Miki Hermann - Academia.edu (original) (raw)

Papers by Miki Hermann

Research paper thumbnail of Chain properties of rule closures

Formal Aspects of Computing, 1990

This article presents an introduction to the generalization of the crossed rule approach to the d... more This article presents an introduction to the generalization of the crossed rule approach to the detection of Knuth Bendix completion procedure divergence. It introduces the closure chains, which are special rule closures constructed by means of particular substitution operations and operators, as a suitable formalism for a progress in this direction. Supporting substitution algebra is developed rst, followed by considerations concerning rule closures in general, and concluded by investigation of closure chain properties.

Research paper thumbnail of On Boolean primitive positive clones

Discrete Mathematics, 2008

Kuznecov introduced the concept of primitive positive clones and proved in 1977 that there are 25... more Kuznecov introduced the concept of primitive positive clones and proved in 1977 that there are 25 Boolean primitive positive clones in a notoriously unavailable article. This paper presents a new proof of his result, relating it to Post's lattice and exhibiting finite bases for those clones.

Research paper thumbnail of Meta-rule synthesis from crossed rewrite systems

Lecture Notes in Computer Science, 1991

... They rely on order-sorted completion modulo -~ [GKK90] in the ease of Dora-based sehematizati... more ... They rely on order-sorted completion modulo -~ [GKK90] in the ease of Dora-based sehematization, and on completion modulo N ~ae87~K86,PS81] for ... The class Gen *( R ) contains two generators, Gen( ( y * z) * i(x) --* y, R) and Gen( i( = ) * ( x * y) --+ y, R ). Both generators in ...

Research paper thumbnail of On the Complexity of Fuzzy Boolean Constraint Satisfaction Problems with Applications to Intelligent Digital Photography

2011 Third Pacific-Asia Conference on Circuits, Communications and System (PACCS), 2011

ABSTRACT We consider fuzzy Boolean constraint satisfaction problems, determine their complexity, ... more ABSTRACT We consider fuzzy Boolean constraint satisfaction problems, determine their complexity, isolate their islands of tractability, and show how they can be applied in digital photography. Keywords - Fuzzy Theory and Models, Fuzzy Mathematics and Applications

Research paper thumbnail of On the Computational Complexity of Monotone Constraint Satisfaction Problems

Lecture Notes in Computer Science, 2009

Constraint Satisfaction Problems (CSP) constitute a convenient way to capture many combinatorial ... more Constraint Satisfaction Problems (CSP) constitute a convenient way to capture many combinatorial problems. The general CSP is known to be NPcomplete, but its complexity depends on a parameter, usually a set of relations, upon which they are constructed. Following the parameter, there exist tractable and intractable instances of CSPs. In this paper we show a dichotomy theorem for every finite domain of CSP including also disjunctions. This dichotomy condition is based on a simple condition, allowing us to classify monotone CSPs as tractable or NP-complete. We also prove that the meta-problem, verifying the tractability condition for monotone constraint satisfaction problems, is fixed-parameter tractable. Moreover, we present a polynomial-time algorithm to answer this question for monotone CSPs over ternary domains. J = {(x, y, z, w) ∈ D 4 | (x = y) ∨ (z = w)} in a classical constraint satisfaction problem reduces it to a monotone constraint satisfaction problem.

Research paper thumbnail of Implementations of Term Rewriting Systems

The Computer Journal, 1991

ABSTRACT

Research paper thumbnail of Complexity of existential positive first-order logic

Let Γ be a (not necessarily finite) structure with a finite relational signature. We prove that d... more Let Γ be a (not necessarily finite) structure with a finite relational signature. We prove that deciding whether a given existential positive sentence holds in Γ is in LOGSPACE or complete for the class CSP(Γ )NP under deterministic polynomial-time many-one reductions. Here, CSP(Γ )NP is the class of problems that can be reduced to the constraint satisfaction problem of Γ under non-deterministic polynomial-time many-one reductions.

