Realizability algebras: a program to well order R

The theory of classical realizability is a framework in which we can develop the proof-program correspondence. Using this framework, we show how to transform into programs the proofs in classical analysis with dependent choice and the existence of a well ordering of the real line. The principal tools are: The notion of realizability algebra, which is a three-sorted variant of the well known combinatory algebra of Curry. An adaptation of the method of forcing used in set theory to prove consistency results. Here, it is used in another way, to obtain programs associated with a well ordering of R and the existence of a non trivial ultrafilter on N

Real algebraic geometry for matrices over commutative rings

We define and study preorderings and orderings on rings of the form $M_n(R)$ where $R$ is a commutative unital ring. We extend the Artin-Lang theorem and Krivine-Stengle Stellens\"atze (both abstract and geometric) from $R$ to $M_n(R)$. While the orderings of $M_n(R)$ are in one-to-one correspondence with the orderings of $R$, this is not true for preorderings. Therefore, our theory is not Morita equivalent to the classical real algebraic geometry

Hybrid realizability for intuitionistic and classical choice

International audienceIn intuitionistic realizability like Kleene's or Kreisel's, the axiom of choice is trivially realized. It is even provable in Martin-Löf's intu-itionistic type theory. In classical logic, however, even the weaker axiom of countable choice proves the existence of non-computable functions. This logical strength comes at the price of a complicated computational interpretation which involves strong recursion schemes like bar recursion. We take the best from both worlds and define a realizability model for arithmetic and the axiom of choice which encompasses both intuitionistic and classical reasoning. In this model two versions of the axiom of choice can co-exist in a single proof: intuitionistic choice and classical countable choice. We interpret intuitionistic choice efficiently, however its premise cannot come from classical reasoning. Conversely, our version of classical choice is valid in full classical logic, but it is restricted to the countable case and its realizer involves bar recursion. Having both versions allows us to obtain efficient extracted programs while keeping the provability strength of classical logic

Existential witness extraction in classical realizability and via a negative translation

We show how to extract existential witnesses from classical proofs using Krivine's classical realizability---where classical proofs are interpreted as lambda-terms with the call/cc control operator. We first recall the basic framework of classical realizability (in classical second-order arithmetic) and show how to extend it with primitive numerals for faster computations. Then we show how to perform witness extraction in this framework, by discussing several techniques depending on the shape of the existential formula. In particular, we show that in the Sigma01-case, Krivine's witness extraction method reduces to Friedman's through a well-suited negative translation to intuitionistic second-order arithmetic. Finally we discuss the advantages of using call/cc rather than a negative translation, especially from the point of view of an implementation.Comment: 52 pages. Accepted in Logical Methods for Computer Science (LMCS), 201

Further Development of the Improved QMD Model and its Applications to Fusion Reaction near Barrier

The Improved Quantum Molecular Dynamics model is further developed by introducing new parameters in interaction potential energy functional based on Skyrme interaction of SkM$^{*}$ and SLy series. The properties of ground states of selected nuclei can be reproduced very well. The Coulomb barriers for a series of reaction systems are studied and compared with the results of the proximity potential. The fusion excitation functions for a series of fusion reactions are calculated and the results are in good agreement with experimental data.Comment: 17 pages, 10 figures, PRC accepte

Realizability Interpretation and Normalization of Typed Call-by-Need λ\lambda-calculus With Control

We define a variant of realizability where realizers are pairs of a term and a substitution. This variant allows us to prove the normalization of a simply-typed call-by-need \lambda$-$calculus with control due to Ariola et al. Indeed, in such call-by-need calculus, substitutions have to be delayed until knowing if an argument is really needed. In a second step, we extend the proof to a call-by-need \lambda$$-calculus equipped with a type system equivalent to classical second-order predicate logic, representing one step towards proving the normalization of the call-by-need classical second-order arithmetic introduced by the second author to provide a proof-as-program interpretation of the axiom of dependent choice

Combinatorial Complexity and Compositional Drift in Protein Interaction Networks

The assembly of molecular machines and transient signaling complexes does not typically occur under circumstances in which the appropriate proteins are isolated from all others present in the cell. Rather, assembly must proceed in the context of large-scale protein-protein interaction (PPI) networks that are characterized both by conflict and combinatorial complexity. Conflict refers to the fact that protein interfaces can often bind many different partners in a mutually exclusive way, while combinatorial complexity refers to the explosion in the number of distinct complexes that can be formed by a network of binding possibilities. Using computational models, we explore the consequences of these characteristics for the global dynamics of a PPI network based on highly curated yeast two-hybrid data. The limited molecular context represented in this data-type translates formally into an assumption of independent binding sites for each protein. The challenge of avoiding the explicit enumeration of the astronomically many possibilities for complex formation is met by a rule-based approach to kinetic modeling. Despite imposing global biophysical constraints, we find that initially identical simulations rapidly diverge in the space of molecular possibilities, eventually sampling disjoint sets of large complexes. We refer to this phenomenon as “compositional drift”. Since interaction data in PPI networks lack detailed information about geometric and biological constraints, our study does not represent a quantitative description of cellular dynamics. Rather, our work brings to light a fundamental problem (the control of compositional drift) that must be solved by mechanisms of assembly in the context of large networks. In cases where drift is not (or cannot be) completely controlled by the cell, this phenomenon could constitute a novel source of phenotypic heterogeneity in cell populations

Thermodynamic graph-rewriting

We develop a new thermodynamic approach to stochastic graph-rewriting. The ingredients are a finite set of reversible graph-rewriting rules called generating rules, a finite set of connected graphs P called energy patterns and an energy cost function. The idea is that the generators define the qualitative dynamics, by showing which transformations are possible, while the energy patterns and cost function specify the long-term probability $\pi$ of any reachable graph. Given the generators and energy patterns, we construct a finite set of rules which (i) has the same qualitative transition system as the generators; and (ii) when equipped with suitable rates, defines a continuous-time Markov chain of which $\pi$ is the unique fixed point. The construction relies on the use of site graphs and a technique of `growth policy' for quantitative rule refinement which is of independent interest. This division of labour between the qualitative and long-term quantitative aspects of the dynamics leads to intuitive and concise descriptions for realistic models (see the examples in S4 and S5). It also guarantees thermodynamical consistency (AKA detailed balance), otherwise known to be undecidable, which is important for some applications. Finally, it leads to parsimonious parameterizations of models, again an important point in some applications

BigraphER: rewriting and analysis engine for bigraphs

BigraphER is a suite of open-source tools providing an effi- cient implementation of rewriting, simulation, and visualisation for bigraphs, a universal formalism for modelling interacting systems that evolve in time and space and first introduced by Milner. BigraphER consists of an OCaml library that provides programming interfaces for the manipulation of bigraphs, their constituents and reaction rules, and a command-line tool capable of simulating Bigraphical Reactive Systems (BRSs) and computing their transition systems. Other features are native support for both bigraphs and bigraphs with sharing, stochastic reaction rules, rule priorities, instantiation maps, parameterised controls, predicate checking, graphical output and integration with the probabilistic model checker PRISM

Zipf's law in Multifragmentation

We discuss the meaning of Zipf's law in nuclear multifragmentation. We remark that Zipf's law is a consequence of a power law fragment size distribution with exponent $\tau \simeq 2$. We also recall why the presence of such distribution is not a reliable signal of a liquid-gas phase transition