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Unary operation
About: Unary operation is a research topic. Over the lifetime, 2277 publications have been published within this topic receiving 29590 citations.
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TL;DR: It is shown that the STP, which subsumes the major part of Vilain and Kautz's point algebra, can be solved in polynomial time and the applicability of path consistency algorithms as preprocessing of temporal problems is studied, to demonstrate their termination and bound their complexities.
1,989 citations
01 Jan 1997
TL;DR: The Undecidable Standard Classes for Pure Predicate Logic, a Treatise on the Transformation of the Classical Decision Problem, and some Results and Open Problems are presented.
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.
791 citations
TL;DR: It is shown here that the weak secondorder theory of two successors is decidable, thus settling a problem of Buchi, and this result is applied to obtain positive solutions to the decision problems for various other theories, e.g., the weaksecond-order theories of order types built up from the finite types.
504 citations
Book•
01 Jan 1987TL;DR: In this paper, the authors introduce basic concepts of algebraic algebraic operations, including lattices, unary and binary operations, and unique factorization, and present a table of notation.
Abstract: Introduction. Preliminaries. Basic concepts. Lattices. Unary and binary operations. Fundamental algebraic results. Unique factorization. Bibliography. Table of notation. Index.
454 citations
TL;DR: The proposed approach to establishing correspondences between two sets of visual features using higher order constraints instead of the unary or pairwise ones used in classical methods is compared to state-of-the-art algorithms on both synthetic and real data.
Abstract: This paper addresses the problem of establishing correspondences between two sets of visual features using higher order constraints instead of the unary or pairwise ones used in classical methods. Concretely, the corresponding hypergraph matching problem is formulated as the maximization of a multilinear objective function over all permutations of the features. This function is defined by a tensor representing the affinity between feature tuples. It is maximized using a generalization of spectral techniques where a relaxed problem is first solved by a multidimensional power method and the solution is then projected onto the closest assignment matrix. The proposed approach has been implemented, and it is compared to state-of-the-art algorithms on both synthetic and real data.
394 citations