13 research outputs found
On quantified propositional logics and the exponential time hierarchy
We study quantified propositional logics from the complexity theoretic point of view. First we introduce alternating dependency quantified boolean formulae (ADQBF) which generalize both quantified and dependency quantified boolean formulae. We show that the truth evaluation for ADQBF is AEXPTIME(poly)-complete. We also identify fragments for which the problem is complete for the levels of the exponential hierarchy. Second we study propositional team-based logics. We show that DQBF formulae correspond naturally to quantified propositional dependence logic and present a general NEXPTIME upper bound for quantified propositional logic with a large class of generalized dependence atoms. Moreover we show AEXPTIME(poly)-completeness for extensions of propositional team logic with generalized dependence atoms.University of AucklandAcademy of Finlan
On the complexity of horn and krom fragments of second-order boolean logic
Second-order Boolean logic is a generalization of QBF, whose constant alternation fragments are known to be complete for the levels of the exponential time hierarchy. We consider two types of restriction of this logic: 1) restrictions to term constructions, 2) restrictions to the form of the Boolean matrix. Of the first sort, we consider two kinds of restrictions: firstly, disallowing nested use of proper function variables, and secondly stipulating that each function variable must appear with a fixed sequence of arguments. Of the second sort, we consider Horn, Krom, and core fragments of the Boolean matrix. We classify the complexity of logics obtained by combining these two types of restrictions. We show that, in most cases, logics with k alternating blocks of function quantifiers are complete for the kth or (k â 1)th level of the exponential time hierarchy. Furthermore, we establish NL-completeness for the Krom and core fragments, when k = 1 and both restrictions of the first sort are in effect
On the Complexity of Team Logic and its Two-Variable Fragment
We study the logic FO(~), the extension of first-order logic with team
semantics by unrestricted Boolean negation. It was recently shown
axiomatizable, but otherwise has not yet received much attention in questions
of computational complexity.
In this paper, we consider its two-variable fragment FO2(~) and prove that
its satisfiability problem is decidable, and in fact complete for the recently
introduced non-elementary class TOWER(poly). Moreover, we classify the
complexity of model checking of FO(~) with respect to the number of variables
and the quantifier rank, and prove a dichotomy between PSPACE- and
ATIME-ALT(exp, poly)-completeness.
To achieve the lower bounds, we propose a translation from modal team logic
MTL to FO2(~) that extends the well-known standard translation from modal logic
ML to FO2. For the upper bounds, we translate to a fragment of second-order
logic
On Deciding Linear Arithmetic Constraints Over p-adic Integers for All Primes
Given an existential formula Ί of linear arithmetic over p-adic integers together with valuation constraints, we study the p-universality problem which consists of deciding whether Ί is satisfiable for all primes p, and the analogous problem for the closely related existential theory of BĂŒchi arithmetic. Our main result is a coNEXP upper bound for both problems, together with a matching
lower bound for existential BĂŒchi arithmetic. On a technical level, our results are obtained from analysing properties of a certain class of p-automata, finite-state automata whose languages encode
sets of tuples of natural numbers
Fixpoint logics on hierarchical structures
Hierarchical graph definitions allow a modular description of graphs using modules for the specification of repeated substructures. Beside this modularity, hierarchical graph definitions also allow to specify graphs of exponential size using polynomial size descriptions. In many cases, this succinctness increases the computational complexity of decision problems. In this paper, the model-checking problem for the modal -calculus and (monadic) least fixpoint logic on hierarchically defined input graphs is investigated. In order to analyze the modal -calculus, parity games on hierarchically defined input graphs are investigated. In most cases precise upper and lower complexity bounds are derived. A restriction on hierarchical graph definitions that leads to more efficient model-checking algorithms is presented
Graph compression using graph grammars
This thesis presents work done on compressed graph representations via hyperedge replacement
grammars. It comprises two main parts. Firstly the RePair compression scheme, known for
strings and trees, is generalized to graphs using graph grammars. Given an object, the scheme
produces a small context-free grammar generating the object (called a âstraight-line grammarâ).
The theoretical foundations of this generalization are presented, followed by a description of a
prototype implementation. This implementation is then evaluated on real-world and synthetic
graphs. The experiments show that several graphs can be compressed stronger by the new
method, than by current state-of-the-art approaches.
The second part considers algorithmic questions of straight-line graph grammars. Two algorithms
are presented to traverse the graph represented by such a grammar. Both algorithms have
advantages and disadvantages: the first one works with any grammar but its runtime per traversal
step is dependent on the input grammar. The second algorithm only needs constant time per
traversal step, but works for a restricted class of grammars and requires quadratic preprocessing
time and space. Finally speed-up algorithms are considered. These are algorithms that can
decide specific problems in time depending only on the size of the compressed representation,
and might thus be faster than a traditional algorithm would on the decompressed structure. The
idea of such algorithms is to reuse computation already done for the rules of the grammar. The
possible speed-ups achieved this way is proportional to the compression ratio of the grammar.
The main results here are a method to answer âregular path queriesâ, and to decide whether
two grammars generate isomorphic trees
Nutzerfreundliche Modellierung mit hybriden Systemen zur symbolischen Simulation in CLP
Die Dissertation beinhaltet die Sprachen MODEL-HS und VYSMO zur modularen, deklarativen Beschreibung hybrider Systeme, die dem Nachweis zeit- und sicherheitskritischer Eigenschaften fĂŒr die symbolische Simulation in CLP dienen. Zum Erlangen sprachtheoretischer Erkenntnisse wie Entscheidbarkeit wurden hybride Systeme neu unter formal nachweisbaren Akzeptanzbedingungen definiert, welche durch praktische Beispiele belegt sind. Weitere Ergebnisse sind eine neue Klassifikation hybrider Systeme, ein Werkzeug ROSSY, Anfragebeschreibungen und deren Transformation in temporal-logische AusdrĂŒcke, Anfragemasken und Anwendungen fĂŒr Studiensysteme und parallele Programme.The dissertation includes the languages MODEL-HS and VYSMO for modular, declarative description of hybrid systems that serve the proof of time- and safety-critical properties for symbolic simulation in CLP. For coming to language-theoretical conclusions like decidability hybrid systems are newly defined under acceptance conditions that can be formally proved and for which practical examples bear witness. A new classification of hybrid systems, a tool ROSSY, query descriptions and their transformation into temporal-logic expressions, query forms and applications for study systems and parallel programs are further results