On bunched typing

2003 ◽  
Vol 13 (4) ◽  
pp. 747-796 ◽  
Author(s):  
PETER O'HEARN
Keyword(s):  
Additive Function ◽  
Linear Logic ◽  
Type System ◽  
The Other ◽  
Basic System ◽  
Interference Control ◽  
Joint Work ◽  
Function Type ◽  
Structural Rules ◽  
Single Category

We study a typing scheme derived from a semantic situation where a single category possesses several closed structures, corresponding to different varieties of function type. In this scheme typing contexts are trees built from two (or more) binary combining operations, or in short, bunches. Bunched typing and its logical counterpart, bunched implications, have arisen in joint work of the author and David Pym. The present paper gives a basic account of the type system, and then focusses on concrete models that illustrate how it may be understood in terms of resource access and sharing. The most basic system has two context-combining operations, and the structural rules of Weakening and Contraction are allowed for one but not the other. This system includes a multiplicative, or substructural, function type −∗ alongside the usual (additive) function type $\rightarrow$; it is dubbed the $\alpha\lambda$-calculus after its binders, $\alpha$ for the $\alpha$dditive binder and $\lambda$ for the multiplicative, or $\lambda$inear, binder. We show that the features of this system are, in a sense, complementary to calculi based on linear logic; it is incompatible with an interpretation where a multiplicative function uses its argument once, but perfectly compatible with a reading based on sharing of resources. This sharing interpretation is derived from syntactic control of interference, a type-theoretic method of controlling sharing of storage, and we show how bunch-based management of Contraction can be used to provide a more flexible type system for interference control.

scholarly journals Syntactic soundness proof of a type-and-capability system with hidden state

2012 ◽  
Vol 23 (1) ◽  
pp. 38-144 ◽  
Author(s):  
FRANÇOIS POTTIER
Keyword(s):  
Linear Logic ◽  
Scale Up ◽  
Type System ◽  
The Other ◽  
Formal Definition ◽  
Proof Technique ◽  
The Core ◽  
Proof Techniques ◽  
Definition Of ◽  
Intuitionistic Linear Logic

AbstractThis paper presents a formal definition and machine-checked soundness proof for a very expressive type-and-capability system, that is, a low-level type system that keeps precise track of ownership and side effects. The programming language has first-class functions and references. The type system's features include the following: universal, existential, and recursive types; subtyping; a distinction between affine and unrestricted data; support for strong updates; support for naming values and heap fragments via singleton and group regions; a distinction between ordinary values (which exist at runtime) and capabilities (which do not); support for dynamic reorganizations of the ownership hierarchy by disassembling and reassembling capabilities; and support for temporarily or permanently hiding a capability via frame and anti-frame rules. One contribution of the paper is the definition of the type-and-capability system itself. We present the system as modularly as possible. In particular, at the core of the system, the treatment of affinity, in the style of dual intuitionistic linear logic, is formulated in terms of an arbitrarymonotonic separation algebra, a novel axiomatization of resources, ownership, and the manner in which they evolve with time. Only the peripheral layers of the system are aware that we are dealing with a specific monotonic separation algebra, whose resources are references and regions. This semi-abstract organization should facilitate further extensions of the system with new forms of resources. The other main contribution is a machine-checked proof of type soundness. The proof is carried out in the Wright and Felleisen's syntactic style. This offers an evidence that this relatively simple-minded proof technique can scale up to systems of this complexity, and constitutes a viable alternative to more sophisticated semantic proof techniques. We do not claim that the syntactic technique is superior: We simply illustrate how it is used and highlight its strengths and shortcomings.

scholarly journals Jump from parallel to sequential proofs: exponentials

2016 ◽  
Vol 28 (7) ◽  
pp. 1204-1252
Author(s):  
PAOLO DI GIAMBERARDINO
Keyword(s):  
Distinctive Feature ◽  
Linear Logic ◽  
The Other ◽  
Cut Elimination ◽  
Strong Normalization ◽  
Structural Rules ◽  
New Class ◽  
Game Semantics ◽  
Proof Nets

In previous works, by importing ideas from game semantics (notably Faggian–Maurel–Curien'sludics nets), we defined a new class of multiplicative/additive polarized proof nets, calledJ-proof nets. The distinctive feature of J-proof nets with respect to other proof net syntaxes, is the possibility of representing proof nets which are partially sequentialized, by usingjumps(that is, untyped extra edges) as sequentiality constraints. Starting from this result, in the present work, we extend J-proof nets to the multiplicative/exponential fragment, in order to take into account structural rules: More precisely, we replace the familiar linear logic notion of exponential box with a less restricting one (calledcone) defined by means of jumps. As a consequence, we get a syntax for polarized nets where, instead of a structure of boxes nested one into the other, we have one of cones which can bepartially overlapping. Moreover, we define cut-elimination for exponential J-proof nets, proving, by a variant of Gandy's method, that even in case of ‘superposed’ cones, reduction enjoys confluence and strong normalization.

