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Coherence Spaces for Resource-Sensitive Computation in ... · Coherence Spaces for...
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Coherence Spaces for Resource-Sensitive Computation in Analysis
Kei Matsumoto(RIMS, Kyoto University)
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Background● Computable analysis studies computation over topological spaces, by giving representations.
– Type two theory of Effectivity– Domain representations
● Their approaches are to track computation by continuous maps over “symbolic” spaces.
The principle: Computable ⇒ ContinuousBaire sp., Scott domains, ...
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Our Proposal
● Our principle: Computable ⇒ Stable [Berry '78]Using instead of Scott-domains coherence spaces [Girard '86]. ● BTW, two morphisms coexists in coherence spaces:
stable & linear maps. ● A new question then arrises:
YX
FX Y
fTopological sp.
Coherence sp.
Tracked by stable map.
What are Linear Computations in Topology?
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Girard's Linear LogicOne of the most influential papers in 80's in both logic and computer science.
● Restructuring both Classical & Intuitionistic Logic● Proof Nets● Resource-Conciousness
Attractive Ideas:
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Girard's Linear Logic
● Restructuring both Classical & Intuitionistic Logic● Proof Nets● Resource-Conciousness
Attractive Ideas:
One of the most influential papers in 80's in both logic and computer science.
![Page 6: Coherence Spaces for Resource-Sensitive Computation in ... · Coherence Spaces for Resource-Sensitive Computation in Analysis Kei Matsumoto (RIMS, Kyoto University) 2 Background ...](https://reader033.fdocuments.us/reader033/viewer/2022042100/5e7c19f5ebce06798c5909f5/html5/thumbnails/6.jpg)
Girard's Linear Logic
● Restructuring both Classical & Intuitionistic Logic● Proof Nets● Resource-Conciousness
Attractive Ideas:
One of the most influential papers in 80's in both logic and computer science.
![Page 7: Coherence Spaces for Resource-Sensitive Computation in ... · Coherence Spaces for Resource-Sensitive Computation in Analysis Kei Matsumoto (RIMS, Kyoto University) 2 Background ...](https://reader033.fdocuments.us/reader033/viewer/2022042100/5e7c19f5ebce06798c5909f5/html5/thumbnails/7.jpg)
Girard's Linear Logic
● Restructuring both Classical & Intuitionistic Logic● Proof Nets● Resource-Conciousness
Attractive Ideas:
One of the most influential papers in 80's in both logic and computer science.
![Page 8: Coherence Spaces for Resource-Sensitive Computation in ... · Coherence Spaces for Resource-Sensitive Computation in Analysis Kei Matsumoto (RIMS, Kyoto University) 2 Background ...](https://reader033.fdocuments.us/reader033/viewer/2022042100/5e7c19f5ebce06798c5909f5/html5/thumbnails/8.jpg)
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Resource-Sensitivity
Imagine:
there's no resource-conciousness
It isn't easy to do
Nothing to comsume or lose for
And no modalities too
Imagine all the people
Living life in Intuitionistic Logic ...
’’
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Resource-Sensitivity
Ex. In Intuitionistic Logic, is true.
a person in the Intuitionistic Logic world
Substitute:● A:= “to pay ¥400”● B:= “to get a pack of cigarettes”● C:= “to get a cup of cake”
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Resource-Sensitivity
Ex. In Intuitionistic Logic, is true.
Substitute:● A:= “to pay ¥400”● B:= “to get a pack of cigarettes”● C:= “to get a cup of cake”
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Resource-Sensitivity
Ex. In Intuitionistic Logic, is true.
Substitute:● A:= “to pay ¥400”● B:= “to get a pack of cigarettes”● C:= “to get a cup of cake”
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Resource-Sensitivity
Ex. In Intuitionistic Logic, is true.
Substitute:● A:= “to pay ¥400”● B:= “to get a pack of cigarettes”● C:= “to get a cup of cake”
Paradox
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Resource-Sensitivity
Ex. In Intuitionistic Logic, is true.
Substitute:● A:= “to pay ¥400”● B:= “to get a pack of cigarettes”● C:= “to get a cup of cake”
Lack of conciousness to comsume assumptions !
A is used twice.
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Resource-Sensitivity
Ex. In Linear Logic, is false. Because: In LL, we must use the assumption exactly once in the proof. Coherence Spaces are proposed as a denotational semantics which reflects this property. Via the Curry-Howard isomorphism, they are also a model of resource-sensitive computations of linear function programs.
New conjunction/ implication
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Main Result from CCA’15Representations based on coherence spaces have an interesting feature: for every real funcitons, we have shown that
● stably realizable ⇔ continuous ● linearly realizable ⇔ uniformly continuous.
