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Heribert Vollmer

Publications and source records attributed to Heribert Vollmer.

At least 37 records · Page 2Linked to original sources
arXiv2013

Paradigms for Parameterized Enumeration

The aim of the paper is to examine the computational complexity and algorithmics of enumeration, the task to output all solutions of a given problem, from the point of view of parameterized complexity. First we define formally different notions of efficient enumeration in the context of parameterized complexity. Second we show how different algorithmic paradigms can be used in order to get parameter-efficient enumeration algorithms in a number of examples. These paradigms use well-known principles from the design of parameterized decision as well as enumeration techniques, like for instance kernelization and self-reducibility. The concept of kernelization, in particular, leads to a characterization of fixed-parameter tractable enumeration problems.

cs.CC↗
arXiv2013

Model Checking for Modal Dependence Logic: An Approach Through Post's Lattice

In this paper we investigate an extended version of modal dependence logic by allowing arbitrary Boolean connectives. Modal dependence logic was recently introduced by Jouko Väänänen by extending modal logic by a the dependence atom dep(.). In this paper we study the computational complexity of the model checking problem. For a complete classification of arbitrary Boolean functions we are using a Lattice approach introduced by Emil Post. This classification is done for all fragments of the logical language allowing modalities $\Diamond$ and $\Box$, the dependence atom, and logical symbols for arbitrary Boolean functions.

cs.CC↗
arXiv2013

Dependence logic with a majority quantifier

We study the extension of dependence logic D by a majority quantifier M over finite structures. We show that the resulting logic is equi-expressive with the extension of second-order logic by second-order majority quantifiers of all arities. Our results imply that, from the point of view of descriptive complexity theory, D(M) captures the complexity class counting hierarchy.

cs.LO↗
arXiv2011

Model Checking CTL is Almost Always Inherently Sequential

The model checking problem for CTL is known to be P-complete (Clarke, Emerson, and Sistla (1986), see Schnoebelen (2002)). We consider fragments of CTL obtained by restricting the use of temporal modalities or the use of negations---restrictions already studied for LTL by Sistla and Clarke (1985) and Markey (2004). For all these fragments, except for the trivial case without any temporal operator, we systematically prove model checking to be either inherently sequential (P-complete) or very efficiently parallelizable (LOGCFL-complete). For most fragments, however, model checking for CTL is already P-complete. Hence our results indicate that, in cases where the combined complexity is of relevance, approaching CTL model checking by parallelism cannot be expected to result in any significant speedup. We also completely determine the complexity of the model checking problem for all fragments of the extensions ECTL, CTL+, and ECTL+.

cs.LO↗
arXiv2011

On the Parameterized Complexity of Default Logic and Autoepistemic Logic

We investigate the application of Courcelle's Theorem and the logspace version of Elberfeld etal. in the context of the implication problem for propositional sets of formulae, the extension existence problem for default logic, as well as the expansion existence problem for autoepistemic logic and obtain fixed-parameter time and space efficient algorithms for these problems. On the other hand, we exhibit, for each of the above problems, families of instances of a very simple structure that, for a wide range of different parameterizations, do not have efficient fixed-parameter algorithms (even in the sense of the large class XPnu), unless P=NP.

cs.CC↗
arXiv2011

Complexity Results for Modal Dependence Logic

Modal dependence logic was introduced recently by Väänänen. It enhances the basic modal language by an operator =(). For propositional variables p_1,...,p_n, =(p_1,...,p_(n-1);p_n) intuitively states that the value of p_n is determined by those of p_1,...,p_(n-1). Sevenster (J. Logic and Computation, 2009) showed that satisfiability for modal dependence logic is complete for nondeterministic exponential time. In this paper we consider fragments of modal dependence logic obtained by restricting the set of allowed propositional connectives. We show that satisfibility for poor man's dependence logic, the language consisting of formulas built from literals and dependence atoms using conjunction, necessity and possibility (i.e., disallowing disjunction), remains NEXPTIME-complete. If we only allow monotone formulas (without negation, but with disjunction), the complexity drops to PSPACE-completeness. We also extend Väänänen's language by allowing classical disjunction besides dependence disjunction and show that the satisfiability problem remains NEXPTIME-complete. If we then disallow both negation and dependence disjunction, satistiability is complete for the second level of the polynomial hierarchy. In this way we completely classify the computational complexity of the satisfiability problem for all restrictions of propositional and dependence operators considered by Väänänen and Sevenster.