Research paper thumbnail of Overview of Existing Recurrent Schematizations

le, but theconstraint formalism lacks schematization power, (2) the constraint formalism is power... more le, but theconstraint formalism lacks schematization power, (2) the constraint formalism is powerfulenough to represent the infinite families but the unification problem for these constraints isundecidable, and (3) the constraint formalism is powerful enough and the unification problemis decidable, but the corresponding constraint solving unification algorithm produces aninfinite family of constraints.In practice, the manipulated sets must be finite, unless there

Research paper thumbnail of Computational Complexity of Simultaneous Elementary Matching Problems

Journal of Automated Reasoning, 1999

The simultaneous elementary E-matching problem for an equational theory E is to decide whether th... more The simultaneous elementary E-matching problem for an equational theory E is to decide whether there is an E-matcher for a given system of equations in which the only function symbols occurring in the terms to be matched are the ones constrained by the equational axioms of E. We study the computational complexity of simultaneous elementary matching problems for the equational theories A of semigroups, AC of commutative semigroups, and ACU of commutative monoids. In each case, we delineate the boundary between NP-completeness and solvability in polynomial time by considering two parameters, the number of equations in the systems and the number of constant symbols in the signature. Moreover, we analyze further the intractable cases of simultaneous elementary AC-matching and ACU-matching by taking also into account the maximum number of occurrences of each variable. Using graph-theoretic techniques, we show that if each variable is restricted to having at most two occurrences, then several cases of simultaneous elementary AC-matching and ACU-matching can be solved in polynomial time.

Research paper thumbnail of Set operations for recurrent term schematizations

Lecture Notes in Computer Science, 1997

Reasoning on programs and automated deduction often require the manipulation of in nite sets of o... more Reasoning on programs and automated deduction often require the manipulation of in nite sets of objects. Many formalisms have been proposed to handle such sets. Here we deal with the formalism of recurrent terms proposed by Chen and Hsiang and subsequently re ned by several authors. These terms contains iterated parts and counter variables to control the iteration, providing an important gain in expressive power. However, little work has been devoted to the study of these terms as a mechanism to represent sets of terms equipped with the corresponding operations union, intersection, inclusion, membership. In this paper, we focus on the set operations relevant for this schematization formalism and we discuss several possible de nitions of these operations. We show how intersection, membership and inclusion can be solved by previously known algorithms and we prove the decidability of the generalisation of two iterated terms, which is the analogy of set union. Moreover, we rene this procedure for computing the generalisation of usual rst-order terms using iterated terms, therefore improving Plotkin's algorithm.

Research paper thumbnail of Counting Complexity of Minimal Cardinality and Minimal Weight Abduction

Lecture Notes in Computer Science, 2008

Abduction is an important method of non-monotonic reasoning with many applications in artificial ... more Abduction is an important method of non-monotonic reasoning with many applications in artificial intelligence and related topics. In this paper, we concentrate on propositional abduction, where the background knowledge is given by a propositional formula. We have recently started to study the counting complexity of propositional abduction. However, several important cases have been left open, namely, the cases when we restrict ourselves to solutions with minimal cardinality or with minimal weight. These cases -possibly combined with priorities -are now settled in this paper. We thus arrive at a complete picture of the counting complexity of propositional abduction.

Research paper thumbnail of On the relation between primitive recursion, schematization, and divergence

Lecture Notes in Computer Science, 1992

The paper presents a new schematization of in nite families of terms called the primal grammars, ... more The paper presents a new schematization of in nite families of terms called the primal grammars, based on the notion of primitive recursive rewrite systems. This schematization is presented by a generating term and a canonical rewrite system. It is proved that the class of primal grammars covers completely the class of crossed rewrite systems. This proof contains a construction of a primal grammar from a crossed rewrite system.

