2,454 research outputs found
The Connectivity of Boolean Satisfiability: Computational and Structural Dichotomies
Boolean satisfiability problems are an important benchmark for questions
about complexity, algorithms, heuristics and threshold phenomena. Recent work
on heuristics, and the satisfiability threshold has centered around the
structure and connectivity of the solution space. Motivated by this work, we
study structural and connectivity-related properties of the space of solutions
of Boolean satisfiability problems and establish various dichotomies in
Schaefer's framework.
On the structural side, we obtain dichotomies for the kinds of subgraphs of
the hypercube that can be induced by the solutions of Boolean formulas, as well
as for the diameter of the connected components of the solution space. On the
computational side, we establish dichotomy theorems for the complexity of the
connectivity and st-connectivity questions for the graph of solutions of
Boolean formulas. Our results assert that the intractable side of the
computational dichotomies is PSPACE-complete, while the tractable side - which
includes but is not limited to all problems with polynomial time algorithms for
satisfiability - is in P for the st-connectivity question, and in coNP for the
connectivity question. The diameter of components can be exponential for the
PSPACE-complete cases, whereas in all other cases it is linear; thus, small
diameter and tractability of the connectivity problems are remarkably aligned.
The crux of our results is an expressibility theorem showing that in the
tractable cases, the subgraphs induced by the solution space possess certain
good structural properties, whereas in the intractable cases, the subgraphs can
be arbitrary
The Connectivity of Boolean Satisfiability: No-Constants and Quantified Variants
For Boolean satisfiability problems, the structure of the solution space is
characterized by the solution graph, where the vertices are the solutions, and
two solutions are connected iff they differ in exactly one variable. Motivated
by research on heuristics and the satisfiability threshold, Gopalan et al. in
2006 studied connectivity properties of the solution graph and related
complexity issues for constraint satisfaction problems in Schaefer's framework.
They found dichotomies for the diameter of connected components and for the
complexity of the st-connectivity question, and conjectured a trichotomy for
the connectivity question that we recently were able to prove.
While Gopalan et al. considered CNF(S)-formulas with constants, we here look
at two important variants: CNF(S)-formulas without constants, and partially
quantified formulas. For the diameter and the st-connectivity question, we
prove dichotomies analogous to those of Gopalan et al. in these settings. While
we cannot give a complete classification for the connectivity problem yet, we
identify fragments where it is in P, where it is coNP-complete, and where it is
PSPACE-complete, in analogy to Gopalan et al.'s trichotomy.Comment: superseded by chapter 3 of arXiv:1510.0670
The Complexity of Rooted Phylogeny Problems
Several computational problems in phylogenetic reconstruction can be
formulated as restrictions of the following general problem: given a formula in
conjunctive normal form where the literals are rooted triples, is there a
rooted binary tree that satisfies the formula? If the formulas do not contain
disjunctions, the problem becomes the famous rooted triple consistency problem,
which can be solved in polynomial time by an algorithm of Aho, Sagiv,
Szymanski, and Ullman. If the clauses in the formulas are restricted to
disjunctions of negated triples, Ng, Steel, and Wormald showed that the problem
remains NP-complete. We systematically study the computational complexity of
the problem for all such restrictions of the clauses in the input formula. For
certain restricted disjunctions of triples we present an algorithm that has
sub-quadratic running time and is asymptotically as fast as the fastest known
algorithm for the rooted triple consistency problem. We also show that any
restriction of the general rooted phylogeny problem that does not fall into our
tractable class is NP-complete, using known results about the complexity of
Boolean constraint satisfaction problems. Finally, we present a pebble game
argument that shows that the rooted triple consistency problem (and also all
generalizations studied in this paper) cannot be solved by Datalog
The Connectivity of Boolean Satisfiability: Dichotomies for Formulas and Circuits
For Boolean satisfiability problems, the structure of the solution space is
characterized by the solution graph, where the vertices are the solutions, and
two solutions are connected iff they differ in exactly one variable. In 2006,
Gopalan et al. studied connectivity properties of the solution graph and
related complexity issues for CSPs, motivated mainly by research on
satisfiability algorithms and the satisfiability threshold. They proved
dichotomies for the diameter of connected components and for the complexity of
the st-connectivity question, and conjectured a trichotomy for the connectivity
question. Recently, we were able to establish the trichotomy [arXiv:1312.4524].