Let us emphasize that these correspondences hold for real functions.Next step: generalize them to a wider class.
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Ⅰ. Review: Coherent Spaces
Ⅱ. Coherence as Uniformity
Ⅲ. Linear Admissibility
Ⅳ. Concluding Comments
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Coherence Spaces
Def. A coherence space is a reflexive graph: ● a countable set of tokens with● a symmetric reflexive. binary rel. on
Write iff and (strict coherence)A clique is a set of tokens which are pairwise coherent. An anticlique is a set of tokens in which every pair is not coherent. ● :the set of all cliuqes. ● :the set of all finite cliques. ● :the set of all maximal cliques.
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Example: Cauchy SequencesLet . Each member of is identified with the dyadic rational as .For each , define
●
●
●
Ex. Define a coherence space
for dyadic Cauchy sequences as:
Maximal cliques ≈ (rapidly converging) Cauchy sequencesRealization of Real Numbers
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Example: Cauchy SequencesLet . Each member of is identified with the dyadic rational as .For each , define
●
●
●
Ex. Define a coherence space
for dyadic Cauchy sequences as:
Maximal cliques ≈ (rapidly converging) Cauchy sequencesRealization of Real Numbers
the “stage”“spotlights”
classified by “colors”
compact intervals are “projected”
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Example: Cauchy SequencesLet . Each member of is identified with the dyadic rational as .For each , define
●
●
●
Ex. Define a coherence space
for dyadic Cauchy sequences as:
Maximal cliques ≈ (rapidly converging) Cauchy sequencesRealization of Real Numbers
the “stage”“spotlights”
classified by “colors”
compact intervals are “projected”
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Example: Cauchy SequencesLet . Each member of is identified with the dyadic rational as .For each , define
●
●
●
Ex. Define a coherence space
for dyadic Cauchy sequences as:
Maximal cliques ≈ (rapidly converging) Cauchy sequencesRealization of Real Numbers
the “stage”“spotlights”
classified by “colors”
compact intervals are “projected”
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Example: Cauchy SequencesLet . Each member of is identified with the dyadic rational as .For each , define
●
●
●
Ex. Define a coherence space
for dyadic Cauchy sequences as:
Maximal cliques ≈ (rapidly converging) Cauchy sequencesRealization of Real Numbers
the “stage”“spotlights”
classified by “colors”
Lights of different colors are overlapped
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Example: Cauchy SequencesLet . Each member of is identified with the dyadic rational as .For each , define
●
●
●
Ex. Define a coherence space
for dyadic Cauchy sequences as:
Maximal cliques ≈ (rapidly converging) Cauchy sequencesRealization of Real Numbers
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Coherence as TopologyThe set of cliques is ordered by ⊆ , endowed with the Scott topology generated by the upper sets of finite cliques:
● Coherence spaces are very simplified domains● Compact Elements = finite cliques
Finite Cliques induce Topology
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Stable & Linear MapsDef. A function is stable if it is -⊆ monotone and satisfies
Model of computations in which the amount of resources to be used is uniquely determined. Def. A function is linear if it is -⊆ monotone and satisfies
Model of computations in which resources are used exactly once. Stable & Linear Maps are Resource-Sensitive.
Collection of resources
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Girard’s FormulaTwo closed structures of coherence spaces:
● The category Stbl of coh. spaces and stable maps is cartesian closed – : the coherence space for stable maps.
● The category Lin of coh. spaces and linear maps is monoidal closed – : the coherence space for linear maps.
They are combined by introducing the “of course” modality: naturally defined by .
Model of Intuit. Logic
Th.
Linear Decompostion of Cartesian closed Structure.
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Ⅰ. Review: Coherent Spaces
Ⅱ. Coherence as Uniformity
Ⅲ. Linear Admissibility
Ⅳ. Concluding Comments
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Review: Uniform Space
Uniform Covers are given horizontally
● A uniform space is a set with a uniformity: a collection of coverings of the set. – Each uniform cover is considered to be consisting of balls of the “same size”– They are partially ordered by the refinement relation and form a filter.
● They also induce a topology in the “vertical” way. – The balls surrounding each point form a neighborhood filter, which generates the uniform topology.
● Every uniformizable space has the finest uniformity: the fine uniform space.
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Review: Uniform Space
Uniform Covers are given horizontally
● A uniform space is a set with a uniformity: a collection of coverings of the set. – Each uniform cover is considered to be consisting of balls of the “same size”– They are partially ordered by the refinement relation and form a filter.
● They also induce a topology in the “vertical” way. – The balls surrounding each point form a neighborhood filter, which generates the uniform topology.