Research paper thumbnail of On proving properties of completion strategies

Lecture Notes in Computer Science, 1991

We develop methods for proving the fairness and correctness properties of rule based completion s... more We develop methods for proving the fairness and correctness properties of rule based completion strategies by means of process logic. The concepts of these properties are formulated generally within process logic and then concretized to rewrite system theory based on transition rules. We develop in parallel the notions of success and failure of a completion strategy, necessary to support the proves of the cited properties. Finally we show the necessity of another property, called justice, in the analysis of completion strategies.

Research paper thumbnail of Computational complexity of simultaneous elementary matching problems

Lecture Notes in Computer Science, 1995

The simultaneous elementary E-matching problem for an equational theory E is to decide whether th... more The simultaneous elementary E-matching problem for an equational theory E is to decide whether there is an E-matcher for a given system of equations in which the only function symbols occurring in the terms to be matched are the ones constrained by the equational axioms of E. We study the computational complexity of simultaneous elementary matching problems for the equational theories A of semigroups, AC of commutative semigroups, and ACU of commutative monoids. In each case, we delineate the boundary between NP-completeness and solvability in polynomial time by considering two parameters, the number of equations in the systems and the number of constant symbols in the signature. Moreover, we analyze further the intractable cases of simultaneous elementary AC-matching and ACU-matching by taking also into account the maximum number of occurrences of each variable. Using graph-theoretic techniques, we show that if each variable is restricted to having at most two occurrences, then several cases of simultaneous elementary AC-matching and ACU-matching can be solved in polynomial time.

Research paper thumbnail of The Inference Problem for Propositional Circumscription of Afine Formulas Is coNP-Complete

Lecture Notes in Computer Science, 2003

We prove that the inference problem of propositional circumscription for afine formulas is coNP-c... more We prove that the inference problem of propositional circumscription for afine formulas is coNP-complete, settling this way a longstanding open question in the complexity of nonmonotonic reasoning. We also show that the considered problem becomes ...

Research paper thumbnail of An Algebraic Approach to the Complexity of Generalized Conjunctive Queries

Lecture Notes in Computer Science, 2005

Kolaitis and Vardi pointed out that constraint satisfaction and conjunctive query containment are... more Kolaitis and Vardi pointed out that constraint satisfaction and conjunctive query containment are essentially the same problem. We study the Boolean conjunctive queries under a more detailed scope, where we investigate their counting problem by means of the algebraic approach through Galois theory, taking advantage of Post's lattice. We prove a trichotomy theorem for the generalized conjunctive query counting problem, showing this way that, contrary to the corresponding decision problems, constraint satisfaction and conjunctive-query containment differ for other computational goals. We also study the audit problem for conjunctive queries asking whether there exists a frozen variable in a given query. This problem is important in databases supporting statistical queries. We derive a dichotomy theorem for this audit problem that sheds more light on audit applicability within database systems.

Research paper thumbnail of Efficient Algorithms for Constraint Description Problems over Finite Totally Ordered Domains: Extended Abstract

Conference on Automated Deduction, 2004

Given a finite set of vectors over a finite totally ordered do- main, we study the problem of com... more Given a finite set of vectors over a finite totally ordered do- main, we study the problem of computing a constraint in conjunctive normal form such that the set of solutions for the produced constraint is identical to the original set. We develop an efficient polynomial-time algorithm for the general case, followed by specific polynomial-time al- gorithms producing Horn, dual

Research paper thumbnail of On the Complexity of Computing Generators of Closed Sets

Lecture Notes in Computer Science, 2008

We investigate the computational complexity of some decision and counting problems related to gen... more We investigate the computational complexity of some decision and counting problems related to generators of closed sets fundamental in Formal Concept Analysis. We recall results from the literature about the problem of checking the existence of a generator with a specified cardinality, and about the problem of determining the number of minimal generators. Moreover, we show that the problem of counting minimum cardinality generators is #·coNP-complete. We also present an incremental-polynomial time algorithm from relational database theory that can be used for computing all minimal generators of an implicationclosed set.