cs.LO↗
arXiv2010

On Second-Order Monadic Monoidal and Groupoidal Quantifiers

We study logics defined in terms of second-order monadic monoidal and groupoidal quantifiers. These are generalized quantifiers defined by monoid and groupoid word-problems, equivalently, by regular and context-free languages. We give a computational classification of the expressive power of these logics over strings with varying built-in predicates. In particular, we show that ATIME(n) can be logically characterized in terms of second-order monadic monoidal quantifiers.

cs.LO↗
arXiv2010

Complexity of Non-Monotonic Logics

Over the past few decades, non-monotonic reasoning has developed to be one of the most important topics in computational logic and artificial intelligence. Different ways to introduce non-monotonic aspects to classical logic have been considered, e.g., extension with default rules, extension with modal belief operators, or modification of the semantics. In this survey we consider a logical formalism from each of the above possibilities, namely Reiter's default logic, Moore's autoepistemic logic and McCarthy's circumscription. Additionally, we consider abduction, where one is not interested in inferences from a given knowledge base but in computing possible explanations for an observation with respect to a given knowledge base. Complexity results for different reasoning tasks for propositional variants of these logics have been studied already in the nineties. In recent years, however, a renewed interest in complexity issues can be observed. One current focal approach is to consider parameterized problems and identify reasonable parameters that allow for FPT algorithms. In another approach, the emphasis lies on identifying fragments, i.e., restriction of the logical language, that allow more efficient algorithms for the most important reasoning tasks. In this survey we focus on this second aspect. We describe complexity results for fragments of logical languages obtained by either restricting the allowed set of operators (e.g., forbidding negations one might consider only monotone formulae) or by considering only formulae in conjunctive normal form but with generalized clause types. The algorithmic problems we consider are suitable variants of satisfiability and implication in each of the logics, but also counting problems, where one is not only interested in the existence of certain objects (e.g., models of a formula) but asks for their number.

cs.CC↗
arXiv2010

The Complexity of Reasoning for Fragments of Default Logic

Default logic was introduced by Reiter in 1980. In 1992, Gottlob classified the complexity of the extension existence problem for propositional default logic as $\SigmaPtwo$-complete, and the complexity of the credulous and skeptical reasoning problem as SigmaP2-complete, resp. PiP2-complete. Additionally, he investigated restrictions on the default rules, i.e., semi-normal default rules. Selman made in 1992 a similar approach with disjunction-free and unary default rules. In this paper we systematically restrict the set of allowed propositional connectives. We give a complete complexity classification for all sets of Boolean functions in the meaning of Post's lattice for all three common decision problems for propositional default logic. We show that the complexity is a hexachotomy (SigmaP2-, DeltaP2-, NP-, P-, NL-complete, trivial) for the extension existence problem, while for the credulous and skeptical reasoning problem we obtain similar classifications without trivial cases.

cs.CC↗
arXiv2010

The Complexity of Reasoning for Fragments of Autoepistemic Logic

Autoepistemic logic extends propositional logic by the modal operator L. A formula that is preceded by an L is said to be "believed". The logic was introduced by Moore 1985 for modeling an ideally rational agent's behavior and reasoning about his own beliefs. In this paper we analyze all Boolean fragments of autoepistemic logic with respect to the computational complexity of the three most common decision problems expansion existence, brave reasoning and cautious reasoning. As a second contribution we classify the computational complexity of counting the number of stable expansions of a given knowledge base. To the best of our knowledge this is the first paper analyzing the counting problem for autoepistemic logic.

cs.LO↗
arXiv2009

The Complexity of Propositional Implication

The question whether a set of formulae G implies a formula f is fundamental. The present paper studies the complexity of the above implication problem for propositional formulae that are built from a systematically restricted set of Boolean connectives. We give a complete complexity classification for all sets of Boolean functions in the meaning of Post's lattice and show that the implication problem is efficentily solvable only if the connectives are definable using the constants {false,true} and only one of {and,or,xor}. The problem remains coNP-complete in all other cases. We also consider the restriction of G to singletons.

cs.CC↗
arXiv2009

The Complexity of Generalized Satisfiability for Linear Temporal Logic

In a seminal paper from 1985, Sistla and Clarke showed that satisfiability for Linear Temporal Logic (LTL) is either NP-complete or PSPACE-complete, depending on the set of temporal operators used. If, in contrast, the set of propositional operators is restricted, the complexity may decrease. This paper undertakes a systematic study of satisfiability for LTL formulae over restricted sets of propositional and temporal operators. Since every propositional operator corresponds to a Boolean function, there exist infinitely many propositional operators. In order to systematically cover all possible sets of them, we use Post's lattice. With its help, we determine the computational complexity of LTL satisfiability for all combinations of temporal operators and all but two classes of propositional functions. Each of these infinitely many problems is shown to be either PSPACE-complete, NP-complete, or in P.

