Weighted Relational Models of Typed Lambda-Calculi

Laird, Jim; Manzonetto, Giulio; McCusker, Guy; Pagani, Michele

doi:10.1109/lics.2013.36

Cited by 64 publications

(97 citation statements)

References 21 publications

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“…Further refinements of the relational semantics, beyond coherence spaces, such as Loader's totality spaces [22] and Ehrhard's Finiteness spaces [13], should yield insights into the operational behaviour of CP and its extensions with features such as non-determinism. Laird et al [19]'s weighted relational semantics interprets processes as semiring-valued matrices. This could be used to model a variant of CP with complexity measures.…”

Section: Discussionmentioning

confidence: 99%

Observed Communication Semantics for Classical Processes

Atkey

2017

Programming Languages and Systems

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Abstract. Classical Linear Logic (CLL) has long inspired readings of its proofs as communicating processes. Wadler's CP calculus is one of these readings. Wadler gave CP an operational semantics by selecting a subset of the cut-elimination rules of CLL to use as reduction rules. This semantics has an appealing close connection to the logic, but does not resolve the status of the other cut-elimination rules, and does not admit an obvious notion of observational equivalence. We propose a new operational semantics for CP based on the idea of observing communication. We use this semantics to define an intuitively reasonable notion of observational equivalence. To reason about observational equivalence, we use the standard relational denotational semantics of CLL. We show that this denotational semantics is adequate for our operational semantics. This allows us to deduce that, for instance, all the cut-elimination rules of CLL are observational equivalences.

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Section: Discussionmentioning

confidence: 99%

Observed Communication Semantics for Classical Processes

Atkey

2017

Programming Languages and Systems

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“…In a nondeterministic case, this means that the semantics of a term is the sum of the semantics of all its possible one-step reducts, see e.g. [Laird et al 2013]. In our setting, the reduction is a stochastic kernel, so this sum becomes an integral, i.e.…”

Section: Soundnessmentioning

confidence: 99%

Measurable cones and stable, measurable functions: a model for probabilistic higher-order programming

Ehrhard

Pagani

Tasson

2017

Proc. ACM Program. Lang.

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We define a notion of stable and measurable map between cones endowed with measurability tests and show that it forms a cpo-enriched cartesian closed category. This category gives a denotational model of an extension of PCF supporting the main primitives of probabilistic functional programming, like continuous and discrete probabilistic distributions, sampling, conditioning and full recursion. We prove the soundness and adequacy of this model with respect to a call-by-name operational semantics and give some examples of its denotations.Panangaden [1999] reframed the work by Kozen in a categorical setting, using the category Kern of stochastic kernels. This category has been presented as the Kleisli category of the so-called Giry's monad [Giry 1982] over the category Meas of measurable spaces and measurable functions. One can precisely state the issue for higher-order types in this framework -both Meas and Kern are cartesian categories but not closed.The quest for a formal syntactic semantics of higher-order probabilistic programming had more success. We mention in particular Park et al. [2008], proposing a probabilistic functional language λ ⃝ based on sampling functions. This language has a type R of sub-probabilistic distributions over the set of real numbers 1 , i.e. measures over the Lebesgue σ -algebra on R with total mass at most 1. Using the usual functional primitives (in particular recursion) together with the uniform distribution over [0, 1] and a sampling construct, the authors encode various methods for generating distributions (like the inverse transform method and rejection sampling) and computing properties about them (such as approximations for expectation values, variances, etc). The amazing feature of λ ⃝ is its rich expressiveness as witnessed by the number of examples and applications detailed in [Park et al. 2008], showing the relevance of the functional paradigm for probabilistic programming.Until now, λ ⃝ lacked a denotation model, [Park et al. 2008] sketching only an operational semantics. In particular, the correctness proof of the encodings follows a syntactic reasoning which is not compositional. Our paper fills this gap, giving a denotational model to a variant of λ ⃝ . As a byproduct, we can check program correctness in a straight way by applying to program denotations the standard laws of calculus (Example 7.3,7.4), even for recursive programs (Example 7.9). This method is justified by the Adequacy Theorem 7.12 stating the correspondence between the operational and the denotational semantics.If we restrict the language to countable data types (like booleans and natural numbers, excluding the real numbers), then the situation is much simpler. Indeed, any distribution over a countable set is discrete, i.e. it can be described as a linear combination of its possible outcomes and there is no need of a notion of measurable space. In previous papers [Ehrhard et al. , 2014Ehrhard and Tasson 2016], we have shown that the category PCoh ! of probabilistic coherence spaces and entire functions...

