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Searching Constant Width Mazes Captures the AC° Hierarchy
 In Proceedings of the 15th Annual Symposium on Theoretical Aspects of Computer Science
, 1997
"... We show that searching a width /' maze is complete for II, i.e., for the /"th level of the AC hierarchy. Equivalently, stconnectivity for width /' grid graphs is complete for II. As an application, we show that there is a data structure solving dynamic stconnectivity for constant width grid gr ..."
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Cited by 24 (4 self)
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We show that searching a width /' maze is complete for II, i.e., for the /"th level of the AC hierarchy. Equivalently, stconnectivity for width /' grid graphs is complete for II. As an application, we show that there is a data structure solving dynamic stconnectivity for constant width grid graphs with time bound O (log log n) per operation on a random access machine. The dynamic algorithm is derived from the parallel one in an indirect way using algebraic tools.
Sheaf Representation for Topoi
, 1997
"... It is shown that every (small) topos is equivalent to the category of global sections of a sheaf of socalled hyperlocal topoi, improving on a result of Lambek & Moerdijk. It follows that every boolean topos is equivalent to the global sections of a sheaf of wellpointed topoi. Completeness theo ..."
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Cited by 2 (1 self)
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It is shown that every (small) topos is equivalent to the category of global sections of a sheaf of socalled hyperlocal topoi, improving on a result of Lambek & Moerdijk. It follows that every boolean topos is equivalent to the global sections of a sheaf of wellpointed topoi. Completeness theorems for higherorder logic result as corollaries. The main result of this paper is the following. Theorem (Sheaf representation for topoi). For any small topos E, there is a sheaf of categories e E on a topological space, such that: (i) E is equivalent to the category of global sections of e E, (ii) every stalk of e E is a hyperlocal topos. Moreover, E is boolean just if every stalk of e E is wellpointed. Before defining the term "hyperlocal," we indicate some of the background of the theorem. The original and most familiar sheaf representations are for commutative rings (see [12, ch. 5] for a survey); e.g. a wellknown theorem due to Grothendieck [9] asserts that every commutative r...
A Complete Axiomatization of HigherOrder Intuitionistic Logic
 CLE ePrints
, 2001
"... Two Hilbert calculi for higherorder logic (or theory of types) are introduced. ..."
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Cited by 2 (2 self)
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Two Hilbert calculi for higherorder logic (or theory of types) are introduced.
Topological Representation of the &ambda;Calculus
, 1998
"... The calculus can be represented topologically by assigning certain spaces to the types and certain continuous maps to the terms. Using a recent result from category theory, the usual calculus of conversion is shown to be deductively complete with respect to such topological semantics. It is al ..."
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The calculus can be represented topologically by assigning certain spaces to the types and certain continuous maps to the terms. Using a recent result from category theory, the usual calculus of conversion is shown to be deductively complete with respect to such topological semantics. It is also shown to be functionally complete, in the sense that there is always a "minimal" topological model, in which every continuous function is definable. These results subsume earlier ones using cartesian closed categories, as well as those employing socalled Henkin and Kripke models. Introduction The calculus originates with Church [6]; it is intended as a formal calculus of functional application and specification. In this paper, we are mainly interested in the version known as simply typed calculus ; as is now wellknown, the untyped version can be treated as a special case of this ([17]). We present here a topological representation of the calculus: types are represented by cert...
Topological Completeness of FirstOrder Modal Logic
"... As McKinsey and Tarski [20] showed, the Stone representation theorem for Boolean algebras extends to algebras with operators to give topological semantics for (classical) propositional modal logic, in which the “necessity ” operation is modeled by taking the interior of an arbitrary subset of a topo ..."
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As McKinsey and Tarski [20] showed, the Stone representation theorem for Boolean algebras extends to algebras with operators to give topological semantics for (classical) propositional modal logic, in which the “necessity ” operation is modeled by taking the interior of an arbitrary subset of a topological space. This topological interpretation was recently extended in a natural way to arbitrary theories of full firstorder logic by Awodey and Kishida [3], using topological sheaves to interpret domains of quantification. This paper proves the system of full firstorder S4 modal logic to be deductively complete with respect to such extended topological semantics. The techniques employed are related to recent work in topos theory, but are new to systems of modal logic. They are general enough to also apply to other modal systems. Keywords: Firstorder modal logic, topological semantics, completeness.