Thomas Ehrhard

Probabilistic coherence spaces are fully abstract for probabilistic PCF

Probabilistic coherence spaces (PCoh) yield a semantics of higher-order probabilistic computation, interpreting types as convex sets and programs as power series. We prove that the equality of interpretations in Pcoh characterizes the operational indistinguishability of programs in PCF with a random primitive. This is the first result of full abstraction for a semantics of probabilistic PCF. The key ingredient relies on the regularity of power series. Along the way to the theorem, we design a weighted intersection type assignment system giving a logical presentation of PCoh.

Probabilistic coherence spaces as a model of higher-order probabilistic computation

We study a probabilistic version of coherence spaces and show that these objects provide a model of linear logic. We build a model of the pure lambda-calculus in this setting and show how to interpret a probabilistic version of the functional language PCF. We give a probabilistic interpretation of the semantics of probabilistic PCF closed terms of ground type. Last we suggest a generalization of this approach, using Banach spaces.

Interpreting a finitary pi-calculus in differential interaction nets

We propose and study a translation of a pi-calculus without sums nor recursion into an untyped version of differential interaction nets. We define a transition system of labeled processes and a transition system of labeled differential interaction nets. We prove that our translation from processes to nets is a bisimulation between these two transition systems. This shows that differential interaction nets are sufficiently expressive for representing concurrency and mobility, as formalized by the pi-calculus. Our study will concern essentially a replication-free fragment of the pi-calculus, but we shall also give indications on how to deal with a restricted form of replication.

Categorical Models for Simply Typed Resource Calculi

We introduce the notion of differential @l-category as an extension of Blute-Cockett-Seelys differential Cartesian categories. We prove that differential @l-categories can be used to model the simply typed versions of: (i) the differential @l-calculus, a @l-calculus extended with a syntactic derivative operator; (ii) the resource calculus, a non-lazy axiomatisation of Boudols @l-calculus with multiplicities. Finally, we provide two concrete examples of differential @l-categories, namely, the category MRel of sets and relations, and the category MFin of finiteness spaces and finitary relations.

The stack calculus

In 1979 Richard Statman proved, using proof-theory, that the purely implicational fragment of Intuitionistic Logic (M!) is PSPACE- complete. He showed a polynomially bounded translation from full In- tuitionistic Propositional Logic into its implicational fragment. By the PSPACE-completeness of S4, proved by Ladner, and the Godel trans- lation from S4 into Intuitionistic Logic, the PSPACE-completeness of M! is drawn. The sub-formula principle for a deductive system for a logic L states that whenever {1,...,k} ⊢Lthere is a proof in which each formula occurrence is either a sub-formula ofor of some of i. In this work we extend Statmans result and show that any proposi- tional (possibly modal) structural logic satisfying a particular statement of the sub-formula principle is PSPACE-complete. As a consequence, EXPTIME-complete propositional logics, such as PDL and the common- knowledge epistemic logic with at least 2 agents satisfy this particular sub-formula principle, if and only if, PSPACE=EXPTIME.We introduce a functional calculus with simple syntax and operational semantics in which the calculi introduced so far in the Curry–Howard correspondence for Classical Logic can be faithfully encoded. Our calculus enjoys confluence without any restriction. Its type system enforces strong normalization of expressions and it is a sound and complete system for full implicational Classical Logic. We give a very simple denotational semantics which allows easy calculations of the interpretation of expressions.This document contains the proceedings of the Seventh International Workshop on Logical and Semantic Frameworks, with Applications, which was held on September 29 and 30, 2012, in Rio de Janeiro, Brazil. It contains 11 regular papers (9 long and 2 short) accepted for presentation at the meeting, as well as extended abstracts of invited talks by Torben Brauner (Roskilde University, Denmark), Maribel Fernandez (Kings College London, United Kingdom), Edward Hermann Haeusler (PUC-Rio, Brazil) and Alexandre Miquel (Ecole Normale Superieure de Lyon, France).

Incremental Update for Graph Rewriting

Graph rewriting formalisms are well-established models for the representation of biological systems such as protein-protein interaction networks. The combinatorial complexity of these models usually prevents any explicit representation of the variables of the system, and one has to rely on stochastic simulations in order to sample the possible trajectories of the underlying Markov chain. The bottleneck of stochastic simulation algorithms is the update of the propensity function that describes the probability that a given rule is to be applied next. In this paper we present an algorithm based on a data structure, called extension basis, that can be used to update the counts of predefined graph observables after a rule of the model has been applied. Extension basis are obtained by static analysis of the graph rewriting rule set. It is derived from the construction of a qualitative domain for graphs and the correctness of the procedure is proven using a purely domain theoretic argument.

Full Abstraction for Probabilistic PCF

We present a probabilistic version of PCF, a well-known simply typed universal functional language. The type hierarchy is based on a single ground type of natural numbers. Even if the language is globally call-by-name, we allow a call-by-value evaluation for ground-type arguments to provide the language with a suitable algorithmic expressiveness. We describe a denotational semantics based on probabilistic coherence spaces, a model of classical Linear Logic developed in previous works. We prove an adequacy and an equational full abstraction theorem showing that equality in the model coincides with a natural notion of observational equivalence.

Resource combinatory algebras

We initiate a purely algebraic study of Ehrhard and Regniers resource λ-calculus, by introducing three equational classes of algebras: resource combinatory algebras, resource lambda-algebras and resource lambda-abstraction algebras. We establish the relations between them, laying down foundations for a model theory of resource λ-calculus. We also show that the ideal completion of a resource combinatory (resp. lambda-, lambda-abstraction) algebra induces a classical combinatory (resp. lambda-, lambda-abstraction) algebra, and that any model of the classical λ-calculus raising from a resource lambda-algebra determines a λ-theory which equates all terms having the same Bohm tree.

The Free Exponential Modality of Probabilistic Coherence Spaces

Probabilistic coherence spaces yield a model of linear logic and lambda-calculus with a linear algebra flavor. Formulas/types are associated with convex sets of