Research paper thumbnail of Complexity of Default Logic on Generalized Conjunctive Queries

Lecture Notes in Computer Science, 2007

Reiter's default logic formalizes nonmonotonic reasoning using default assumptions. The semantics... more Reiter's default logic formalizes nonmonotonic reasoning using default assumptions. The semantics of a given instance of default logic is based on a fixpoint equation defining an extension. Three different reasoning problems arise in the context of default logic, namely the existence of an extension, the presence of a given formula in an extension, and the occurrence of a formula in all extensions. Since the end of 1980s, several complexity results have been published concerning these default reasoning problems for different syntactic classes of formulas. We derive in this paper a complete classification of default logic reasoning problems by means of universal algebra tools using Post's clone lattice. In particular we prove a trichotomy theorem for the existence of an extension, classifying this problem to be either polynomial, NP-complete, or Σ2P-complete, depending on the set of underlying Boolean connectives. We also prove similar trichotomy theorems for the two other algorithmic problems in connection with default logic reasoning.

Research paper thumbnail of Complexity of Clausal Constraints Over Chains

Theory of Computing Systems, 2008

We investigate the complexity of the satisfiability problem of constraints over finite totally or... more We investigate the complexity of the satisfiability problem of constraints over finite totally ordered domains. In our context, a clausal constraint is a disjunction of inequalities of the form x ≥ d and x ≤ d. We classify the complexity of constraints based on clausal patterns. A pattern abstracts away from variables and contains only information about the domain elements and the type of inequalities occurring in a constraint. Every finite set of patterns gives rise to a (clausal) constraint satisfaction problem in which all constraints in instances must have an allowed pattern. We prove that every such problem is either polynomially decidable or NPcomplete, and give a polynomial-time algorithm for recognizing the tractable cases. Some of these tractable cases are new and have not been previously identified in the literature. * The work has been supported byÉGIDE 06606ZF andÖAD Amadeus 18/2004. †

Research paper thumbnail of Chain properties of rule closures

Formal Aspects of Computing, 1990

This article presents an introduction to the generalization of the crossed rule approach to the d... more This article presents an introduction to the generalization of the crossed rule approach to the detection of Knuth Bendix completion procedure divergence. It introduces the closure chains, which are special rule closures constructed by means of particular substitution operations and operators, as a suitable formalism for a progress in this direction. Supporting substitution algebra is developed rst, followed by considerations concerning rule closures in general, and concluded by investigation of closure chain properties.

Research paper thumbnail of On Boolean primitive positive clones

Discrete Mathematics, 2008

Kuznecov introduced the concept of primitive positive clones and proved in 1977 that there are 25... more Kuznecov introduced the concept of primitive positive clones and proved in 1977 that there are 25 Boolean primitive positive clones in a notoriously unavailable article. This paper presents a new proof of his result, relating it to Post's lattice and exhibiting finite bases for those clones.

Research paper thumbnail of Meta-rule synthesis from crossed rewrite systems

Lecture Notes in Computer Science, 1991

... They rely on order-sorted completion modulo -~ [GKK90] in the ease of Dora-based sehematizati... more ... They rely on order-sorted completion modulo -~ [GKK90] in the ease of Dora-based sehematization, and on completion modulo N ~ae87~K86,PS81] for ... The class Gen *( R ) contains two generators, Gen( ( y * z) * i(x) --* y, R) and Gen( i( = ) * ( x * y) --+ y, R ). Both generators in ...

