Randomness and uniform distribution modulo one

Becher, Verónica; Grigorieff, Serge

doi:10.1016/j.ic.2021.104857

Cited by 4 publications

(6 citation statements)

References 13 publications

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“…In this section, we define the coinductive type of infinitary trees with finite branching factor (cotrees) as an algebraic CPO. We then use cotrees to encode samplers for discrete distributions in the random bit model [36,41] and show how AlgCo enables weakest pre-expectation [24,25,33] style reasoning about them (Section 6.1), culminating in Theorem 6.18 showing that sequences of samples are equidistributed [7] wrt. the weakest pre-expectation semantics of the cotrees used to generate them.…”

Section: Coinductive Binary Treesmentioning

confidence: 99%

Inductive Reasoning for Coinductive Types

Bagnall¹,

Stewart²,

Banerjee³

2023

Preprint

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We present AlgCo (Algebraic Coinductives), a practical framework for inductive reasoning over commonly used coinductive types such as conats, streams, and infinitary trees with finite branching factor. The key idea is to exploit the domain-theoretic notion of algebraic CPO to define continuous operations over coinductive types indirectly via primitive recursion on "dense" collections of their elements. This enables a convenient strategy for reasoning about algebraic coinductives by straightforward proofs by induction. We implement the AlgCo framework in Coq and demonstrate its power by verifying a stream variant of the sieve of Eratosthenes, a regular expression library based on coinductive trie encodings of formal languages, and weakest pre-expectation style semantics for coinductive sampling processes over discrete probability distributions in the random bit model.

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Section: Coinductive Binary Treesmentioning

confidence: 99%

Inductive Reasoning for Coinductive Types

Bagnall¹,

Stewart²,

Banerjee³

2023

Preprint

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“…Making this reduction work formally means precisely characterizing the input source of randomness and the distributional correctness of the output sampler. We specify the input randomness in Section 4.1, drawing on the classic theory of uniform distribution modulo 1 [8,42,73]. We characterize distributional correctness by proving that our samplers satisfy an equidistribution theorem (Section 4) wrt.…”

Section: Correctness Of Samplersmentioning

confidence: 99%

“…randomness (Section 4.1), a sampler for 𝑐 is correct if the sampler produces a sequence 𝑥 𝑛 : N → Σ such that for any observation 𝑄, the proportion of samples falling within 𝑄 asymptotically converges to the expected value of [𝑄] (the probability of 𝑄) according to 𝑐's cwp semantics. In other words, a sampler is correct when the samples it produces are equidistributed [8] wrt. cwp.…”

Section: Correctness Of Samplingmentioning

confidence: 99%

“…has deep connections to Martin-Löf randomness [50] and Schnorr randomness [25] (cf. [8,Theorem 3]). We can now state the equidistribution theorem: Theorem 4.2 (cpGCL eqidistribution).…”

Section: Equidistributionmentioning

confidence: 99%

“…itwp but doesn't directly guarantee properties of samplers generated by the resulting itrees. Drawing on basic measure theory [33] and the theory of uniform-distribution modulo 1 [8,42,73], the next section extends the result of Theorem 3.14 to show that sequences of generated samples are equidistributed wrt. the cwp semantics of their source programs.…”

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confidence: 99%

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Formally Verified Samplers From Probabilistic Programs With Loops and Conditioning

Bagnall¹,

Stewart²,

Banerjee³

2022

Preprint

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We present Zar: a formally verified compiler pipeline from discrete probabilistic programs with unbounded loops in the conditional probabilistic guarded command language (cpGCL) to proved-correct executable samplers in the random bit model. We exploit the key idea that all discrete probability distributions can be reduced to unbiased coin-flipping schemes. The compiler pipeline first translates a cpGCL program into choice-fix trees, an intermediate representation suitable for reduction of biased probabilistic choices. Choice-fix trees are then translated to coinductive interaction trees for execution within the random bit model. The correctness of the composed translations establishes the sampling equidistribution theorem: compiled samplers are correct wrt. the conditional weakest pre-expectation semantics of cpGCL source programs. Zar is implemented and fully verified in the Coq proof assistant. We extract verified samplers to OCaml and Python and empirically validate them on a number of illustrative examples.

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Poisson Generic Sequences

Álvarez

Becher

Mereb

2022

International Mathematics Research Notices

Self Cite

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Years ago, Zeev Rudnick defined the Poisson generic real numbers by counting the number of occurrences of long blocks of digits in the initial segments of the expansions of the real numbers in a fixed integer base. Peres and Weiss proved that almost all real numbers, with respect to the Lebesgue measure, are Poisson generic, but they did not publish their proof. In this note, we first transcribe Peres and Weiss’ proof and then we show that there are computable Poisson generic instances and that all Martin–Löf random real numbers are Poisson generic.

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Randomness and uniform distribution modulo one

Cited by 4 publications

References 13 publications

Inductive Reasoning for Coinductive Types

Inductive Reasoning for Coinductive Types

Formally Verified Samplers From Probabilistic Programs With Loops and Conditioning

Poisson Generic Sequences

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