Here, we consider connectivity issues of satisfiability problems defined by
Boolean circuits and propositional formulas that use gates, resp. connectives,
from a fixed set of Boolean functions. We obtain dichotomies for the diameter
and the two connectivity problems: on one side, the diameter is linear in the
number of variables, and both problems are in P, while on the other side, the
diameter can be exponential, and the problems are PSPACE-complete. For
partially quantified formulas, we show an analogous dichotomy.Comment: 20 pages, several improvement
Applications of Finite Model Theory: Optimisation Problems, Hybrid Modal Logics and Games.
There exists an interesting relationships between two seemingly distinct fields: logic from the field of Model Theory, which deals with the truth of statements about discrete structures; and Computational Complexity, which deals with the classification of problems by how much of a particular computer resource is required in order to compute a solution. This relationship is known as Descriptive Complexity and it is the primary application of the tools from Model Theory when they are restricted to the finite; this restriction is commonly called Finite Model Theory.
In this thesis, we investigate the extension of the results of Descriptive Complexity from classes of decision problems to classes of optimisation problems. When dealing with decision problems the natural mapping from true and false in logic to yes and no instances of a problem is used but when dealing with optimisation problems, other features of a logic need to be used. We investigate what these features are and provide results in the form of logical frameworks that can be used for describing optimisation problems in particular classes, building on the existing research into this area.
Another application of Finite Model Theory that this thesis investigates is the relative expressiveness of various fragments of an extension of modal logic called hybrid modal logic. This is achieved through taking the Ehrenfeucht-Fraïssé game from Model Theory and modifying it so that it can be applied to hybrid modal logic. Then, by developing winning strategies for the players in the game, results are obtained that show strict hierarchies of expressiveness for fragments of hybrid modal logic that are generated by varying the quantifier depth and the number of proposition and nominal symbols available
Connectivity of Boolean Satisfiability
For Boolean satisfiability problems, the structure of the solution space is
characterized by the solution graph, where the vertices are the solutions, and
two solutions are connected iff they differ in exactly one variable. For this
implicitly defined graph, we here study the st-connectivity and connectivity
problems.
Building on the work of Gopalan et al. ("The Connectivity of Boolean
Satisfiability: Computational and Structural Dichotomies", 2006/2009), we first
investigate satisfiability problems given by CSPs, more exactly CNF(S)-formulas
with constants (as considered in Schaefer's famous 1978 dichotomy theorem); we
prove a computational dichotomy for the st-connectivity problem, asserting that
it is either solvable in polynomial time or PSPACE-complete, and an aligned
structural dichotomy, asserting that the maximal diameter of connected
components is either linear in the number of variables, or can be exponential;
further, we show a trichotomy for the connectivity problem, asserting that it
is either in P, coNP-complete, or PSPACE-complete.
Next we investigate two important variants: CNF(S)-formulas without
constants, and partially quantified formulas; in both cases, we prove analogous
dichotomies for st-connectivity and the diameter; for for the connectivity
problem, we show a trichotomy in the case of quantified formulas, while in the
case of formulas without constants, we identify fragments of a possible
trichotomy.
Finally, we consider the connectivity issues for B-formulas, which are
arbitrarily nested formulas built from some fixed set B of connectives, and for
B-circuits, which are Boolean circuits where the gates are from some finite set
B; we prove a common dichotomy for both connectivity problems and the diameter;
for partially quantified B-formulas, we show an analogous dichotomy.Comment: PhD thesis, 82 pages, contains all results from the previous papers
arXiv:1312.4524, arXiv:1312.6679, and arXiv:1403.6165, plus additional
findings. arXiv admin note: text overlap with arXiv:cs/0609072 by other
author
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