Showing papers on "Undecidable problem published in 1997"

The classical decision problem

Abstract: 1. Introduction: The Classical Decision Problem.- 1.1 The Original Problem.- 1.2 The Transformation of the Classical Decision Problem.- 1.3 What Is and What Isn't in this Book.- I. Undecidable Classes.- 2. Reductions.- 2.1 Undecidability and Conservative Reduction.- 2.1.1 The Church-Turing Theorem and Reduction Classes.- 2.1.2 Trakhtenbrot's Theorem and Conservative Reductions.- 2.1.3 Inseparability and Model Complexity.- 2.2 Logic and Complexity.- 2.2.1 Propositional Satisfiability.- 2.2.2 The Spectrum Problem and Fagin's Theorem.- 2.2.3 Capturing Complexity Classes.- 2.2.4 A Decidable Prefix-Vocabulary Class.- 2.3 The Classifiability Problem.- 2.3.1 The Problem.- 2.3.2 Well Partially Ordered Sets.- 2.3.3 The Well Quasi Ordering of Prefix Sets.- 2.3.4 The Well Quasi Ordering of Arity Sequences.- 2.3.5 The Classifiability of Prefix-Vocabulary Sets.- 2.4 Historical Remarks.- 3. Undecidable Standard Classes for Pure Predicate Logic.- 3.1 The Kahr Class.- 3.1.1 Domino Problems.- 3.1.2 Formalization of Domino Problems by $$[\forall \exists \forall , (0,\omega )]$$-Formulae.- 3.1.3 Graph Interpretation of $$[\forall \exists \forall , (0,\omega )]$$-Formulae.- 3.1.4 The Remaining Cases Without $$\exists *$$.- 3.2 Existential Interpretation for $$[{{\forall }^{3}}\exists *, (0,1)]$$.- 3.3 The Gurevich Class.- 3.3.1 The Proof Strategy.- 3.3.2 Reduction to Diagonal-Freeness.- 3.3.3 Reduction to Shift-Reduced Form.- 3.3.4 Reduction toFi-Elimination Form.- 3.3.5 Elimination of MonadicFi.- 3.3.6 The Kostyrko-Genenz and Suranyi Classes.- 3.4 Historical Remarks.- 4. Undecidable Standard Classes with Functions or Equality.- 4.1 Classes with Functions and Equality.- 4.2 Classes with Functions but Without Equality.- 4.3 Classes with Equality but Without Functions: the Goldfarb Classes 161 4.3.1 Formalization of Natural Numbers in $$[{{\forall }^{3}}\exists *, (\omega ,\omega ),(0)]$$=.- 4.3.2 Using Only One Existential Quantifiers.- 4.3.3 Encoding the Non-Auxiliary Binary Predicates.- 4.3.4 Encoding the Auxiliary Binary Predicates of NUM*.- 4.4 Historical Remarks.- 5. Other Undecidable Cases.- 5.1 Krom and Horn Formulae.- 5.1.1 Krom Prefix Classes Without Functions or Equality.- 5.1.2 Krom Prefix Classes with Functions or Equality.- 5.2 Few Atomic Subformulae.- 5.2.1 Few Function and Equality Free Atoms.- 5.2.2 Few Equalities and Inequalities.- 5.2.3 Horn Clause Programs With One Krom Rule.- 5.3 Undecidable Logics with Two Variables.- 5.3.1 First-Order Logic with the Choice Operator.- 5.3.2 Two-Variable Logic with Cardinality Comparison.- 5.4 Conjunctions of Prefix-Vocabulary Classes.- 5.4.1 Reduction to the Case of Conjunctions.- 5.4.2 Another Classifiability Theorem.- 5.4.3 Some Results and Open Problems.- 5.5 Historical Remarks.- II. Decidable Classes and Their Complexity.- 6. Standard Classes with the Finite Model Property.- 6.1 Techniques for Proving Complexity Results.- 6.1.1 Domino Problems Revisited.- 6.1.2 Succinct Descriptions of Inputs.- 6.2 The Classical Solvable Cases.- 6.2.1 Monadic Formulae.- 6.2.2 The Bernays-Schonfinkel-Ramsey Class.- 6.2.3 The Godel-Kalmar-Schutte Class: a Probabilistic Proof.- 6.3 Formulae with One ?.- 6.3.1 A Satisfiability Test for [?*??*, all, all].- 6.3.2 The Ackermann Class.- 6.3.3 The Ackermann Class with Equality.- 6.4 Standard Classes of Modest Complexity.- 6.4.1 The Relational Classes in P, NP and Co-NP.- 6.4.2 Fragments of the Theory of One Unary Function.- 6.4.3 Other Functional Classes.- 6.5 Finite Model Property vs. Infinity Axioms.- 6.6 Historical Remarks.- 7. Monadic Theories and Decidable Standard Classes with Infinity Axioms.- 7.1 Automata, Games and Decidability of Monadic Theories.- 7.1.1 Monadic Theories.- 7.1.2 Automata on Infinite Words and the Monadic Theory of One Successor.- 7.1.3 Tree Automata, Rabin's Theorem and Forgetful De terminacy.- 7.1.4 The Forgetful Determinacy Theorem for Graph Games.- 7.2 The Monadic Second-Order Theory of One Unary Function.- 7.2.1 Decidability Results for One Unary Function.- 7.2.2 The Theory of One Unary Function is not Elementary Recursive.- 7.3 The Shelah Class.- 7.3.1 Algebras with One Unary Operation.- 7.3.2 Canonic Sentences.- 7.3.3 Terminology and Notation.- 7.3.4 1-Satisfiability.- 7.3.5 2-Satisfiability.- 7.3.6 Refinements.- 7.3.7 Villages.- 7.3.8 Contraction.- 7.3.9 Towns.- 7.3.10 The Final Reduction.- 7.4 Historical Remarks.- 8. Other Decidable Cases.- 8.1 First-Order Logic with Two Variables.- 8.2 Unification and Applications to the Decision Problem.- 8.2.1 Unification.- 8.2.2 Herbrand Formulae.- 8.2.3 Positive First-Order Logic.- 8.3 Decidable Classes of Krom Formulae.- 8.3.1 The Chain Criterion.- 8.3.2 The Aanderaa-Lewis Class.- 8.3.3 The Maslov Class.- 8.4 Historical Remarks.- A. Appendix: Tiling Problems.- A.1 Introduction.- A.2 The Origin Constrained Domino Problem.- A.3 Robinson's Aperiodic Tile Set.- A.4 The Unconstrained Domino Problem.- A.5 The Periodic Problem and the Inseparability Result.- Annotated Bibliography.

