72,048 research outputs found
Layered Fixed Point Logic
We present a logic for the specification of static analysis problems that
goes beyond the logics traditionally used. Its most prominent feature is the
direct support for both inductive computations of behaviors as well as
co-inductive specifications of properties. Two main theoretical contributions
are a Moore Family result and a parametrized worst case time complexity result.
We show that the logic and the associated solver can be used for rapid
prototyping and illustrate a wide variety of applications within Static
Analysis, Constraint Satisfaction Problems and Model Checking. In all cases the
complexity result specializes to the worst case time complexity of the
classical methods
Schaefer's theorem for graphs
Schaefer's theorem is a complexity classification result for so-called
Boolean constraint satisfaction problems: it states that every Boolean
constraint satisfaction problem is either contained in one out of six classes
and can be solved in polynomial time, or is NP-complete.
We present an analog of this dichotomy result for the propositional logic of
graphs instead of Boolean logic. In this generalization of Schaefer's result,
the input consists of a set W of variables and a conjunction \Phi\ of
statements ("constraints") about these variables in the language of graphs,
where each statement is taken from a fixed finite set \Psi\ of allowed
quantifier-free first-order formulas; the question is whether \Phi\ is
satisfiable in a graph.
We prove that either \Psi\ is contained in one out of 17 classes of graph
formulas and the corresponding problem can be solved in polynomial time, or the
problem is NP-complete. This is achieved by a universal-algebraic approach,
which in turn allows us to use structural Ramsey theory. To apply the
universal-algebraic approach, we formulate the computational problems under
consideration as constraint satisfaction problems (CSPs) whose templates are
first-order definable in the countably infinite random graph. Our method to
classify the computational complexity of those CSPs is based on a
Ramsey-theoretic analysis of functions acting on the random graph, and we
develop general tools suitable for such an analysis which are of independent
mathematical interest.Comment: 54 page
Existential Second-Order Logic Over Graphs: A Complete Complexity-Theoretic Classification
Descriptive complexity theory aims at inferring a problem's computational
complexity from the syntactic complexity of its description. A cornerstone of
this theory is Fagin's Theorem, by which a graph property is expressible in
existential second-order logic (ESO logic) if, and only if, it is in NP. A
natural question, from the theory's point of view, is which syntactic fragments
of ESO logic also still characterize NP. Research on this question has
culminated in a dichotomy result by Gottlob, Kolatis, and Schwentick: for each
possible quantifier prefix of an ESO formula, the resulting prefix class either
contains an NP-complete problem or is contained in P. However, the exact
complexity of the prefix classes inside P remained elusive. In the present
paper, we clear up the picture by showing that for each prefix class of ESO
logic, its reduction closure under first-order reductions is either FO, L, NL,
or NP. For undirected, self-loop-free graphs two containment results are
especially challenging to prove: containment in L for the prefix and containment in FO for the prefix
for monadic . The complex argument by
Gottlob, Kolatis, and Schwentick concerning polynomial time needs to be
carefully reexamined and either combined with the logspace version of
Courcelle's Theorem or directly improved to first-order computations. A
different challenge is posed by formulas with the prefix : We show that they express special constraint satisfaction problems
that lie in L.Comment: Technical report version of a STACS 2015 pape
A Definability Dichotomy for Finite Valued CSPs
Finite valued constraint satisfaction problems are a formalism for describing many natural optimisation problems, where constraints on the values that variables can take come with rational weights and the aim is to find an assignment of minimal cost. Thapper and Zivny have recently established a complexity dichotomy for valued constraint languages. They show that each such languages either gives rise to a polynomial-time solvable optimisation problem, or to an NP-hard one, and establish a criterion to distinguish the two cases. We refine the dichotomy by showing that all optimisation problems in the first class are definable in fixed-point language with counting, while all languages in the second class are not definable, even in infinitary logic with counting. Our definability dichotomy is not conditional on any complexity-theoretic assumption
Fuzzy positive primitive formulas
Can non-classical logic contribute to the analysis of complexity in computer science? In this paper, we give an step towards the solution of this open problem, taking a logical model-theoretic approach to the analysis of complexity in fuzzy constraint satisfaction. We study fuzzy positive-primitive sentences, and we present an algebraic characterization of classes axiomatized by these kind of sentences in terms of homomorphisms and finite direct products. The ultimate goal is to study the expressiveness and reasoning mechanisms of non-classical languages, with respect to constraint satisfaction problems and, in general, in modelling decision scenario
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