605 research outputs found
Connection Matrices and the Definability of Graph Parameters
In this paper we extend and prove in detail the Finite Rank Theorem for
connection matrices of graph parameters definable in Monadic Second Order Logic
with counting (CMSOL) from B. Godlin, T. Kotek and J.A. Makowsky (2008) and
J.A. Makowsky (2009). We demonstrate its vast applicability in simplifying
known and new non-definability results of graph properties and finding new
non-definability results for graph parameters. We also prove a Feferman-Vaught
Theorem for the logic CFOL, First Order Logic with the modular counting
quantifiers
Algebraic Methods in the Congested Clique
In this work, we use algebraic methods for studying distance computation and
subgraph detection tasks in the congested clique model. Specifically, we adapt
parallel matrix multiplication implementations to the congested clique,
obtaining an round matrix multiplication algorithm, where
is the exponent of matrix multiplication. In conjunction
with known techniques from centralised algorithmics, this gives significant
improvements over previous best upper bounds in the congested clique model. The
highlight results include:
-- triangle and 4-cycle counting in rounds, improving upon the
triangle detection algorithm of Dolev et al. [DISC 2012],
-- a -approximation of all-pairs shortest paths in
rounds, improving upon the -round -approximation algorithm of Nanongkai [STOC 2014], and
-- computing the girth in rounds, which is the first
non-trivial solution in this model.
In addition, we present a novel constant-round combinatorial algorithm for
detecting 4-cycles.Comment: This is work is a merger of arxiv:1412.2109 and arxiv:1412.266
Evaluating Datalog via Tree Automata and Cycluits
We investigate parameterizations of both database instances and queries that
make query evaluation fixed-parameter tractable in combined complexity. We show
that clique-frontier-guarded Datalog with stratified negation (CFG-Datalog)
enjoys bilinear-time evaluation on structures of bounded treewidth for programs
of bounded rule size. Such programs capture in particular conjunctive queries
with simplicial decompositions of bounded width, guarded negation fragment
queries of bounded CQ-rank, or two-way regular path queries. Our result is
shown by translating to alternating two-way automata, whose semantics is
defined via cyclic provenance circuits (cycluits) that can be tractably
evaluated.Comment: 56 pages, 63 references. Journal version of "Combined Tractability of
Query Evaluation via Tree Automata and Cycluits (Extended Version)" at
arXiv:1612.04203. Up to the stylesheet, page/environment numbering, and
possible minor publisher-induced changes, this is the exact content of the
journal paper that will appear in Theory of Computing Systems. Update wrt
version 1: latest reviewer feedbac
Quantum automata, braid group and link polynomials
The spin--network quantum simulator model, which essentially encodes the
(quantum deformed) SU(2) Racah--Wigner tensor algebra, is particularly suitable
to address problems arising in low dimensional topology and group theory. In
this combinatorial framework we implement families of finite--states and
discrete--time quantum automata capable of accepting the language generated by
the braid group, and whose transition amplitudes are colored Jones polynomials.
The automaton calculation of the polynomial of (the plat closure of) a link L
on 2N strands at any fixed root of unity is shown to be bounded from above by a
linear function of the number of crossings of the link, on the one hand, and
polynomially bounded in terms of the braid index 2N, on the other. The growth
rate of the time complexity function in terms of the integer k appearing in the
root of unity q can be estimated to be (polynomially) bounded by resorting to
the field theoretical background given by the Chern-Simons theory.Comment: Latex, 36 pages, 11 figure
Polynomial Invariants for Affine Programs
We exhibit an algorithm to compute the strongest polynomial (or algebraic)
invariants that hold at each location of a given affine program (i.e., a
program having only non-deterministic (as opposed to conditional) branching and
all of whose assignments are given by affine expressions). Our main tool is an
algebraic result of independent interest: given a finite set of rational square
matrices of the same dimension, we show how to compute the Zariski closure of
the semigroup that they generate
Colored operads, series on colored operads, and combinatorial generating systems
We introduce bud generating systems, which are used for combinatorial
generation. They specify sets of various kinds of combinatorial objects, called
languages. They can emulate context-free grammars, regular tree grammars, and
synchronous grammars, allowing us to work with all these generating systems in
a unified way. The theory of bud generating systems uses colored operads.
Indeed, an object is generated by a bud generating system if it satisfies a
certain equation in a colored operad. To compute the generating series of the
languages of bud generating systems, we introduce formal power series on
colored operads and several operations on these. Series on colored operads are
crucial to express the languages specified by bud generating systems and allow
us to enumerate combinatorial objects with respect to some statistics. Some
examples of bud generating systems are constructed; in particular to specify
some sorts of balanced trees and to obtain recursive formulas enumerating
these.Comment: 48 page
Weighted Automata and Monadic Second Order Logic
Let S be a commutative semiring. M. Droste and P. Gastin have introduced in
2005 weighted monadic second order logic WMSOL with weights in S. They use a
syntactic fragment RMSOL of WMSOL to characterize word functions (power series)
recognizable by weighted automata, where the semantics of quantifiers is used
both as arithmetical operations and, in the boolean case, as quantification.
Already in 2001, B. Courcelle, J.Makowsky and U. Rotics have introduced a
formalism for graph parameters definable in Monadic Second order Logic, here
called MSOLEVAL with values in a ring R. Their framework can be easily adapted
to semirings S. This formalism clearly separates the logical part from the
arithmetical part and also applies to word functions.
In this paper we give two proofs that RMSOL and MSOLEVAL with values in S
have the same expressive power over words. One proof shows directly that
MSOLEVAL captures the functions recognizable by weighted automata. The other
proof shows how to translate the formalisms from one into the other.Comment: In Proceedings GandALF 2013, arXiv:1307.416
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