Computability and complexity: from a programming perspective

Abstract: A programming approach to computability and complexity theory yields proofs of central results that are sometimes more natural than the classical ones; and some new results as well. These notes contain some high points from the recent book [14], emphasising what is different or novel with respect to more traditional treatments. Topics include: Kleene’s s-m-n theorem applied to compiling and compiler generation. Proof that constant time factors do matter: for a natural computation model, problems solvable in linear time have a proper hierarchy, ordered by coefficient values. (In contrast to the “linear speedup” property of Turing machines.) Results on which problems possess optimal algorithms, including Levin’s Search theorem (for the first time in book form). Characterisations in programming terms of a wide range of complexity classes. These are intrinsic: without externally imposed space or time computation bounds. Boolean program problems complete for PTIME, NPTIME, PSPACE.

Undecidability Results on Two-Variable Logics

Abstract: It is a classical result of Mortimer's that L2, first-order logic with two variables, is decidable for satisfiability (whereas L3 is undecidable). We show that going beyond L2 by adding any one of the following leads to an undecidable logic: very weak forms of recursion, such as transitive closure or monadic fixed-point operations. cardinality comparison quantifiers.

Non-failure analysis for logic programs

Abstract: We provide a method whereby, given mode and (upper approximation) type information, we can detect procedures and goals that can be guaranteed to not fail (i.e., to produce at least one solution or not terminate). The technique is based on an intuitively very simple notion, that of a (set of) tests "covering" the type of a set of variables. We show that the problem of determining a covering is undecidable in general, and give decidability and complexity results for the Herbrand and linear arithmetic constraint systems. We give sound algorithms for determining covering that are precise and efiicient in practice. Based on this information, we show how to identify goals and procedures that can be guaranteed to not fail at runtime. Applications of such non-failure information include programming error detection, program transiormations and parallel execution optimization, avoiding speculative parallelism and estimating lower bounds on the computational costs of goals, which can be used for granularity control. Finally, we report on an implementation of our method and show that better results are obtained than with previously proposed approaches.

Implication and referential constraints: a new formal reasoning

Abstract: The authors address the issue of reasoning with two classes of commonly used semantic integrity constraints in database and knowledge-base systems: implication constraints and referential constraints. They first consider a central problem in this respect, the IRC-refuting problem, which is to decide whether a conjunctive query always produces an empty relation on (finite) database instances satisfying a given set of implication and referential constraints. Since the general problem is undecidable, they only consider acyclic referential constraints. Under this assumption, they prove that the IRC-refuting problem is decidable, and give a novel necessary and sufficient condition for it. Under the same assumption, they also study several other problems encountered in semantic query optimization, such as the semantics-based query containment problem, redundant join problem, and redundant selection-condition problem, and show that they are polynomially equivalent or reducible to the IRC-refuting problem. Moreover, they give results on reducing the complexity for some special cases of the IRC-refuting problem.

