Differentials and Distances in Probabilistic Coherence Spaces

In probabilistic coherence spaces, a denotational model of probabilistic functional languages, morphisms are analytic and therefore smooth. We explore two related applications of the corresponding derivatives. First we show how derivatives allow to compute the expectation of execution time in the weak head reduction of probabilistic PCF (pPCF). Next we apply a general notion of "local" differential of morphisms to the proof of a Lipschitz property of these morphisms allowing in turn to relate the observational distance on pPCF terms to a distance the model is naturally equipped with. This suggests that extending probabilistic programming languages with derivatives, in the spirit of the differential lambda-calculus, could be quite meaningful

Differentials and distances in probabilistic coherence spaces

In probabilistic coherence spaces, a denotational model of probabilistic functional languages, morphisms are analytic and therefore smooth. We explore two related applications of the corresponding derivatives. First we show how derivatives allow to compute the expectation of execution time in the weak head reduction of probabilistic PCF (pPCF). Next we apply a general notion of "local" differential of morphisms to the proof of a Lipschitz property of these morphisms allowing in turn to relate the observational distance on pPCF terms to a distance the model is naturally equipped with. This suggests that extending probabilistic programming languages with derivatives, in the spirit of the differential lambda-calculus, could be quite meaningful

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

From Differential Linear Logic to Coherent Differentiation

In this survey, we present in a unified way the categorical and syntactical settings of coherent differentiation introduced recently, which shows that the basic ideas of differential linear logic and of the differential lambda-calculus are compatible with determinism. Indeed, due to the Leibniz rule of the differential calculus, differential linear logic and the differential lambda-calculus feature an operation of addition of proofs or terms operationally interpreted as a strong form of nondeterminism. The main idea of coherent differentiation is that these sums can be controlled and kept in the realm of determinism by means of a notion of summability, upon enforcing summability restrictions on the derivatives which can be written in the models and in the syntax

On the linear structure of cones

For encompassing the limitations of probabilistic coherence spaces which do not seem to provide natural interpretations of continuous data types such as the real line, Ehrhard and al. introduced a model of probabilistic higher order computation based on (positive) cones, and a class of totally monotone functions that they called "stable". Then Crubillé proved that this model is a conservative extension of the earlier probabilistic coherence space model. We continue these investigations by showing that the category of cones and linear and Scott-continuous functions is a model of intuitionistic linear logic. To define the tensor product, we use the special adjoint functor theorem, and we prove that this operation is and extension of the standard tensor product of probabilistic coherence spaces. We also show that these latter are dense in cones, thus allowing to lift the main properties of the tensor product of probabilistic coherence spaces to general cones. Last we define in the same way an exponential of cones and extend measurability to these new operations

The Sources of Economic Energy.

The time and space so loved by philosophers and poets, burden and delight of physicists and astronomists, for a long time have been more for economists elements of inconvenience than of analysis. All this finds a justification in the mechanistic logic which also regulates the great economic theories. But the “Newtonian†general economic theory, fascinating though it is and irreplaceable in conferring rigour to theoretical formulations and reducing to a simplified form the apparently (or real) chaos of the great systems, in its necessarily high flying it is unsuitable to interpreting the local level where, instead, it is indispensable to keep one’s feet on the ground, to move in the territory following the infinite combinations of the surrounding countryside, to worm oneself into the maze of economic and social interrelations which make it unique and unrepeatable.The long journey began with the Solow type neoclassical growth models, characterised by the production function with decreasing returns and with perfect market forms, passing through endogenous growth models, now reaches territorialised forms, which have the advantage of being less abstract than neoclassical models, in that they operate in imperfect markets, but which do not manage to keep the growth rate under control, which is always given as positive. From “implosive†models we pass to “explosive†models.Forceably including local interrelations into classical production functions is not successful in overcoming the basic contradictions between Newtonian determinism and localistic indeterminism, with the result that the classical elegance is lost without acquiring localistic concreteness Now the new physicists are trying again with the String Theory, above all in the M version or the Theory of Everything. But this fascinating theory does not yet allow us to understand some fundamental things, for example it does not tell us why particles align in a certain way, in a certain order and with a certain potential. Adapting concepts and paths elaborated by post-Newtonian physics, the economist could do much less and a bit more. Much less because he is not required to solve in any way the mysteries of the universe, a bit more because, perhaps, he can describe without contradictions, using known economic science, what physicists, in their field, are not able to describe: he can tell us, using formal models why at a certain point in time and space a determined productive set composed of a well defined number of “economics quanta†relative to material and immaterial elements, of which is known the magnitude, order and force, behaves like a string and begins to “vibrate†setting off the chain reaction of economic development.

Tropical Mathematics and the Lambda Calculus I: Metric and Differential Analysis of Effectful Programs

We study the interpretation of the lambda-calculus in a framework based on tropical mathematics, and we show that it provides a unifying framework for two well-developed quantitative approaches to program semantics: on the one hand program metrics, based on the analysis of program sensitivity via Lipschitz conditions, on the other hand resource analysis, based on linear logic and higher-order program differentiation. To do that we focus on the semantic arising from the relational model weighted over the tropical semiring, and we discuss its application to the study of "best case" program behavior for languages with probabilistic and non-deterministic effects. Finally, we show that a general foundation for this approach is provided by an abstract correspondence between tropical algebra and Lawvere's theory of generalized metric spaces