Normalizing the Taylor expansion of non-deterministic {\lambda}-terms, via parallel reduction of resource vectors

It has been known since Ehrhard and Regnier's seminal work on the Taylor expansion of $\lambda$-terms that this operation commutes with normalization: the expansion of a $\lambda$-term is always normalizable and its normal form is the expansion of the B\"ohm tree of the term. We generalize this result to the non-uniform setting of the algebraic $\lambda$-calculus, i.e. $\lambda$-calculus extended with linear combinations of terms. This requires us to tackle two difficulties: foremost is the fact that Ehrhard and Regnier's techniques rely heavily on the uniform, deterministic nature of the ordinary $\lambda$-calculus, and thus cannot be adapted; second is the absence of any satisfactory generic extension of the notion of B\"ohm tree in presence of quantitative non-determinism, which is reflected by the fact that the Taylor expansion of an algebraic $\lambda$-term is not always normalizable. Our solution is to provide a fine grained study of the dynamics of $\beta$-reduction under Taylor expansion, by introducing a notion of reduction on resource vectors, i.e. infinite linear combinations of resource $\lambda$-terms. The latter form the multilinear fragment of the differential $\lambda$-calculus, and resource vectors are the target of the Taylor expansion of $\lambda$-terms. We show the reduction of resource vectors contains the image of any $\beta$-reduction step, from which we deduce that Taylor expansion and normalization commute on the nose. We moreover identify a class of algebraic $\lambda$-terms, encompassing both normalizable algebraic $\lambda$-terms and arbitrary ordinary $\lambda$-terms: the expansion of these is always normalizable, which guides the definition of a generalization of B\"ohm trees to this setting

Full Abstraction for the Resource Lambda Calculus with Tests, through Taylor Expansion

We study the semantics of a resource-sensitive extension of the lambda calculus in a canonical reflexive object of a category of sets and relations, a relational version of Scott's original model of the pure lambda calculus. This calculus is related to Boudol's resource calculus and is derived from Ehrhard and Regnier's differential extension of Linear Logic and of the lambda calculus. We extend it with new constructions, to be understood as implementing a very simple exception mechanism, and with a "must" parallel composition. These new operations allow to associate a context of this calculus with any point of the model and to prove full abstraction for the finite sub-calculus where ordinary lambda calculus application is not allowed. The result is then extended to the full calculus by means of a Taylor Expansion formula. As an intermediate result we prove that the exception mechanism is not essential in the finite sub-calculus

An introduction to Differential Linear Logic: proof-nets, models and antiderivatives

Differential Linear Logic enriches Linear Logic with additional logical rules for the exponential connectives, dual to the usual rules of dereliction, weakening and contraction. We present a proof-net syntax for Differential Linear Logic and a categorical axiomatization of its denotational models. We also introduce a simple categorical condition on these models under which a general antiderivative operation becomes available. Last we briefly describe the model of sets and relations and give a more detailed account of the model of finiteness spaces and linear and continuous functions

Taylor subsumes Scott, Berry, Kahn and Plotkin

The speculative ambition of replacing the old theory of program approximation based on syntactic continuity with the theory of resource consumption based on Taylor expansion and originating from the differential γ-calculus is nowadays at hand. Using this resource sensitive theory, we provide simple proofs of important results in γ-calculus that are usually demonstrated by exploiting Scott's continuity, Berry's stability or Kahn and Plotkin's sequentiality theory. A paradigmatic example is given by the Perpendicular Lines Lemma for the Böhm tree semantics, which is proved here simply by induction, but relying on the main properties of resource approximants: strong normalization, confluence and linearity

Strong normalization property for second order linear logic

AbstractThe paper contains the first complete proof of strong normalization (SN) for full second order linear logic (LL): Girard’s original proof uses a standardization theorem which is not proven. We introduce sliced pure structures (sps), a very general version of Girard’s proof-nets, and we apply to sps Gandy’s method to infer SN from weak normalization (WN). We prove a standardization theorem for sps: if WN without erasing steps holds for an sps, then it enjoys SN. A key step in our proof of standardization is a confluence theorem for sps obtained by using only a very weak form of correctness, namely acyclicity slice by slice. We conclude by showing how standardization for sps allows to prove SN of LL, using as usual Girard’s reducibility candidates

A Perspective on Fisheries Sector Interventions for Livelihood Promotion

The distinctive features of fisheries resources, fishers and their geographic contexts, on the one hand, and broad stylized features of the existing lacklustre performance of this sector, on the other, call for specialized and sustained efforts to promote livelihood of usually poor, backward and unorganized fisher communities, which are nevertheless and often the most intimate stakeholder of this sector and its underlying resources. To develop a perspective on intervention strategies for livelihood promotion of most intimate stakeholders – that is, the fisher folk, in a sustainable manner, this paper uses clues from recent economic theories and management tools on property rights, Coase Theorem, stakeholder cooperation and public-private-community partnership in an effort towards resolving the multi-dimensional problems of this sector. It stratifies and brings out the pros and cons of the existing fishing efforts into four categories of models – the traditional marketing model, state-led models of livelihood promotion and fisheries development (including cases of para-statal cooperatives), entrepreneur or leader-driven models, and technology-driven models, through selected illustrations from different parts of the country and covering both marine and inland (including brackish water) segments of fisheries. The paper, after identifying the major ingredients for sustainable livelihood development around fisheries, finally articulates Dr. APJ Kalam’s concept of PURA to recommend a rural entrepreneur-led hybrid model of fisheries development to solicit sustainable and growth oriented cooperation among the suppliers of land (i.e., stakeholders to fishery resources, which are available through Nature), labor (including fishers) and capital (including professionals). The ultimate goal of this paper is to derive inspiration from Coase Theorem and the Japanese model of Keiretsu to empower the producers and suppliers of fish – namely, the fisher folk and to place them at the centre stage of control of rural entrepreneur-led private organizations, wherein the fisher community will not be deemed as mere consumers or vendors of fish, but will enter as dignified co-producer partners with significant shares in residual claim and residual control in those organizations.