On Equality Up-to Constraints over Finite Trees, Context Unification, and One-Step Rewriting

Abstract: We introduce equality up-to constraints over finite trees and investigate their expressiveness. Equality up-to constraints subsume equality constraints, subtree constraints, and one-step rewriting constraints. We establish a close correspondence between equality up-to constraints over finite trees and context unification. Context unification subsumes string unification and is subsumed by linear second-order unification. We obtain the following three new results. The satisfiability problem of equality up-to constraints is equivalent to context unification, which is an open problem. The positive existential fragment of the theory of one-step rewriting is decidable. The ∃*∀*∃* fragment of the theory of context unification is undecidable.

Discrete-Time Control for Rectangular Hybrid Automata

Abstract: Rectangular hybrid automata model digital control programs of analog plant environments. We study rectangular hybrid automata where the plant state evolves continuously in real-numbered time, and the controller samples the plant state and changes the control state discretely, only at the integer points in time. We prove that rectangular hybrid automata have finite bisimilarity quotients when all control transitions happen at integer times, even if the constraints on the derivatives of the variables vary between control states. This is sharply in contrast with the conventional model where control transitions may happen at any real time, and already the reachability problem is undecidable. Based on the finite bisimilarity quotients, we give an exponential algorithm for the symbolic sampling-controller synthesis of rectangular automata. We show our algorithm to be optimal by proving the problem to be EXPTIME-hard. We also show that rectangular automata form a maximal class of systems for which the sampling-controller synthesis problem can be solved algorithmically.

E-Unification by Means of Tree Tuple Synchronized Grammars

Abstract: The goal of this paper is both to give a E-unification procedure that always terminates, and to decide unifiability. For this, we assume that the equational theory is specified by a confluent and constructor-based rewrite system, and that four additional restrictions are satisfied. We give a procedure that represents the (possibly infinite) set of solutions thanks to a new kind of grammar, called tree tuple synchronized grammar, and that can decide unifiability thanks to an emptiness test. Moreover we show that if only three of the four additional restrictions are satisfied then unifiability is undecidable.

Decidable and Undecidable Problems in Matrix Theory

Abstract: This work is a survey on decidable and undecidable problems in matrix theory. The problems studied are simply formulated, however most of them are undecidable. The method to prove undecidabilities is the one found by Paterson [Pat] in 1970 to prove that the mortality of finitely generated matrix monoids is undecidable. This method is based on the undecidability of the Post Correspondence Problem. We shall present a new proof to this mortality problem, which still uses the method of Paterson, but is a bit simpler.

Order-sorted feature theory unification

Abstract: Order-sorted feature (OSF) terms provide an adequate representation for objects as flexible records. They are sorted, attributed, possibly nested structures, ordered thanks to a subsort ordering. Sorts definitions offer the functionality of classes imposing structural constraints on objects. These constraints involve variable sorting and equations among feature paths, including self-reference. Formally, sort definitions may be seen as axioms forming an OSF theory. OSF theory unification is the process of normalizing an OSF term taking into account sort definitions, enforcing structural constraints imposed by an OSF theory. It allows objects to inherit, and thus abide by, constraints from their classes. We propose a formal system that logically models record objects with (possibly recursive) class definitions accommodating multiple inheritance. We show that OSF theory unification is undecidable in general. However, we give a set of confluent normalization rules which is complete for detecting the inconsistency of an object with respect to an OSF theory. Furthermore, a subset consisting of all rules but one is confluent and terminating. This yields a practical complete normalization strategy, as well as an effective compilation scheme.

The first-order theory of one step rewriting in linear noetherian systems is undecidable

Abstract: We construct a finite linear finitely terminating rewrite rule system with undecidable theory of one step rewriting.

Undecidable Extensions of Buchi Arithmetic and Cobham-Semenov Theorem

Abstract: Let k and I be two multiplicatively independent integers, and let L C N' be a I-recognizable set which is not definable in (N; +). We prove that the elementary theory of (N; +, Vk, L), where Vk (x) denotes the greatest power of k dividing x, is undecidable. This result leads to a new proof of the CobhamSemenov theorem. ?