Research paper thumbnail of On the Complexity of Fuzzy Boolean Constraint Satisfaction Problems with Applications to Intelligent Digital Photography

2011 Third Pacific-Asia Conference on Circuits, Communications and System (PACCS), 2011

ABSTRACT We consider fuzzy Boolean constraint satisfaction problems, determine their complexity, ... more ABSTRACT We consider fuzzy Boolean constraint satisfaction problems, determine their complexity, isolate their islands of tractability, and show how they can be applied in digital photography. Keywords - Fuzzy Theory and Models, Fuzzy Mathematics and Applications

Research paper thumbnail of On the Computational Complexity of Monotone Constraint Satisfaction Problems

Lecture Notes in Computer Science, 2009

Constraint Satisfaction Problems (CSP) constitute a convenient way to capture many combinatorial ... more Constraint Satisfaction Problems (CSP) constitute a convenient way to capture many combinatorial problems. The general CSP is known to be NPcomplete, but its complexity depends on a parameter, usually a set of relations, upon which they are constructed. Following the parameter, there exist tractable and intractable instances of CSPs. In this paper we show a dichotomy theorem for every finite domain of CSP including also disjunctions. This dichotomy condition is based on a simple condition, allowing us to classify monotone CSPs as tractable or NP-complete. We also prove that the meta-problem, verifying the tractability condition for monotone constraint satisfaction problems, is fixed-parameter tractable. Moreover, we present a polynomial-time algorithm to answer this question for monotone CSPs over ternary domains. J = {(x, y, z, w) ∈ D 4 | (x = y) ∨ (z = w)} in a classical constraint satisfaction problem reduces it to a monotone constraint satisfaction problem.

Research paper thumbnail of Implementations of Term Rewriting Systems

The Computer Journal, 1991

ABSTRACT

Research paper thumbnail of Complexity of existential positive first-order logic

Let Γ be a (not necessarily finite) structure with a finite relational signature. We prove that d... more Let Γ be a (not necessarily finite) structure with a finite relational signature. We prove that deciding whether a given existential positive sentence holds in Γ is in LOGSPACE or complete for the class CSP(Γ )NP under deterministic polynomial-time many-one reductions. Here, CSP(Γ )NP is the class of problems that can be reduced to the constraint satisfaction problem of Γ under non-deterministic polynomial-time many-one reductions.

Research paper thumbnail of Overview of Existing Recurrent Schematizations

le, but theconstraint formalism lacks schematization power, (2) the constraint formalism is power... more le, but theconstraint formalism lacks schematization power, (2) the constraint formalism is powerfulenough to represent the infinite families but the unification problem for these constraints isundecidable, and (3) the constraint formalism is powerful enough and the unification problemis decidable, but the corresponding constraint solving unification algorithm produces aninfinite family of constraints.In practice, the manipulated sets must be finite, unless there

Research paper thumbnail of Computational Complexity of Simultaneous Elementary Matching Problems

Journal of Automated Reasoning, 1999

The simultaneous elementary E-matching problem for an equational theory E is to decide whether th... more The simultaneous elementary E-matching problem for an equational theory E is to decide whether there is an E-matcher for a given system of equations in which the only function symbols occurring in the terms to be matched are the ones constrained by the equational axioms of E. We study the computational complexity of simultaneous elementary matching problems for the equational theories A of semigroups, AC of commutative semigroups, and ACU of commutative monoids. In each case, we delineate the boundary between NP-completeness and solvability in polynomial time by considering two parameters, the number of equations in the systems and the number of constant symbols in the signature. Moreover, we analyze further the intractable cases of simultaneous elementary AC-matching and ACU-matching by taking also into account the maximum number of occurrences of each variable. Using graph-theoretic techniques, we show that if each variable is restricted to having at most two occurrences, then several cases of simultaneous elementary AC-matching and ACU-matching can be solved in polynomial time.

