Hybrid - a definitional two-level approach to reasoning with higher-order abstract syntax

Combining higher-order abstract syntax and (co)-induction in a logical framework is well known to be problematic.We describe the theory and the practice of a tool called Hybrid, within Isabelle/HOL and Coq, which aims to address many of these difficulties. It allows object logics to be represented using higher-order abstract syntax, and reasoned about using tactical theorem proving and principles of (co)induction. Moreover, it is definitional, which guarantees consistency within a classical type theory. The idea is to have a de Bruijn representation of \u3bb-terms providing a definitional layer that allows the user to represent object languages using higher-order abstract syntax, while offering tools for reasoning about them at the higher level. In this paper we describe how to use Hybrid in a multi-level reasoning fashion, similar in spirit to other systems such as Twelf and Abella. By explicitly referencing provability in a middle layer called a specification logic, we solve the problem of reasoning by (co)induction in the presence of non-stratifiable hypothetical judgments, which allow very elegant and succinct specifications of object logic inference rules. We first demonstrate the method on a simple example, formally proving type soundness (subject reduction) for a fragment of a pure functional language, using a minimal intuitionistic logic as the specification logic. We then prove an analogous result for a continuation-machine presentation of the operational semantics of the same language, encoded this time in an ordered linear logic that serves as the specification layer. This example demonstrates the ease with which we can incorporate new specification logics, and also illustrates a significantly more complex object logic whose encoding is elegantly expressed using features of the new specification logic

Speciation‐by‐Depth on Coral Reefs: Sympatric Divergence with Gene Flow or Cryptic Transient Isolation?

The distributions of many sister species in the sea overlap geographically but are partitioned along depth gradients. The genetic changes leading to depth segregation may evolve in geographic isolation as a prerequisite to coexistence or may emerge during primary divergence leading to new species. These alternatives can now be distinguished via the power endowed by the thousands of scorable loci provided by second‐generation sequence data. Here, we revisit the case of two depth‐segregated, genetically isolated ecotypes of the nominal Caribbean candelabrum coral Eunicea flexuosa. Previous analyses based on a handful of markers could not distinguish between models of genetic exchange after a period of isolation (consistent with secondary contact) and divergence with gene flow (consistent with primary divergence). Analyses of the history of isolation, genetic exchange and population size based on 15,640 new SNP markers derived from RNAseq data best support models where divergence began 800K BP and include epochs of divergence with gene flow, but with an intermediate period of transient isolation. Results also supported the previous conclusion that recent exchange between the ecotypes occurs asymmetrically from the Shallow lineage to the Deep. Parallel analyses of data from two other corals with depth‐segregated populations (Agaricia fragilis and Pocillopora damicornis) suggest divergence leading to depth‐segregated populations may begin with a period of symmetric exchange, but that an epoch of population isolation precedes more complete isolation marked by asymmetric introgression. Thus, while divergence‐with‐gene flow may account for much of the differentiation that separates closely related, depth‐segregated species, it remains to be seen whether any critical steps in the speciation process only occur when populations are isolated

An Improved Implementation and Abstract Interface for Hybrid

Hybrid is a formal theory implemented in Isabelle/HOL that provides an interface for representing and reasoning about object languages using higher-order abstract syntax (HOAS). This interface is built around an HOAS variable-binding operator that is constructed definitionally from a de Bruijn index representation. In this paper we make a variety of improvements to Hybrid, culminating in an abstract interface that on one hand makes Hybrid a more mathematically satisfactory theory, and on the other hand has important practical benefits. We start with a modification of Hybrid's type of terms that better hides its implementation in terms of de Bruijn indices, by excluding at the type level terms with dangling indices. We present an improved set of definitions, and a series of new lemmas that provide a complete characterization of Hybrid's primitives in terms of properties stated at the HOAS level. Benefits of this new package include a new proof of adequacy and improvements to reasoning about object logics. Such proofs are carried out at the higher level with no involvement of the lower level de Bruijn syntax.Comment: In Proceedings LFMTP 2011, arXiv:1110.668

Algebraic totality, towards completeness

Finiteness spaces constitute a categorical model of Linear Logic (LL) whose objects can be seen as linearly topologised spaces, (a class of topological vector spaces introduced by Lefschetz in 1942) and morphisms as continuous linear maps. First, we recall definitions of finiteness spaces and describe their basic properties deduced from the general theory of linearly topologised spaces. Then we give an interpretation of LL based on linear algebra. Second, thanks to separation properties, we can introduce an algebraic notion of totality candidate in the framework of linearly topologised spaces: a totality candidate is a closed affine subspace which does not contain 0. We show that finiteness spaces with totality candidates constitute a model of classical LL. Finally, we give a barycentric simply typed lambda-calculus, with booleans ${\mathcal{B}}$ and a conditional operator, which can be interpreted in this model. We prove completeness at type ${\mathcal{B}}^n\to{\mathcal{B}}$ for every n by an algebraic method