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“…In [10] (see also [9]), it is shown how to generalize this construction to any complete 3 semi-ring R and yet obtain a model of linear logic. In particular, the composition of two matrices φ ∈ R A×B and ψ ∈ R B×C is given by the usual matrix multiplication:…”

Section: Introductionmentioning

confidence: 99%

Strong Normalizability as a Finiteness Structure via the Taylor Expansion of $$\lambda $$ λ -terms

Pagani

Tasson

Vaux

2016

Lecture Notes in Computer Science

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Abstract. In the folklore of linear logic, a common intuition is that the structure of finiteness spaces, introduced by Ehrhard, semantically reflects the strong normalization property of cut-elimination. We make this intuition formal in the context of the non-deterministic λ-calculus by introducing a finiteness structure on resource terms, which is such that a λ-term is strongly normalizing iff the support of its Taylor expansion is finitary. An application of our result is the existence of a normal form for the Taylor expansion of any strongly normalizable non-deterministic λ-term. IntroductionIt is well-known that sets and relations can be presented as a category of modules and linear functions over the boolean semi-ring, giving one of the simplest semantics of linear logic. In [10] (see also [9]), it is shown how to generalize this construction to any complete 3 semi-ring R and yet obtain a model of linear logic. In particular, the composition of two matrices φ ∈ R A×B and ψ ∈ R B×C is given by the usual matrix multiplication:The semi-ring R must be complete because the above sum might be infinite. This is an issue, because it prevents us from considering standard vector spaces, which are usually constructed over "non-complete" fields like reals or complexes. In order to overcome this problem, Ehrhard introduced the notion of finiteness space [3]. A finiteness space is a pair of a set A and a set A (called finiteness structure in Definition 7) of subsets of A which is closed under a notion of duality. The point is that, for any field K (resp. any semi-ring), the set of vectors in K A whose support 4 is in A constitutes a vector space (resp. a module) over ⋆ Supported by French ANR Project Coquas (number ANR 12 JS02 006 01). 3 A semi-ring is complete if any sum, even infinite, is well-defined. 4 The support of v ∈ K A is the set of those a ∈ A such that the scalar va is non-null.K. Moreover, any two matrices φ ∈ K A×B and ψ ∈ K B×C whose supports are in resp. A ⊸ B and B ⊸ C (the finiteness structures associated with the linear arrow) compose, because the duality condition on the supports makes the terms in the sum of Equation (1) be zero almost everywhere. This gives rise to a category which is a model of linear logic and its differential extension.The notion of finiteness space seems strictly related to the property of normalization. Already in [3], it is remarked that the coKleisli category of the exponential comonad is a model of simply typed λ-calculus, but it is not cpo-enriched and thus cannot interpret (at least in a standard way) fixed-point combinators, so neither PCF nor untyped λ-calculus. Moreover, in the setting of differential nets, Pagani showed that the property of having a finitary interpretation corresponds to an acyclicity criterion (called visible acyclicity [12]) which entails the normalization property of the cut-elimination procedure [11], while there are examples of visibly cyclic differential nets which do not normalize.The goal of this paper is to shed further light on the link betw...

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Weighted Relational Models of Typed Lambda-Calculi

Cited by 64 publications

References 21 publications

Observed Communication Semantics for Classical Processes

Observed Communication Semantics for Classical Processes

Measurable cones and stable, measurable functions: a model for probabilistic higher-order programming

Strong Normalizability as a Finiteness Structure via the Taylor Expansion of $$\lambda $$ λ -terms

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