● Every uniformizable space has the finest uniformity: the fine uniform space.
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Review: Uniform Space
Uniform Covers are given horizontally
● A uniform space is a set with a uniformity: a collection of coverings of the set. – Each uniform cover is considered to be consisting of balls of the “same size”– They are partially ordered by the refinement relation and form a filter.
● They also induce a topology in the “vertical” way. – The balls surrounding each point form a neighborhood filter, which generates the uniform topology.
● Every uniformizable space has the finest uniformity: the fine uniform space.
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Review: Uniform Space
neighbourhood filter
Neighbors are given vertically
● A uniform space is a set with a uniformity: a collection of coverings of the set. – Each uniform cover is considered to be consisting of balls of the “same size”– They are partially ordered by the refinement relation and form a filter.
● They also induce a topology in the “vertical” way. – The balls surrounding each point form a neighborhood filter, which generates the uniform topology.
● Every uniformizable space has the finest uniformity: the fine uniform space.
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Review: Uniform Space
Th. Let be the fine space compatible with a topological space .
We’ve seen that situation!
● A uniform space is a set with a uniformity: a collection of coverings of the set. – Each uniform cover is considered to be consisting of balls of the “same size”– They are partially ordered by the refinement relation and form a filter.
● They also induce a topology in the “vertical” way. – The balls surrounding each point form a neighborhood filter, which generates the uniform topology.
● Every uniformizable space has the finest uniformity: the fine uniform space.
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Girard’s FormulaTwo closed structures of coherence spaces:
● The category Stbl of coh. spaces and stable maps is cartesian closed – : the coherence space for stable maps.
● The category Lin of coh. spaces and linear maps is monoidal closed – : the coherence space for linear maps.
They are combined by introducing the “of course” modality: naturally defined by .
Model of Intuit. Logic
Th.
Linear Decompostion of Cartesian closed Structure.
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Coherence as UniformityRecall: Incoherence implies disjointness: A partition of is an anticlique which induces the disjoint covering. Condition: a coherence space is disjointly coverable if every token can be extended to a partition:
Anticliques induce Uniformity
Th. 1) Partitions of form a subbasis for a uniformity. 2) Partitions of form a basis for a uniformity. 3) Both uniformities induce the Scott topology as the uniform topologies.4) The induced uniformity on is fine.
Prop. (homeomorphic.)
Coh. Sp. for anticliques
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Cauchy Sequences AgainEx. Define a coherence space
for dyadic Cauchy sequences as:
• In any partition, spotlights of different colors must project disjoint sets by the second condition of the incoherence. • This is impossible essentially due to Sierpiński's theorem. • The partitions then generate the uniformity compatible with the real line, the representation is a uniformly open map.
Each is a partition of , because a maximal clique must contain “all colors”. There are no other partitions consisting of “several colors”.
( incoherence)
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Linear ⇒ Uniform ContinuousRecall that
Th. Every linear map is uniformly continuous with respect to the uniformities induced by partitions. Cor. Every stable map is continuous.
(although it is a reinvention of the wheel...)
Th. Th.
We then combine these results.Assume that preserves maximality of cliques.
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Ⅰ. Review: Coherent Spaces
Ⅱ. Coherence as Uniformity
Ⅲ. Linear Admissibility
Ⅳ. Concluding Comments
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Standard RepresentationsLet be a Hausdorff uniform space with a countable basis consisting of countable coverings. Fact. Such a space is known to be separable metrizable. Def. The standard representation of is given by the coherence space defined by and
Each Maximal clique of specifies at most one point.
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Linear AdmissibilityStandard representation does not depend on the choice of basis :This generalizes to the notion of admissibility. An Idea. A representation is linear admissible if 1. for every uniform cover of there exists a uniform cover of which refines , and2. for every subspace and its representation satisfying (1) the inclusion map is tracked by some linear map .
uniform cont.
A naive idea is to mimic Admissibility in TTE, but it doesn’t work!!
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Linear AdmissibilityStandard representation does not depend on the choice of basis :This generalizes to the notion of admissibility. An Idea. A representation is linear admissible if 1. for every uniform cover of there exists a uniform cover of which refines , and2. for every subspace and its representation satisfying (1) the inclusion map is tracked by some linear map .
uniform cont.
A naive idea is to mimic Admissibility in TTE, but it doesn’t work!!
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Linear AdmissibilityStandard representation does not depend on the choice of basis :This generalizes to the notion of admissibility. Def. A representation is linear admissible if 1. for every uniform cover of there exists a partition of which refines , and2. for every subspace and its representation satisfying (1) the inclusion map is tracked by some linear map .