Undecidability of the First Order Theory of One-Step Right Ground Rewriting

Abstract: The problem of decidability of the first order theory of one-step rewriting was stated in [CCD93]. One can find the problem on the lists of open problems in rewriting in [DJK93] and [DJK95]. In 1995 Ralf Treinen proved that the theory is undecidable.

E-unification by means of tree tuple synchronized grammars

Abstract: The goal of this paper is both to give an E-unification procedure that always terminates, and to decide unifiability. For this, we assume that the equational theory is specified by a confluent and constructor-based rewrite system, and that four additional restrictions are satisfied. We give a procedure that represents the (possibly infinite) set of solutions thanks to a tree tuple synchronized grammar, and that can decide upon unifiability thanks to an emptiness test. Moreover, we show that if only three of the four additional restrictions are satisfied then unifiability is undecidable.

Grid Structure and Undecidable Constraint Theories

Abstract: We express conditions for a term to be a finite grid-like structure. Together with definitions of term properties by excluding “forbidden patterns” we obtain three new undecidability results in three areas: the ∃*∀*-fragment of the theory of one-step rewriting for linear and shallow rewrite systems, the emptiness for automata with equality tests between first cousins (i.e. only tests at depth 2 below the current node are available), and the ∃*∀*-fragment of the theory of set constraints.

Complexity of Modal Logics of Relations

Abstract: We consider two families of modal logics of relations: arrow logic and cylindric modal logic and several natural expansions of these, interpreted on a range of (relativised) model­classes. We give a systematic study of the complexity of the validity problem of these logics, obtaining price tags for various features as assumptions on the universe of the models, similarity types, and number of variables involved. The general picture is that the process of relativisation turns an undecidable logic into one whose validity problem is exptime­complete. There are interesting deviations to this though, which we also discuss. The numerous results in this paper are all directed to obtain a better understanding why relativisation can turn an undecidable modal logic of relations into a decidable one. We connect the semantic way of ``taming logic'' by relativisation with the syntactic approach of isolating decidable so­called guarded fragments by showing that validity of loosely guarded formulas is preserved under relativisation.

The Undecidability of Simultaneous Rigid E-Unification with Two Variables

Abstract: Recently it was proved that the problem of simultaneous rigid E-unification, or SREU, is undecidable. Here we show that 4 rigid equations with ground left-hand sides and 2 variables already imply undecidability. As a corollary we improve the undecidability result of the 3*-fragment of intuitionistic logic with equality. Our proof shows undecidability of a very restricted subset of the 33-fragment. Together with other results, it contributes to a complete characterization of decidability of the prenex fragment of intuitionistic logic with equality, in terms of the quantifier prefix.

On the Unification Problem for Cartesian Closed Categories

Abstract: An axiomatization of the isomorphisms that hold in all Cartesian closed categories (CCCs), discovered independently by S.V. Soloviev (1983) and by K.B. Bruce and G. Longo (1985), leads to seven equalities. It is shown that the unification problem for this theory is undecidable, thus setting an open question. It is also shown that an important subcase, namely unification modulo the linear isomorphisms, is NP-complete. Furthermore, the problem of matching in CCCs is NP-complete when the subject term is irreducible. CCC-matching and unification form the basis for an elegant and practical solution to the problem of retrieving functions from a library indexed by types investigated by M. Rittri (1990, 1991). It also has potential applications to the problem of polymorphic higher-order unification, which in turn is relevant to theorem proving, logic programming, and type reconstruction in higher-order languages. >

Some undecidable problems related to the Herbrand theorem

Abstract: We improve upon a number of recent undecidability results related to the so­called Herbrand Skeleton Problem, the Simultaneous Rigid E-­Unification Problem and the prenex fragment of intuitionistic logic with equality

Looking for an Analogue of Rice's Theorem in Circuit Complexity Theory

Abstract: Rice's Theorem says that every nontrivial semantic property of programs is undecidable. It this spirit we show the following: Every nontrivial absolute (gap, relative) counting property of circuits is UP-hard with respect to polynomial-time Turing reductions.

Containing of Regular Languages in Non-Regular Timing Diagram Languages is Decidable

Abstract: Parametric timing constraints are expressed naturally in timing diagram logics. Algorithmic verification of parametrically constrained timing properties is a difficult problem known to be undecidable in most general cases. This paper establishes that a class of parametrically constrained timing properties can be verified algorithmically against finite-state systems; alternatively stated containment by a regular language is shown decidable for a class of language properties (regular and non-regular) expressible in our timing diagram logic.

On the Decidability of Some Equivalence Problems for L Algebraic Series

Abstract: The equivalence problem for Lindenmayerian algebraic series is discussed. While the problem in general is undecidable, decidable special cases of interest are presented.