Cryptic temporal changes in stock composition explain the decline of a flounder (<i>Platichthys</i> spp.) assemblage

Unobserved diversity, such as undetected genetic structure or the presence of cryptic species, is of concern for the conservation and management of global biodiversity in the face of threatening anthropogenic processes. For instance, unobserved diversity can lead to overestimation of maximum sustainable yields and therefore to overharvesting of the more vulnerable stock components within unrecognized mixed‐stock fisheries. We used DNA from archival (otolith) samples to reconstruct the temporal (1976–2011) genetic makeup of two mixed‐stock flounder fisheries in the Åland Sea (AS) and the Gulf of Finland (GoF). Both fisheries have hitherto been managed as a single stock of European flounders (Platichthys flesus), but were recently revealed to target two closely related species: the pelagic‐spawning P.&nbsp;flesus and the newly described, demersal‐spawning P.&nbsp;solemdali. While the AS and GoF fisheries were assumed to consist exclusively of P.&nbsp;solemdali, P.&nbsp;flesus dominated the GoF flounder assemblage (87% of total) in 1983, had disappeared (0%) by 1993, and remained in low proportions (10%–11%) thereafter. In the AS, P.&nbsp;solemdali dominated throughout the sampling period (&gt;70%), and P.&nbsp;flesus remained in very low proportions after 1983. The disappearance of P.&nbsp;flesus from the GoF coincides in time with a dramatic (~60%) decline in commercial landings and worsening environmental conditions in P.&nbsp;flesus’ northernmost spawning ground, the Eastern Gotland Basin, in the preceding 4–6&nbsp;years. These results are compatible with the hypothesis that P.&nbsp;flesus in the GoF is a sink population relying on larval subsidies from southern spawning grounds and the cause of their disappearance is a cessation of larval supply. Our results highlight the importance of uncovering unobserved genetic diversity and studying spatiotemporal changes in the relative contribution of different stock components, as well as the underlying environmental causes, to manage marine resources in the age of rapid anthropogenic change

On the Proof Theory of Regular Fixed Points

International audienceWe consider encoding finite automata as least fixed points in a proof theoretical framework equipped with a general induction scheme, and study automata inclusion in that setting. We provide a coinductive characterization of inclusion that yields a natural bridge to proof-theory. This leads us to generalize these observations to regular formulas, obtaining new insights about inductive theorem proving and cyclic proofs in particular

The deindustrialisation/tertiarisation hypothesis reconsidered: a subsystem application to the OECD7

The diffusion of outsourcing, both national and international, and vertical FDIs among manufacturing firms, along with the higher integra- tion of business services in manufacturing, has recently led to question the empirical evidence supporting the Deindustrialisation/Tertiarisation (DT) hypothesis. Rather than a \real" phenomenon, it has been argued, DT would be an \apparent" one, mainly due to the reorganization of production across national and sectoral boundaries. The empirical studies that have dealt with the topic so far have not been able to effectively rule out such possibility, because of two main limitations: the sectoral level of the analysis and/or the national focus. In order to overcome them, the paper carries out an appreciative investigation of the actual extent of the DT occurred in the OECD area over the '80s and the '90s by moving from a sector to a subsystem perspective, thus retaining both direct and indirect relations, and by referring to a \pseudo-World" of 7 OECD countries, thus taking into account the \global" dimension of the phenomenon. The results strongly support the DT hypothesis: although the weight of business sector services in the manufacturing subsystem increased, acting as a counterbalancing tendency to the manufacturing decline, subsystem shares significantly decreased, thus confirming DT as a more fundamental trend of modern economies

Outsourcing and structural change: shifting firm and sectoral boundaries

The paper aims at investigating the structural change implications of outsourcing. In trying to bridge the organizational/industrial and the sectoral/structural analysis of outsourcing, it discusses the rational and the methodological pros and cons of a “battery” of outsourcing measurements for structural change analysis. Their functioning is then illustrated through a concise application of them to the OECD area over the ’80s and the early ’90s. A combined used of them emerges as recommendable in checking for the role of outsourcing with respect to that of other structural change determinants