Th. (1) Every uniformly continuous maps is then linearly realizable w.r.t. linear admissible representations.(2) Every standard representation is linear admissible.
strongly uniform cont.
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Chain-Connectedness
Def. A uniform space is chain-connected if
A typical example: (though it is totally disconnected)In particular, every two members of a uniform cover is “chainly connected”
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Chain-Connectedness
Def. A uniform space is chain-connected if
A typical example: (though it is totally disconnected)In particular, every two members of a uniform cover is “chainly connected”
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Chain-Connectedness
Def. A uniform space is chain-connected if
A typical example: (though it is totally disconnected)In particular, every two members of a uniform cover is “chainly connected”
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Chain-Connectedness
Def. A uniform space is chain-connected if
A typical example: (though it is totally disconnected)In particular, every two members of a uniform cover is “chainly connected”
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Principal LemmaSuppose that is a chain-connected separable metrizable space and that is a uniform basis consiting of open coverings. N.B. Every uniform space has a basis of open coverings.Lem. The standard representation is a topologically and uniformly open map. ● Topological openness is immediate from the assumption.● Uniform openness is essentially due to “one-coloredness” of partitions.
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Principal LemmaSuppose that is a chain-connected separable metrizable space and that is a uniform basis consiting of open coverings. N.B. Every uniform space has a basis of open coverings.Lem. The standard representation is a topologically and uniformly open map. ● Topological openness is immediate from the assumption.● Uniform openness is essentially due to “one-coloredness” of partitions.
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Principal LemmaSuppose that is a chain-connected separable metrizable space and that is a uniform basis consiting of open coverings. N.B. Every uniform space has a basis of open coverings.Lem. The standard representation is a topologically and uniformly open map. ● Topological openness is immediate from the assumption.● Uniform openness is essentially due to “one-coloredness” of partitions.
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Principal LemmaSuppose that is a chain-connected separable metrizable space and that is a uniform basis consiting of open coverings. N.B. Every uniform space has a basis of open coverings.Lem. The standard representation is a topologically and uniformly open map. ● Topological openness is immediate from the assumption.● Uniform openness is essentially due to “one-coloredness” of partitions.
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Principal LemmaSuppose that is a chain-connected separable metrizable space and that is a uniform basis consiting of open coverings. N.B. Every uniform space has a basis of open coverings.Lem. The standard representation is a topologically and uniformly open map. ● Topological openness is immediate from the assumption.● Uniform openness is essentially due to “one-coloredness” of partitions.
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Main Result
Th. If is chain-connected, ● : stably realizable ⇔ it is continuous.● : linearly realizable ⇔ it is uniformly continuous.Cor. If is linearly realizable then it is uniformly continuous on each chain-connected component.
Let and be separable metrizable uniform spaces with linear admissible representations, and .
Conversely, every uniformly continuous function is linearly realizable. To complete the corollary, we need to reduce the components by identifying some of them.
Recall: every separable metrizable space has the standard representation, hence has linear admissible representations.
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Ⅰ. Review: Coherent Spaces
Ⅱ. Coherence as Uniformity
Ⅲ. Linear Admissibility
Ⅳ. Concluding Comments
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Towards Linear Realizability● A linear combinatory algebra (LCA) U lin is defined from the “universal coherence space”. ● Coherent Representations = Modest Sets over Ulin● Mod (Ulin) is a model of linear logic. ● I'm essentially a realist...but still a bit a dreamer
● so I imagine there's a kind of “Linear Analysis” as the decomposition of Computable Analysis. – An analogy of the discovery of Linear Logic. – Every mathematical space has “admissible” representations in some sense, and functions are all linearly computable...
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Towards Complexity in Analysis● It seems hopeless that the category of l inear admissible representations is monoidal closed because function spaces are not separable metrizable.● Nonetheless, I still believe that linearity is strongly related to uniform structures because:
Th (Férée-Gomaa-Hoyrup' 13). For any real functional
linear in our terminology
F is “uniformly continuous” w.r.t. a kind of uniformity on C[0,1]: We can explain this phenomenon in our model: Uniformity on C[0,1] a point in [0,1] & a uniformity of R
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A Possible Way: Quasi-Uniformity● In some sense, separable metrisability of linear admissible representations seems inevitable:
– Because of countability of coherence spaces. ● We must have a countable basis of countable coverings.
● But... the Hausdorff property seems not necessary. ● just used for well-definedness of the representation.
● The use of non-Hausdorff metric seems a possible answer.● Every second-countable T1-space is separable quasi-metrizable.
– It is very likely that they have the standard representations for quasi-uniform spaces.