeling operators, come from an amalgamated free probability construction, and
we compute the corresponding amalgamated free moments. We show that the free
moments are completely determined by certain scalar-valued functions.Comment: 69 page
Synchronization and Control in Intrinsic and Designed Computation: An Information-Theoretic Analysis of Competing Models of Stochastic Computation
We adapt tools from information theory to analyze how an observer comes to
synchronize with the hidden states of a finitary, stationary stochastic
process. We show that synchronization is determined by both the process's
internal organization and by an observer's model of it. We analyze these
components using the convergence of state-block and block-state entropies,
comparing them to the previously known convergence properties of the Shannon
block entropy. Along the way, we introduce a hierarchy of information
quantifiers as derivatives and integrals of these entropies, which parallels a
similar hierarchy introduced for block entropy. We also draw out the duality
between synchronization properties and a process's controllability. The tools
lead to a new classification of a process's alternative representations in
terms of minimality, synchronizability, and unifilarity.Comment: 25 pages, 13 figures, 1 tabl
Markov semigroups, monoids, and groups
A group is Markov if it admits a prefix-closed regular language of unique
representatives with respect to some generating set, and strongly Markov if it
admits such a language of unique minimal-length representatives over every
generating set. This paper considers the natural generalizations of these
concepts to semigroups and monoids. Two distinct potential generalizations to
monoids are shown to be equivalent. Various interesting examples are presented,
including an example of a non-Markov monoid that nevertheless admits a regular
language of unique representatives over any generating set. It is shown that
all finitely generated commutative semigroups are strongly Markov, but that
finitely generated subsemigroups of virtually abelian or polycyclic groups need
not be. Potential connections with word-hyperbolic semigroups are investigated.
A study is made of the interaction of the classes of Markov and strongly Markov
semigroups with direct products, free products, and finite-index subsemigroups
and extensions. Several questions are posed.Comment: 40 pages; 3 figure
Spin networks, quantum automata and link invariants
The spin network simulator model represents a bridge between (generalized)
circuit schemes for standard quantum computation and approaches based on
notions from Topological Quantum Field Theories (TQFT). More precisely, when
working with purely discrete unitary gates, the simulator is naturally modelled
as families of quantum automata which in turn represent discrete versions of
topological quantum computation models. Such a quantum combinatorial scheme,
which essentially encodes SU(2) Racah--Wigner algebra and its braided
counterpart, is particularly suitable to address problems in topology and group
theory and we discuss here a finite states--quantum automaton able to accept
the language of braid group in view of applications to the problem of
estimating link polynomials in Chern--Simons field theory.Comment: LateX,19 pages; to appear in the Proc. of "Constrained Dynamics and
Quantum Gravity (QG05), Cala Gonone (Italy) September 12-16 200
New results on pushdown module checking with imperfect information
Model checking of open pushdown systems (OPD) w.r.t. standard branching
temporal logics (pushdown module checking or PMC) has been recently
investigated in the literature, both in the context of environments with
perfect and imperfect information about the system (in the last case, the
environment has only a partial view of the system's control states and stack
content). For standard CTL, PMC with imperfect information is known to be
undecidable. If the stack content is assumed to be visible, then the problem is
decidable and 2EXPTIME-complete (matching the complexity of PMC with perfect
information against CTL). The decidability status of PMC with imperfect
information against CTL restricted to the case where the depth of the stack
content is visible is open. In this paper, we show that with this restriction,
PMC with imperfect information against CTL remains undecidable. On the other
hand, we individuate an interesting subclass of OPDS with visible stack content
depth such that PMC with imperfect information against the existential fragment
of CTL is decidable and in 2EXPTIME. Moreover, we show that the program
complexity of PMC with imperfect information and visible stack content against
CTL is 2EXPTIME-complete (hence, exponentially harder than the program
complexity of PMC with perfect information, which is known to be
EXPTIME-complete).Comment: In Proceedings GandALF 2011, arXiv:1106.081
Many Roads to Synchrony: Natural Time Scales and Their Algorithms
We consider two important time scales---the Markov and cryptic orders---that
monitor how an observer synchronizes to a finitary stochastic process. We show
how to compute these orders exactly and that they are most efficiently
calculated from the epsilon-machine, a process's minimal unifilar model.
Surprisingly, though the Markov order is a basic concept from stochastic
process theory, it is not a probabilistic property of a process. Rather, it is
a topological property and, moreover, it is not computable from any
finite-state model other than the epsilon-machine. Via an exhaustive survey, we
close by demonstrating that infinite Markov and infinite cryptic orders are a
dominant feature in the space of finite-memory processes. We draw out the roles
played in statistical mechanical spin systems by these two complementary length
scales.Comment: 17 pages, 16 figures:
http://cse.ucdavis.edu/~cmg/compmech/pubs/kro.htm. Santa Fe Institute Working
Paper 10-11-02
Inducing Probabilistic Grammars by Bayesian Model Merging
We describe a framework for inducing probabilistic grammars from corpora of
positive samples. First, samples are {\em incorporated} by adding ad-hoc rules
to a working grammar; subsequently, elements of the model (such as states or
nonterminals) are {\em merged} to achieve generalization and a more compact
representation. The choice of what to merge and when to stop is governed by the
Bayesian posterior probability of the grammar given the data, which formalizes
a trade-off between a close fit to the data and a default preference for
simpler models (`Occam's Razor'). The general scheme is illustrated using three
types of probabilistic grammars: Hidden Markov models, class-based -grams,
and stochastic context-free grammars.Comment: To appear in Grammatical Inference and Applications, Second
International Colloquium on Grammatical Inference; Springer Verlag, 1994. 13
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