cs.LO↗
arXiv2008

Extensional Uniformity for Boolean Circuits

Imposing an extensional uniformity condition on a non-uniform circuit complexity class C means simply intersecting C with a uniform class L. By contrast, the usual intensional uniformity conditions require that a resource-bounded machine be able to exhibit the circuits in the circuit family defining C. We say that (C,L) has the "Uniformity Duality Property" if the extensionally uniform class C \cap L can be captured intensionally by means of adding so-called "L-numerical predicates" to the first-order descriptive complexity apparatus describing the connection language of the circuit family defining C. This paper exhibits positive instances and negative instances of the Uniformity Duality Property.

cs.LO↗
arXiv2008

The Tractability of Model-Checking for LTL: The Good, the Bad, and the Ugly Fragments

In a seminal paper from 1985, Sistla and Clarke showed that the model-checking problem for Linear Temporal Logic (LTL) is either NP-complete or PSPACE-complete, depending on the set of temporal operators used. If, in contrast, the set of propositional operators is restricted, the complexity may decrease. This paper systematically studies the model-checking problem for LTL formulae over restricted sets of propositional and temporal operators. For almost all combinations of temporal and propositional operators, we determine whether the model-checking problem is tractable (in P) or intractable (NP-hard). We then focus on the tractable cases, showing that they all are NL-complete or even logspace solvable. This leads to a surprising gap in complexity between tractable and intractable cases. It is worth noting that our analysis covers an infinite set of problems, since there are infinitely many sets of propositional operators.

cs.LO↗
arXiv2004

The Complexity of Boolean Constraint Isomorphism

In 1978, Schaefer proved his famous dichotomy theorem for generalized satisfiability problems. He defined an infinite number of propositional satisfiability problems (nowadays usually called Boolean constraint satisfaction problems) and showed that all these satisfiability problems are either in P or NP-complete. In recent years, similar results have been obtained for quite a few other problems for Boolean constraints.Almost all of these problems are variations of the satisfiability problem. In this paper, we address a problem that is not a variation of satisfiability, namely, the isomorphism problem for Boolean constraints. Previous work by Böhler et al. showed that the isomorphism problem is either coNP-hard or reducible to the graph isomorphism problem (a problem that is in NP, but not known to be NP-hard), thus distinguishing a hard case and an easier case. However, they did not classify which cases are truly easy, i.e., in P. This paper accomplishes exactly that. It shows that Boolean constraint isomorphism is coNP-hard (and GI-hard), or equivalent to graph isomorphism, or in P, and it gives simple criteria to determine which case holds.

cs.CC↗
arXiv2002

Equivalence and Isomorphism for Boolean Constraint Satisfaction

A Boolean constraint satisfaction instance is a conjunction of constraint applications, where the allowed constraints are drawn from a fixed set B of Boolean functions. We consider the problem of determining whether two given constraint satisfaction instances are equivalent and prove a Dichotomy Theorem by showing that for all sets C of allowed constraints, this problem is either polynomial-time solvable or coNP-complete, and we give a simple criterion to determine which case holds. A more general problem addressed in this paper is the isomorphism problem, the problem of determining whether there exists a renaming of the variables that makes two given constraint satisfaction instances equivalent in the above sense. We prove that this problem is coNP-hard if the corresponding equivalence problem is coNP-hard, and polynomial-time many-one reducible to the graph isomorphism problem in all other cases.

cs.CC↗
arXiv1998

The descriptive complexity approach to LOGCFL

Building upon the known generalized-quantifier-based first-order characterization of LOGCFL, we lay the groundwork for a deeper investigation. Specifically, we examine subclasses of LOGCFL arising from varying the arity and nesting of groupoidal quantifiers. Our work extends the elaborate theory relating monoidal quantifiers to NC1 and its subclasses. In the absence of the BIT predicate, we resolve the main issues: we show in particular that no single outermost unary groupoidal quantifier with FO can capture all the context-free languages, and we obtain the surprising result that a variant of Greibach's ``hardest context-free language'' is LOGCFL-complete under quantifier-free BIT-free projections. We then prove that FO with unary groupoidal quantifiers is strictly more expressive with the BIT predicate than without. Considering a particular groupoidal quantifier, we prove that first-order logic with majority of pairs is strictly more expressive than first-order with majority of individuals. As a technical tool of independent interest, we define the notion of an aperiodic nondeterministic finite automaton and prove that FO translations are precisely the mappings computed by single-valued aperiodic nondeterministic finite transducers.

cs.CC↗