Resources and their use and consumption form a central part of our life. Many branches of science and engineering are concerned with the question of which given resource objects can be converted into which target resource objects. For example, information theory studies the conversion of a noisy communication channel instance into an exchange of information. Inspired by work in quantum information theory, we develop a general mathematical toolbox for this type of question. The convertibility of resources into other ones and the possibility of combining resources is accurately captured by the mathematics of ordered commutative monoids. As an intuitive example, we consider chemistry, where chemical reaction equations such as

\mathrm{2H_2 + O_2} \lra \mathrm{2H_2O,}

While closely related to both Girard's linear logic and to Deutsch's constructor theory, our framework also produces results very reminiscent of the utility theorem of von Neumann and Morgenstern in decision theory and of a theorem of Lieb and Yngvason on the foundations of thermodynamics.

Concerning pure algebra, our observation is that some pieces of algebra can be developed in a context in which equality is not necessarily symmetric, i.e. in which the equality relation is replaced by an ordering relation. For example, notions like cancellativity or torsion-freeness are still sensible and very natural concepts in our ordered setting.

We prove the conservation theorem for differential nets – the graph-theoretical syntax of the differential extension of Linear Logic (Ehrhard and Regnier's DiLL). The conservation theorem states that the property of having infinite reductions (here infinite chains of cut elimination steps) is preserved by non-erasing steps. This turns the quest for strong normalisation (SN) into one for non-erasing weak normalisation (WN), and indeed we use this result to prove SN of simply typed DiLL (with promotion). Along the way to the theorem we achieve a number of additional results having their own interest, such as a standardisation theorem and a slightly modified system of nets, DiLL∂ϱ.

In Interactive realizability for second-order Heyting Arithmetic with EM1 and SK1 (the excluded middle and Skolem axioms restricted to Σ10-formulas), realizers are written in a classical version of Girard's System F. Since the usual reducibility semantics does not apply to such a system, we introduce a constructive forcing/reducibility semantics: though realizers are not computable functionals in the sense of Girard, they can be forced to be computable. We apply this semantics to show how to extract witnesses for realizable Π20-formulas. In particular, a constructive and efficient method is introduced. It is based on a new ‘(state-extending-continuation)-passing-style translation’ whose properties are described with the constructive forcing/reducibility semantics.

We relate the so-called powercone models of mixed non-deterministic and probabilistic choice proposed by Tix, Keimel, Plotkin, Mislove, Ouaknine, Worrell, Morgan and McIver, to our own models of previsions. Under suitable topological assumptions, we show that they are isomorphic. We rely on Keimel's cone-theoretic variants of the classical Hahn–Banach separation theorems, using functional analytic methods, and on the Schröder–Simpson Theorem.

In this paper, we treat the specification problem in Krivine classical realizability (Krivine 2009 Panoramas et synthèses 27), in the case of arithmetical formulæ. In the continuity of previous works from Miquel and the first author (Guillermo 2008 Jeux de réalisabilité en arithmétique classique, Ph.D. thesis, Université Paris 7; Guillermo and Miquel 2014 Mathematical Structures in Computer Science, Epub ahead of print), we characterize the universal realizers of a formula as being the winning strategies for a game (defined according to the formula). In the first sections, we recall the definition of classical realizability, as well as a few technical results. In Section 5, we introduce in more details the specification problem and the intuition of the game-theoretic point of view we adopt later. We first present a game 1, that we prove to be adequate and complete if the language contains no instructions ‘quote’ (Krivine 2003 Theoretical Computer Science 308 259–276), using interaction constants to do substitution over execution threads. We then show that as soon as the language contain ‘Quote,’ the game is no more complete, and present a second game 2 that is both adequate and complete in the general case. In the last Section, we draw attention to a model-theoretic point of view and use our specification result to show that arithmetical formulæ are absolute for realizability models.