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A logic for metric and topology
 Journal of Symbolic Logic
, 2005
"... Abstract. We propose a logic for reasoning about metric spaces with the induced topologies. It combines the ‘qualitative ’ interior and closure operators with ‘quantitative’ operators ‘somewhere in the sphere of radius r, ’ including or excluding the boundary. We supply the logic with both the inten ..."
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Cited by 14 (11 self)
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Abstract. We propose a logic for reasoning about metric spaces with the induced topologies. It combines the ‘qualitative ’ interior and closure operators with ‘quantitative’ operators ‘somewhere in the sphere of radius r, ’ including or excluding the boundary. We supply the logic with both the intended metric space semantics and a natural relational semantics, and show that the latter (i) provides finite partial representations of (in general) infinite metric models and (ii) reduces the standard ‘εdefinitions ’ of closure and interior to simple constraints on relations. These features of the relational semantics suggest a finite axiomatisation of the logic and provide means to prove its EXPTIMEcompleteness (even if the rational numerical parameters are coded in binary). An extension with metric variables satisfying linear rational (in)equalities is proved to be decidable as well. Our logic can be regarded as a ‘wellbehaved ’ common denominator of logical systems constructed in temporal, spatial, and similaritybased quantitative and qualitative representation and reasoning. Interpreted on the real line (with its Euclidean metric), it is a natural fragment of decidable temporal logics for specification and verification of realtime systems. On the real plane, it is closely related to quantitative and qualitative formalisms for spatial
SemiQualitative Reasoning About Distances: A Preliminary Report
, 2000
"... We introduce a family of languages intended for representing knowledge and reasoning about metric (and more general distance) spaces. While the simplest language can speak only about distances between individual objects and Boolean relations between sets, the more expressive ones are capable of capt ..."
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Cited by 11 (8 self)
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We introduce a family of languages intended for representing knowledge and reasoning about metric (and more general distance) spaces. While the simplest language can speak only about distances between individual objects and Boolean relations between sets, the more expressive ones are capable of capturing notions such as `somewhere in (or somewhere out of) the sphere of a certain radius', `everywhere in a certain ring', etc. The computational complexity of the satisfiability problem for formulas in our languages ranges from NPcompleteness to undecidability and depends on the class of distance spaces in which they are interpreted. Besides the class of all metric spaces, we consider, for example, the spaces R \Theta R and N \Theta N with their natural metrics. 1 Introduction The concept of `distance between objects' is one of the most fundamental abstractions both in science and in everyday life. Imagine for instance (only imagine) that you are going to buy a house in London. You then i...
Hilbert’s twentyfourth problem
 American Mathematical Monthly
, 2001
"... 1. INTRODUCTION. For geometers, Hilbert’s influential work on the foundations of geometry is important. For analysts, Hilbert’s theory of integral equations is just as important. But the address “Mathematische Probleme ” [37] that David Hilbert (1862– 1943) delivered at the second International Cong ..."
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Cited by 11 (4 self)
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1. INTRODUCTION. For geometers, Hilbert’s influential work on the foundations of geometry is important. For analysts, Hilbert’s theory of integral equations is just as important. But the address “Mathematische Probleme ” [37] that David Hilbert (1862– 1943) delivered at the second International Congress of Mathematicians (ICM) in Paris has tremendous importance for all mathematicians. Moreover, a substantial part of
The Complexity of ResourceBounded FirstOrder Classical Logic
 11th Symposium on Theoretical Aspects of Computer Science
, 1994
"... . We give a finer analysis of the difficulty of proof search in classical firstorder logic, other than just saying that it is undecidable. To do this, we identify several measures of difficulty of theorems, which we use as resource bounds to prune infinite proof search trees. In classical firstord ..."
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Cited by 8 (1 self)
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. We give a finer analysis of the difficulty of proof search in classical firstorder logic, other than just saying that it is undecidable. To do this, we identify several measures of difficulty of theorems, which we use as resource bounds to prune infinite proof search trees. In classical firstorder logic without interpreted symbols, we prove that for all these measures, the search for a proof of bounded difficulty (i.e, for a simple proof) is \Sigma p 2 complete. We also show that the same problem when the initial formula is a set of Horn clauses is only NPcomplete, and examine the case of firstorder logic modulo an equational theory. These results allow us not only to give estimations of the inherent difficulty of automated theorem proving problems, but to gain some insight into the computational relevance of several automated theorem proving methods. Topics: computational complexity, logics, computational issues in AI (automated theorem proving). 1 Introduction Firstorder ...
Some Undecidable Problems Related to the Herbrand Theorem
, 1997
"... We improve upon a number of recent undecidability results related to the socalled Herbrand Skeleton Problem, the Simultaneous Rigid EUnification Problem and the prenex fragment of intuitionistic logic with equality. Partially supported by grants from NSF, ONR and the Faculty of Science and Techn ..."
