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The complexity of computing a Nash equilibrium
, 2006
"... We resolve the question of the complexity of Nash equilibrium by showing that the problem of computing a Nash equilibrium in a game with 4 or more players is complete for the complexity class PPAD. Our proof uses ideas from the recentlyestablished equivalence between polynomialtime solvability of n ..."
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Cited by 324 (23 self)
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We resolve the question of the complexity of Nash equilibrium by showing that the problem of computing a Nash equilibrium in a game with 4 or more players is complete for the complexity class PPAD. Our proof uses ideas from the recentlyestablished equivalence between polynomialtime solvability of normalform games and graphical games, and shows that these kinds of games can implement arbitrary members of a PPADcomplete class of Brouwer functions. 1
Categorial Type Logics
 Handbook of Logic and Language
, 1997
"... Contents 1 Introduction: grammatical reasoning 1 2 Linguistic inference: the Lambek systems 5 2.1 Modelinggrammaticalcomposition ............................ 5 2.2 Gentzen calculus, cut elimination and decidability . . . . . . . . . . . . . . . . . . . . 9 2.3 Discussion: options for resource mana ..."
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Cited by 299 (6 self)
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Contents 1 Introduction: grammatical reasoning 1 2 Linguistic inference: the Lambek systems 5 2.1 Modelinggrammaticalcomposition ............................ 5 2.2 Gentzen calculus, cut elimination and decidability . . . . . . . . . . . . . . . . . . . . 9 2.3 Discussion: options for resource management . . . . . . . . . . . . . . . . . . . . . . 13 3 The syntaxsemantics interface: proofs and readings 16 3.1 Term assignment for categorial deductions . . . . . . . . . . . . . . . . . . . . . . . . 17 3.2 Natural language interpretation: the deductive view . . . . . . . . . . . . . . . . . . . 21 4 Grammatical composition: multimodal systems 26 4.1 Mixedinference:themodesofcomposition........................ 26 4.2 Grammaticalcomposition:unaryoperations ....................... 30 4.2.1 Unary connectives: logic and structure . . . . . . . . . . . . . . . . . . . . . . . 31 4.2.2 Applications: imposing constraints, structural relaxation
Relativization: A revisionistic retrospective
 Bulletin of the EATCS
, 1992
"... In this column we examine the role of relativization in complexity theory in light of recent nonrelativizing results involving interactive protocols. We begin with the twicetold tale of the relativization principle and ponder upon its possible demise. Then, we discuss whether usual assumptions are ..."
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Cited by 7 (0 self)
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In this column we examine the role of relativization in complexity theory in light of recent nonrelativizing results involving interactive protocols. We begin with the twicetold tale of the relativization principle and ponder upon its possible demise. Then, we discuss whether usual assumptions
A Method for enforcing integrability in shape from shading algorithms
 IEEE Transactions on Pattern Analysis and Machine Intelligence
, 1988
"... AbstractSeveral recently developed techniques for reconstructing surface shape from shading information estimate surface slopes without ensuring that they are integrable. This paper presents an approach for enforcing integrability, a particular implementation of the approach, an example of its ap ..."
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Cited by 283 (6 self)
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AbstractSeveral recently developed techniques for reconstructing surface shape from shading information estimate surface slopes without ensuring that they are integrable. This paper presents an approach for enforcing integrability, a particular implementation of the approach, an example of its application to extending an existing shapefromshading algorithm, and experimental results showing the improvement that results from enforcing integrability. A possibly nonintegrable estimate of surface slopes is represented by a finite set of basis functions, and integrability is enforced by calculating the orthogonal projection onto a vector subspace spanning the set of integrable slopes. This projection maps closed convex sets into closed convex sets and, hence, is attractive as a constraint in iterative algorithms. The same technique is also useful for noniterative algorithms since it provides a leastsquares fit of integrable slopes to nonintegrable slopes in one pass of the algorithm. The special case of Fou
Proof Complexity of Relativized Statements
, 2012
"... The firstorder formulas that do not have finite models give rise to uniform families of propositional unsatisfiable formulas, one for each finite cardinality. Motivated by the question of characterizing the class of such formulas whose propositional translations have short resolution refutations, w ..."
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, we study their closure properties. Our starting point is the observation that the class of relativized formulas that have polynomiallength refutations in R(c) is closed under quantifierfree interpretations. The analogous question for unrelativized formulas remains open and has an intriguing status
Constructivism and Proof Theory
, 2003
"... Introduction to the constructive point of view in the foundations of mathematics, in
particular intuitionism due to L.E.J. Brouwer, constructive recursive mathematics
due to A.A. Markov, and Bishop’s constructive mathematics. The constructive interpretation
and formalization of logic is described. F ..."
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Cited by 204 (4 self)
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principles which are valid for choice sequences are discussed.
The second half of the article deals with some aspects of proof theory, i.e.,
the study of formal proofs as combinatorial objects. Gentzen’s fundamental contributions
are outlined: his introduction of the socalled Gentzen systems which use
Relativizing Chaitin’s halting probability
 J. Math. Log
"... Abstract. As a natural example of a 1random real, Chaitin proposed the halting probability Ω of a universal prefixfree machine. We can relativize this example by considering a universal prefixfree oracle machine U. Let Ω A U be the halting probability of U A; this gives a natural uniform way of p ..."
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Cited by 34 (9 self)
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Abstract. As a natural example of a 1random real, Chaitin proposed the halting probability Ω of a universal prefixfree machine. We can relativize this example by considering a universal prefixfree oracle machine U. Let Ω A U be the halting probability of U A; this gives a natural uniform way
Semantic database modeling: Survey, applications, and research issues
 ACM Computing Surveys
, 1987
"... Most common database management systems represent information in a simple recordbased format. Semantic modeling provides richer data structuring capabilities for database applications. In particular, research in this area has articulated a number of constructs that provide mechanisms for representi ..."
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Cited by 261 (3 self)
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Most common database management systems represent information in a simple recordbased format. Semantic modeling provides richer data structuring capabilities for database applications. In particular, research in this area has articulated a number of constructs that provide mechanisms for representing structurally complex interrelations among data typically arising in commercial applications. In general terms, semantic modeling complements work on knowledge representation (in artificial intelligence) and on the new generation of database models based on the objectoriented paradigm of programming languages. This paper presents an indepth discussion of semantic data modeling. It reviews the philosophical motivations of semantic models, including the need for highlevel modeling abstractions and the reduction of semantic overloading of data type constructors. It then provides a tutorial introduction to the primary components of semantic models, which are the explicit representation of objects, attributes of and relationships among objects, type constructors for building complex types, ISA relationships, and derived schema components. Next, a survey of the prominent semantic models in the literature is presented. Further, since a broad area of research has developed around semantic modeling, a number of related topics based on these models are discussed, including data languages, graphical interfaces, theoretical investigations, and physical implementation strategies.
Notes on Polynomially Bounded Arithmetic
"... We characterize the collapse of Buss' bounded arithmetic in terms of the provable collapse of the polynomial time hierarchy. We include also some general modeltheoretical investigations on fragments of bounded arithmetic. Contents 0 Introduction and motivation. 1 1 Preliminaries. 3 1.1 The p ..."
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Cited by 60 (1 self)
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We characterize the collapse of Buss' bounded arithmetic in terms of the provable collapse of the polynomial time hierarchy. We include also some general modeltheoretical investigations on fragments of bounded arithmetic. Contents 0 Introduction and motivation. 1 1 Preliminaries. 3 1
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