Research paper thumbnail of Set operations for recurrent term schematizations

Lecture Notes in Computer Science, 1997

Reasoning on programs and automated deduction often require the manipulation of in nite sets of o... more Reasoning on programs and automated deduction often require the manipulation of in nite sets of objects. Many formalisms have been proposed to handle such sets. Here we deal with the formalism of recurrent terms proposed by Chen and Hsiang and subsequently re ned by several authors. These terms contains iterated parts and counter variables to control the iteration, providing an important gain in expressive power. However, little work has been devoted to the study of these terms as a mechanism to represent sets of terms equipped with the corresponding operations union, intersection, inclusion, membership. In this paper, we focus on the set operations relevant for this schematization formalism and we discuss several possible de nitions of these operations. We show how intersection, membership and inclusion can be solved by previously known algorithms and we prove the decidability of the generalisation of two iterated terms, which is the analogy of set union. Moreover, we rene this procedure for computing the generalisation of usual rst-order terms using iterated terms, therefore improving Plotkin's algorithm.

Research paper thumbnail of Counting Complexity of Minimal Cardinality and Minimal Weight Abduction

Lecture Notes in Computer Science, 2008

Abduction is an important method of non-monotonic reasoning with many applications in artificial ... more Abduction is an important method of non-monotonic reasoning with many applications in artificial intelligence and related topics. In this paper, we concentrate on propositional abduction, where the background knowledge is given by a propositional formula. We have recently started to study the counting complexity of propositional abduction. However, several important cases have been left open, namely, the cases when we restrict ourselves to solutions with minimal cardinality or with minimal weight. These cases -possibly combined with priorities -are now settled in this paper. We thus arrive at a complete picture of the counting complexity of propositional abduction.

Research paper thumbnail of On the relation between primitive recursion, schematization, and divergence

Lecture Notes in Computer Science, 1992

The paper presents a new schematization of in nite families of terms called the primal grammars, ... more The paper presents a new schematization of in nite families of terms called the primal grammars, based on the notion of primitive recursive rewrite systems. This schematization is presented by a generating term and a canonical rewrite system. It is proved that the class of primal grammars covers completely the class of crossed rewrite systems. This proof contains a construction of a primal grammar from a crossed rewrite system.

Research paper thumbnail of On proving properties of completion strategies

Lecture Notes in Computer Science, 1991

We develop methods for proving the fairness and correctness properties of rule based completion s... more We develop methods for proving the fairness and correctness properties of rule based completion strategies by means of process logic. The concepts of these properties are formulated generally within process logic and then concretized to rewrite system theory based on transition rules. We develop in parallel the notions of success and failure of a completion strategy, necessary to support the proves of the cited properties. Finally we show the necessity of another property, called justice, in the analysis of completion strategies.

Research paper thumbnail of Computational complexity of simultaneous elementary matching problems

Lecture Notes in Computer Science, 1995

The simultaneous elementary E-matching problem for an equational theory E is to decide whether th... more The simultaneous elementary E-matching problem for an equational theory E is to decide whether there is an E-matcher for a given system of equations in which the only function symbols occurring in the terms to be matched are the ones constrained by the equational axioms of E. We study the computational complexity of simultaneous elementary matching problems for the equational theories A of semigroups, AC of commutative semigroups, and ACU of commutative monoids. In each case, we delineate the boundary between NP-completeness and solvability in polynomial time by considering two parameters, the number of equations in the systems and the number of constant symbols in the signature. Moreover, we analyze further the intractable cases of simultaneous elementary AC-matching and ACU-matching by taking also into account the maximum number of occurrences of each variable. Using graph-theoretic techniques, we show that if each variable is restricted to having at most two occurrences, then several cases of simultaneous elementary AC-matching and ACU-matching can be solved in polynomial time.

Research paper thumbnail of The Inference Problem for Propositional Circumscription of Afine Formulas Is coNP-Complete

Lecture Notes in Computer Science, 2003

We prove that the inference problem of propositional circumscription for afine formulas is coNP-c... more We prove that the inference problem of propositional circumscription for afine formulas is coNP-complete, settling this way a longstanding open question in the complexity of nonmonotonic reasoning. We also show that the considered problem becomes ...