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Cited by 7 (6 self)
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We improve upon a number of recent undecidability results related to the socalled Herbrand Skeleton Problem, the Simultaneous Rigid EUnification Problem and the prenex fragment of intuitionistic logic with equality. Partially supported by grants from NSF, ONR and the Faculty of Science and Technology of Uppsala University. 1 Introduction We study classical firstorder logic with equality but without any other relation symbols. The letters ' and / are reserved for quantifierfree formulas. The signature of a syntactic object S (a term, a set of terms, a formula, etc.) is the collection of function symbols in S augmented, in the case when S contains no constants, with a constant c. The language of S is the language of the signature of S. Any syntactic object is ground if it contains no variables. A substitution is ground if its range is ground, and it is said to be in a given language if the terms in its range are in that language. A set of substitutions is ground if each membe...
Undecidability of firstorder intuitionistic and modal logics with two variables
 Bulletin of Symbolic Logic
, 2005
"... Abstract. We prove that the twovariable fragment of firstorder intuitionistic logic is undecidable, even without constants and equality. We also show that the twovariable fragment of a quantified modal logic L with expanding firstorder domains is undecidable whenever there is a Kripke frame for L ..."
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Abstract. We prove that the twovariable fragment of firstorder intuitionistic logic is undecidable, even without constants and equality. We also show that the twovariable fragment of a quantified modal logic L with expanding firstorder domains is undecidable whenever there is a Kripke frame for L with a point having infinitely many successors (such are, in particular, the firstorder extensions of practically all standard modal logics like K, K4, GL, S4, S5, K4.1, S4.2, GL.3, etc.). For many quantified modal logics, including those in the standard nomenclature above, even the monadic twovariable fragments turn out to be undecidable. §1. Introduction. Ever since the undecidability of firstorder classical logic became known [5], there has been a continuing interest in establishing the ‘borderline ’ between its decidable and undecidable fragments; see [2] for a detailed exposition. One approach to this classification problem is to consider fragments with finitely many individual variables. The
Deciding monodic fragments by temporal resolution
 In Proc. CADE20
, 2005
"... Abstract. In this paper we study the decidability of various fragments of monodic firstorder temporal logic by temporal resolution. We focus on two resolution calculi, namely, monodic temporal resolution and finegrained temporal resolution. For the first, we state a very general decidability result ..."
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Abstract. In this paper we study the decidability of various fragments of monodic firstorder temporal logic by temporal resolution. We focus on two resolution calculi, namely, monodic temporal resolution and finegrained temporal resolution. For the first, we state a very general decidability result, which is independent of the particular decision procedure used to decide the firstorder part of the logic. For the second, we introduce refinements using orderings and selection functions. This allows us to transfer existing results on decidability by resolution for firstorder fragments to monodic firstorder temporal logic and obtain new decision procedures. The latter is of immediate practical value, due to the availability of TeMP, an implementation of finegrained temporal resolution. 1
Common Knowledge and Quantification
"... The paper consists of two parts. The first one is a concise introduction to epistemic (both propositional and predicate) logic with common knowledge operator. As the full predicate logics of common knowledge are not even recursively enumerable, in the second part we introduce and investigate the mon ..."
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Cited by 3 (0 self)
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The paper consists of two parts. The first one is a concise introduction to epistemic (both propositional and predicate) logic with common knowledge operator. As the full predicate logics of common knowledge are not even recursively enumerable, in the second part we introduce and investigate the monodic fragment of these logics which allows applications of the epistemic operators to formulas with at most one free variable. We provide the monodic fragments of the most important common knowledge predicate logics with finite Hilbertstyle axiomatizations, prove their completeness, and single out a number of decidable subfragments. On the other hand, we show that the addition of equality to the monodic fragment makes it not recursively enumerable. 1 Introduction Ever since it became common knowledge that the intelligent behavior of an agent is based not only on her knowledge about the world but also on the knowledge about both her own and other agents' knowledge, logical formalisms desig...
Can't decide? Undecide!
"... In my mathematical youth, when I first learned of Gödel’s Theorem, and computational undecidability, I was at once fascinated and strangely reassured of our limited place in the grand universe: incredibly mathematics itself establishes limits on mathematical knowledge. At the same time, as one digs ..."
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Cited by 3 (0 self)
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In my mathematical youth, when I first learned of Gödel’s Theorem, and computational undecidability, I was at once fascinated and strangely reassured of our limited place in the grand universe: incredibly mathematics itself establishes limits on mathematical knowledge. At the same time, as one digs into the formalisms, this area can seem remote from most areas of mathematics and irrelevant to the efforts of most workaday mathematicians. But that’s just not so! Undecidable problems surround us, everywhere, even in recreational mathematics!