Formalization of Shannon’s Theorems

Affeldt, Reynald; Hagiwara, Manabu; Sénizergues, Jonas

doi:10.1007/s10817-013-9298-1

Cited by 19 publications

(8 citation statements)

References 7 publications

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“…We interpret our monadic encoding in terms of Ramsey’s probability monad [ 42 ], which decomposes a complex distribution into composition of primitive ones bound together via conditional distributions. To capture this interpretation within Coq, we then use the encoding of this monad from the infotheo library [ 1 , 2 ], and provide a function that evaluates computations into distributions by recursively mapping them to the probability monad. Here, represents infotheo ’s encoding of distributions over a finite support , defined as being composed of a measure function , and a proof that the sum of the measure over the support A produces 1.…”

Section: Encoding Amqs In Coqmentioning

confidence: 99%

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Certifying Certainty and Uncertainty in Approximate Membership Query Structures

Gopinathan

Sergey

2020

Computer Aided Verification

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Approximate Membership Query structures (AMQs) rely on randomisation for time-and space-efficiency, while introducing a possibility of false positive and false negative answers. Correctness proofs of such structures involve subtle reasoning about bounds on probabilities of getting certain outcomes. Because of these subtleties, a number of unsound arguments in such proofs have been made over the years. In this work, we address the challenge of building rigorous and reusable computer-assisted proofs about probabilistic specifications of AMQs. We describe the framework for systematic decomposition of AMQs and their properties into a series of interfaces and reusable components. We implement our framework as a library in the Coq proof assistant and showcase it by encoding in it a number of non-trivial AMQs, such as Bloom filters, counting filters, quotient filters and blocked constructions, and mechanising the proofs of their probabilistic specifications. We demonstrate how AMQs encoded in our framework guarantee the absence of false negatives by construction. We also show how the proofs about probabilities of false positives for complex AMQs can be obtained by means of verified reduction to the implementations of their simpler counterparts. Finally, we provide a library of domain-specific theorems and tactics that allow a high degree of automation in probabilistic proofs.

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Section: Encoding Amqs In Coqmentioning

confidence: 99%

“…Our mechanised development [24] is entirely axiom-free, and is compatible with Coq 8.11.0 [11] and MathComp 1.10 [31]. It relies on the infotheo library [2] for encoding discrete probabilities.…”

Section: Introductionmentioning

confidence: 99%

Certifying Certainty and Uncertainty in Approximate Membership Query Structures

Gopinathan

Sergey

2020

Computer Aided Verification

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“…As we will point out in Section 3, the codings Deflate requires do not need to be Huffman codings, but they need to satisfy a canonicity condition. From the general topic of data compression, there is a formalization of Shannon's theorems in Coq [5].…”

Section: Related Workmentioning

confidence: 99%

“…We now have to save the coding 1 , and again, it is sufficient to save the code 1 lengths. These code 1 lengths for the 19 codepoints 1 are saved as 3 bit leastsignificant-bit first numbers, but in the following order: 16,17,18,0,8,7,9,6,10,5,11,4,12,3,13,2,14,1,15. Again, the codepoint 2 0 denotes that the corresponding codepoint 1 does not occur.…”

Section: B An Overview Of Deflatementioning

confidence: 99%

An Implementation of Deflate in Coq

Senjak

Hofmann

2016

FM 2016: Formal Methods

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The widely-used compression format "Deflate" is defined in RFC 1951 and is based on prefix-free codings and backreferences. There are unclear points about the way these codings are specified, and several sources for confusion in the standard. We tried to fix this problem by giving a rigorous mathematical specification, which we formalized in Coq. We produced a verified implementation in Coq which achieves competitive performance on inputs of several megabytes. In this paper we present the several parts of our implementation: a fully verified implementation of canonical prefix-free codings, which can be used in other compression formats as well, and an elegant formalism for specifying sophisticated formats, which we used to implement both a compression and decompression algorithm in Coq which we formally prove inverse to each otherthe first time this has been achieved to our knowledge. The compatibility to other Deflate implementations can be shown empirically. We furthermore discuss some of the difficulties, specifically regarding memory and runtime requirements, and our approaches to overcome them.

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“…Affeldt et al [9] formally verify basic definitions and theorems in information theory by using SSReflect, and also Shannon's channel coding theorem and source coding theorem [10] which are famous theorems of all results of information theory. These formalized theorems are valuable not only toward the formally verification of integrity between coding and decoding algorithms and the software implemented ones, but also further facilitating of the formal verification by utilizing the formalized types.…”

Section: Introductionmentioning

confidence: 99%

Formal Verification of Robertson-Type Uncertainty Relation

Masuhara¹,

Kuriyama²,

Yoshida³

et al. 2015

JQIS

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Formal verification using interactive theorem provers have been noticed as a method of verification of proofs that are too big for humans to check the validity of them. The purpose of this work is to verify the validity of Robertson-type uncertainty relation toward verifying unconditional security of quantum key distributions. We verify the validity of the relation by using proof assistant Coq and it is turned out that the theorem regarding the relation formally holds. The source code for Coq which represents the validity of the theorem is printed in Appendix.

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Formalization of Shannon’s Theorems

Cited by 19 publications

References 7 publications

Certifying Certainty and Uncertainty in Approximate Membership Query Structures

Certifying Certainty and Uncertainty in Approximate Membership Query Structures

An Implementation of Deflate in Coq

Formal Verification of Robertson-Type Uncertainty Relation

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