Research paper thumbnail of An Algebraic Approach to the Complexity of Generalized Conjunctive Queries

Lecture Notes in Computer Science, 2005

Kolaitis and Vardi pointed out that constraint satisfaction and conjunctive query containment are... more Kolaitis and Vardi pointed out that constraint satisfaction and conjunctive query containment are essentially the same problem. We study the Boolean conjunctive queries under a more detailed scope, where we investigate their counting problem by means of the algebraic approach through Galois theory, taking advantage of Post's lattice. We prove a trichotomy theorem for the generalized conjunctive query counting problem, showing this way that, contrary to the corresponding decision problems, constraint satisfaction and conjunctive-query containment differ for other computational goals. We also study the audit problem for conjunctive queries asking whether there exists a frozen variable in a given query. This problem is important in databases supporting statistical queries. We derive a dichotomy theorem for this audit problem that sheds more light on audit applicability within database systems.

Research paper thumbnail of Efficient Algorithms for Constraint Description Problems over Finite Totally Ordered Domains: Extended Abstract

Conference on Automated Deduction, 2004

Given a finite set of vectors over a finite totally ordered do- main, we study the problem of com... more Given a finite set of vectors over a finite totally ordered do- main, we study the problem of computing a constraint in conjunctive normal form such that the set of solutions for the produced constraint is identical to the original set. We develop an efficient polynomial-time algorithm for the general case, followed by specific polynomial-time al- gorithms producing Horn, dual

Research paper thumbnail of On the Complexity of Computing Generators of Closed Sets

Lecture Notes in Computer Science, 2008

We investigate the computational complexity of some decision and counting problems related to gen... more We investigate the computational complexity of some decision and counting problems related to generators of closed sets fundamental in Formal Concept Analysis. We recall results from the literature about the problem of checking the existence of a generator with a specified cardinality, and about the problem of determining the number of minimal generators. Moreover, we show that the problem of counting minimum cardinality generators is #·coNP-complete. We also present an incremental-polynomial time algorithm from relational database theory that can be used for computing all minimal generators of an implicationclosed set.

Research paper thumbnail of Complexity of Default Logic on Generalized Conjunctive Queries

Lecture Notes in Computer Science, 2007

Reiter's default logic formalizes nonmonotonic reasoning using default assumptions. The semantics... more Reiter's default logic formalizes nonmonotonic reasoning using default assumptions. The semantics of a given instance of default logic is based on a fixpoint equation defining an extension. Three different reasoning problems arise in the context of default logic, namely the existence of an extension, the presence of a given formula in an extension, and the occurrence of a formula in all extensions. Since the end of 1980s, several complexity results have been published concerning these default reasoning problems for different syntactic classes of formulas. We derive in this paper a complete classification of default logic reasoning problems by means of universal algebra tools using Post's clone lattice. In particular we prove a trichotomy theorem for the existence of an extension, classifying this problem to be either polynomial, NP-complete, or Σ2P-complete, depending on the set of underlying Boolean connectives. We also prove similar trichotomy theorems for the two other algorithmic problems in connection with default logic reasoning.

Research paper thumbnail of Complexity of Clausal Constraints Over Chains

Theory of Computing Systems, 2008

We investigate the complexity of the satisfiability problem of constraints over finite totally or... more We investigate the complexity of the satisfiability problem of constraints over finite totally ordered domains. In our context, a clausal constraint is a disjunction of inequalities of the form x ≥ d and x ≤ d. We classify the complexity of constraints based on clausal patterns. A pattern abstracts away from variables and contains only information about the domain elements and the type of inequalities occurring in a constraint. Every finite set of patterns gives rise to a (clausal) constraint satisfaction problem in which all constraints in instances must have an allowed pattern. We prove that every such problem is either polynomially decidable or NPcomplete, and give a polynomial-time algorithm for recognizing the tractable cases. Some of these tractable cases are new and have not been previously identified in the literature. * The work has been supported byÉGIDE 06606ZF andÖAD Amadeus 18/2004. †