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Page 7142 of Mathematical Reviews Vol. , Issue 95m [page]

1995 Mathematical Reviews  
Kosta Dosen (F-TOUL3-IR; Toulouse) 95m:03015 03B20 03B15 Polacik, Tomasz (PL-SILS-IM; Katowice) Operators defined by propositional quantification and their interpretation over Cantor space.  ...  These systems have modal operators 0, indexed by individual variables x, and their characteristic axiom schemata, originating from G.  ... 

Semi-decidability of May, Must and Probabilistic Testing in a Higher-type Setting

Martín Escardó
2009 Electronical Notes in Theoretical Computer Science  
Choices can only be performed at powertypes, and the different powertypes have different operational interpretations of choice.  ...  The types are closed under finite products and function spaces, and certain powertype constructors, interpreted as powerdomain monads, which capture various kinds of non-determinism.  ...  Quantification and integration over the Cantor space. For the operational semantics defined in Section 5 below, we need quantification and integration over the Cantor type Cantor = (Nat → Bool).  ... 
doi:10.1016/j.entcs.2009.07.092 fatcat:3wzlzrjh6zawhj5cijidfmuiaq

Synthetic Topology

Martín Escardó
2004 Electronical Notes in Theoretical Computer Science  
We are aware of other applications of synthetic topology, e.g. to locales, convergence spaces, sequential spaces, equilogical spaces, and some sheaf and realizability toposes, but this will be reported  ...  In fact, we start by developing synthetic topology of programming-language data types in Part I, without assuming any background in classical topology and without introducing any.  ...  The functional program for universal quantification over the Cantor space provided here is due to the author but it is related to a program formerly discovered by Berger [13] , which we present in Chapter  ... 
doi:10.1016/j.entcs.2004.09.017 fatcat:qb7ckcpuvzgbrmkq2fdsom7vae

Synthetic Topologyof Data Types and Classical Spaces

M ESCARDO
2004 Electronical Notes in Theoretical Computer Science  
We are aware of other applications of synthetic topology, e.g. to locales, convergence spaces, sequential spaces, equilogical spaces, and some sheaf and realizability toposes, but this will be reported  ...  In fact, we start by developing synthetic topology of programming-language data types in Part I, without assuming any background in classical topology and without introducing any.  ...  The functional program for universal quantification over the Cantor space provided here is due to the author but it is related to a program formerly discovered by Berger [13] , which we present in Chapter  ... 
doi:10.1016/s1571-0661(04)05135-7 fatcat:2z7ew7j3erarxioan4uqjnjssy

Why topology in the minimalist foundation must be pointfree

Maria Emilia Maietti, Giovanni Sambin
2013 Logic and Logical Philosophy  
We heartily thank Francesco Ciraulo, Per Martin-Löf and Claudio Sacerdoti Coen for fruitful discussions on the topics of this paper.  ...  the set of positive natural numbers {x ∈ N | x ≥ 1}, and to define operations on such sets, we need to think of propositions as types of their proofs: small propositions are seen as sets of their proofs  ...  , Cantor space, . . . ) do not give rise to a point-wise topology since their points do not form a set.  ... 
doi:10.12775/llp.2013.010 fatcat:djpf6wtw4fd2lc3wpobxmbmsue

Measure theory and higher order arithmetic [article]

Alexander P. Kreuzer
2015 arXiv   pre-print
We investigate the statement that the Lebesgue measure defined on all subsets of the Cantor space exists. As base system we take ACA_0^ω + (μ).  ...  The system ACA_0^ω is the higher order extension of Friedman's system ACA_0, and (μ) denotes Feferman's μ, that is a uniform functional for arithmetical comprehension defined by f(μ(f))=0 if ∃ n f(n)=0  ...  We will denote by (λ) the statement the Lebesgue measure defined on all subsets of the Cantor-space exists.  ... 
arXiv:1312.1531v2 fatcat:tctxumkoyvb6rapk3hdhc7553e

Infinite sets that admit fast exhaustive search

Martin Escardo
2007 22nd Annual IEEE Symposium on Logic in Computer Science (LICS 2007)  
How fast can exhaustive search over infinite sets be performed?  ...  consult his paper [18] for a more accurate and detailed account.  ...  I also thank Dag Normann for having answered questions regarding the history and technical ramifications of the subject, and for sending me a copy of Tait's unpublished manuscript -but the reader should  ... 
doi:10.1109/lics.2007.25 dblp:conf/lics/Escardo07 fatcat:ntbideqoyzhlzge62whz7y32xa

Quantum set algebra for quantum set theory [article]

