11 research outputs found
The Sum-Product Algorithm For Quantitative Multiplicative Linear Logic
We consider an extension of multiplicative linear logic which encompasses bayesian networks and expresses samples sharing and marginalisation with the polarised rules of contraction and weakening. We introduce the necessary formalism to import exact inference algorithms from bayesian networks, giving the sum-product algorithm as an example of calculating the weighted relational semantics of a multiplicative proof-net improving runtime performance by storing intermediate results
An Analysis of Symmetry in Quantitative Semantics
In this paper, we build on a recent bicategorical model called thin spans of
groupoids, introduced by Clairambault and Forest. Notably, thin spans feature a
decomposition of symmetry into two sub-groupoids of polarized -- positive and
negative -- symmetries. We first construct a variation of the original
exponential of thin spans, based on sequences rather than families. Then we
give a syntactic characterisation of the interpretation of simply-typed
lambda-terms in thin spans, in terms of rigid intersection types and rigid
resource terms. Finally, we formally relate thin spans with the weighted
relational model and generalized species of structure. This allows us to show
how some quantities in those models reflect polarized symmetries: in particular
we show that the weighted relational model counts witnesses from generalized
species of structure, divided by the cardinal of a group of positive
symmetries
Fixed points in quantitative semantics
We describe an interpretation of recursive computation in a symmetric monoidal category with infinite biproducts and cofree commutative comonoids (for instance, the category of free modules over a complete semiring). Such categories play a significant role in ``quantitative'' models of computation: they bear a canonical complete monoid enrichment, but may not be cpo-enriched, making standard techniques for reasoning about fixed points unavailable. By constructing a bifree algebra for the cofree exponential, we obtain fixed points for morphisms in its co-Kleisli category without requiring any order-theoretic structure. These fixed points corresponding to infinite sums of finitary approximants indexed over the nested finite multisets, each representing a unique call-pattern for computation of the fixed point. We illustrate this construction by using it to give a denotational semantics for PCF with non-deterministic choice and scalar weights from a complete semiring, proving that this is computationally adequate with respect to an operational semantics which evaluates a term by taking a weighted sum of the residues of its terminating reduction paths
Weighted models for higher-order computation
We study a class of quantitative models for higher-order computation: Lafont categories with (infinite) biproducts. Each of these has a complete āinternal semiringā and can be enriched over its modules. We describe a semantics of nondeterministic PCF weighted over this semiring in which fixed points are obtained from the bifree algebra over its exponential structure. By characterizing them concretely as infinite sums of approximants indexed over nested finite multisets, we prove computational adequacy. We can construct examples of our semantics by weighting existing models such as categories of games over a complete semiring. This transition from qualitative to quantitative semantics is characterized as a āchange of baseā of enriched categories arising from a monoidal functor from coherence spaces to modules over a complete semiring. For example, the game semantics of Idealized Algol is coherence space enriched and thus gives rise to to a weighted model, which is fully abstract.</p
A Set-Theoretical Definition of Application
This paper is in two parts. Part 1 is the previously unpublished 1972 memorandum [41], with editorial changes and some minor corrections. Part 2
presents what happened next, together with some further development of
the material. The first part begins with an elementary set-theoretical model
of the Ī»Ī²-calculus. Functions are modelled in a similar way to that normally
employed in set theory, by their graphs; difficulties are caused in this enterprise by the axiom of foundation. Next, based on that model, a model of the
Ī»Ī²Ī·-calculus is constructed by means of a natural deduction method. Finally, a theorem is proved giving some general properties of those non-trivial
models of the Ī»Ī²Ī·-calculus which are continuous complete lattices.
In the second part we begin with a brief discussion of models of the
Ī»-calculus in set theories with anti-foundation axioms. Next we review
the model of the Ī»Ī²-calculus of Part 1 and also the closely related- but
different!- models of Scott [51, 52] and of Engeler [19, 20]. Then we discuss general frameworks in which elementary constructions of models can
be given. Following Longo [36], one can employ certain Scott-Engeler algebras. Following Coppo, Dezani-Ciancaglini, Honsell and Longo [13], one
can obtain filter models from their Extended Applicative Type Structures.
We give an extended discussion of various ways of constructing models of
the Ī»Ī²Ī·-calculus, and the connections between them. Finally we give extensions of the theorem to complete partial orders. Throughout we concentrate
on means of constructing models. We hardly consider any analysis of their
properties; we do not at all consider their application
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Probabilistic concurrent game semantics
This thesis presents a variety of models for probabilistic programming languages in the framework of concurrent games.
Our starting point is the model of concurrent games with symmetry of Castellan, Clairambault and Winskel. We show that they form a symmetric monoidal closed bicategory, and that this can be turned into a cartesian closed bicategory using a linear exponential pseudo-comonad inspired by linear logic.
Then, we enrich this with probability, relying heavily on Winskel's model of probabilistic concurrent strategies. We see that the bicategorical structure is not perturbed by the addition of probability. We apply this model to two probabilistic languages: a probabilistic untyped Ī»-calculus, and Probabilistic PCF. For the former, we relate the semantics to the probabilistic Nakajima trees of Leventis, thus obtaining a characterisation of observational equivalence for programs in terms of strategies. For the latter, we show a definability result in the spirit of the game semantics tradition. This solves an open problem, as it is notoriously difficult to model Probabilistic PCF with sequential game semantics.
Finally, we introduce a model for measurable game semantics, in which games and strategies come equipped with measure-theoretic structure allowing for an accurate description of computation with continuous data types. The objective of this model is to support computation with arbitrary probability measures on the reals. In the last part of this thesis we see how this can be done by equipping strategies with parametrised families of probability measures (also known as stochastic kernels), and we construct a bicategory of measurable concurrent games and probabilistic measurable strategies