David Ritz Finkelstein
2014 arXiv   pre-print
Quantum field theory can be physically regularized by modularizing it on several levels of aggregation.  ...  Since computation is already thoroughly modularized, physical experiments are treated here as quantum relativistic cellular computations with spins for cells, address, memory, and control registers.  ...  I define the quantum binary field with domain port space X and binary range 2 to have the port space 2 X , the Grassmann algebra over X .  ... 
arXiv:1403.3725v1 fatcat:3etni7bgxncntioclxx3mix34i

Exhaustible sets in higher-type computation

Martin Escardo, Andrew Pitts
2008 Logical Methods in Computer Science  
The Cantor space of infinite sequences of binary digits is known to be searchable.  ...  We also show that, in the non-empty case, they are precisely the computable images of the Cantor space.  ...  and closed sets by their characteristic functions), and compact sets (using the function space S S X and representing compact sets by their universal quantification functionals).  ... 
doi:10.2168/lmcs-4(3:3)2008 fatcat:ptu4qbbbgjaeboe3rtxros5uoq

Measure theory and higher order arithmetic

Alexander P. Kreuzer
2015 Proceedings of the American Mathematical Society  
We investigate the statement that the Lebesgue measure defined on all subsets of the Cantor-space exists. As a base system we take ACA ω 0 +(μ).  ...  The system ACA ω 0 is the higher order extension of Friedman's system ACA 0 , and (μ) denotes Feferman's μ, that is, a uniform functional for arithmetical comprehension defined by f (μ(f )) = 0 if ∃n f  ...  measure defined on all subsets of the Cantor-space exists.  ... 
doi:10.1090/proc/12671 fatcat:ff3tab76nnffzmmfrqsxedjv7u

Girard's !() as a reversible fixed-point operator [article]

Peter Hines
2013 arXiv   pre-print
We demonstrate that it may be thought of as a fixed-point operation for reversible logic & computation.  ...  () exponential in the Geometry of Interaction system, with particular emphasis on the fact that the GoI interpretation 'forgets types'.  ...  Although the system described was logically degenerate (i.e. conjunction and disjunction were identified, as were universal and existential quantification, and propositions and their negations), the computational  ... 
arXiv:1309.0361v1 fatcat:lll6fjokojdsvfled6qzhcnucu

Applications of type theory [chapter]

Bernd Mahr
1993 Lecture Notes in Computer Science  
-calculus which truly solve the equation of reflexive domains, and e T-logic, a first-order theory of propositions with selfreference and impredicative quantification allowing for intcnsional models of  ...  Proofs and derivations are conventionally defined and give rise to tile notion of consequence R 1> J where R is a set of rules and J is a judgement that can be derivc~l using tile rules of R.  ...  quantification over propositions.  ... 
doi:10.1007/3-540-56610-4_75 fatcat:atgs6zm335d3xmwn5md3ty7o2u

What sequential games, the tychonoff theorem and the double-negation shift have in common

Martín Escardó, Paulo Oliva
2010 Proceedings of the third ACM SIGPLAN workshop on Mathematically structured functional programming - MSFP '10  
The functional makes sense for finite and infinite (lazy) lists, and in the binary case it amounts to an operation that is available in any (strong) monad.  ...  operation.  ...  When we run this, using the interpreter this time, we get: A non-trivial application of quantification over the Cantor space is given in [16] .  ... 
doi:10.1145/1863597.1863605 dblp:conf/icfp/EscardoO10 fatcat:y4rnbuktjzds3avoyxawcvd5s4

Natural Density and The Quantifier 'Most' [article]

Selçuk Topal, Ahmet Çevik
2019 arXiv   pre-print
We consider sentences of the form "Most A are B", where A and B are plural nouns and the interpretations of A and B are infinite subsets of N .  ...  This paper proposes a formalization of the class of sentences quantified by most, which is also interpreted as proportion of or majority of depending on the domain of discourse.  ...  Moss, John Corcoran and Georges Grekos for many useful discussions and patiently answering our questions.  ... 
arXiv:1901.10394v2 fatcat:dnsqipunpzcmja7xrfzgka5tme

The Limits of Computation

Andrew Powell
2021 Axiomathes  
Moreover, it is claimed that typed systems of the lambda calculus give rise naturally to a functional interpretation of rich systems of types and to a hierarchy of ordinal recursive functionals of arbitrary  ...  type that can be reduced by substitution to natural number functions.  ...  over types (written ), 8 which corresponds to "second-order" propositional logic with quantification over propositions.  ... 
doi:10.1007/s10516-021-09561-8 fatcat:w6nlldlpxvblpjgxp37u6jejke
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