Symmetry vs. Duality in Logic

2014 ◽  
Vol 8 (4) ◽  
pp. 83-97 ◽  
Author(s):  
Giulia Battilotti
Keyword(s):  
Sequent Calculus ◽  
Logical Consequence ◽  
Quantum States ◽  
The Other ◽  
Human Reasoning ◽  
Symmetric Mode ◽  
The Unconscious ◽  
Structural Rules ◽  
Model Based ◽  
Long Time

The author discusses the problem of symmetry, namely of the orientation of the logical consequence. The author shows that the problem is surprisingly entangled with the problem of “being infinite”. The author presents a model based on quantum states and shows that it features satisfy the requirements of the symmetric mode of Bi-logic, a logic introduced in the '70s by the psychoanalyst I. Matte Blanco to describe the logic of the unconscious. The author discusess symmetry, in the model, to include correlations, in order to obtain a possible approach to displacement. In this setting, the author finds a possible reading of the structural rules of sequent calculus, whose role in computation, on one side, and in the representation of human reasoning, on the other, has been debated for a long time.

Sigmaxinella cearense sp. nov. from sandstone reefs off Fortaleza (Ceara State, Brazil) (Desmacellidae, Mycalina, Poecilosclerida, Demospongiae)

Zootaxa ◽  
2006 ◽  
Vol 1369 (1) ◽  
pp. 43 ◽  
Author(s):  
SULA SALANI ◽  
TITO M.D.C. LOTUFO ◽  
EDUARDO HAJDU
Keyword(s):  
New Species ◽  
Atlantic Ocean ◽  
First Record ◽  
South Atlantic ◽  
The Other ◽  
Single Specimen ◽  
The South ◽  
Australian Species ◽  
Mean Width ◽  
Single Category

Sigmaxinella cearense sp. nov. is described here on the basis of a single specimen collected in 2004 at Parque Estadual Marinho da Pedra da Risca do Meio, off Fortaleza, Ceará, Brazil. This is the first record of Sigmaxinella for the Atlantic Ocean. The new species differs from the other eleven Sigmaxinella species by the absence of raphides/microxea, and is the only species with a single category of styles as megascleres (mean length 435µm, mean width 12µm), and sigmas (mean length 21µm). The new species appears to be closer to Australian species of Sigmaxinella, on the basis of overall spicule morphology (mainly dimensions and categories of megascleres). This could be suggestive of a transpacific track for the colonization of the South Atlantic, instead of the shorter Agulhas track.Key words: Sigmaxinella, taxonomy, Parque Estadual da Pedra da Risca do Meio, sponge, new species, sandstone reef, Brazil

Exchange rules

2001 ◽  
Vol 66 (2) ◽  
pp. 509-516 ◽  
Author(s):  
Mario Piazza
Keyword(s):  
Linear Logic ◽  
Expressive Power ◽  
Theoretical Argument ◽  
Structural Rules

AbstractIn this paper, we show by a proof-theoretical argument that in a logic without structural rules, that is in noncommutative linear logic with exponentials, every formula A for which exchange rules (and weakening and contraction as well) are admissible is provably equivalent to? A. This property shows that the expressive power of “noncommutative exponentials” is much more important than that of “commutative exponentials”.

scholarly journals Bistructures, Bidomains and Linear Logic

1994 ◽  
Vol 1 (9) ◽  
Author(s):  
Gordon Plotkin ◽  
Glynn Winskel
Keyword(s):  
Partial Order ◽  
Full Subcategory ◽  
Linear Logic ◽  
The Other ◽  
Causal Dependency ◽  
Input And Output ◽  
Event Structures ◽  
Kleisli Category ◽  
Higher Types ◽  
Cartesian Closed

Bistructures are a generalisation of event structures to represent spaces of functions at higher types; the partial order of causal dependency is replaced by two orders, one associated with input and the other output in the behaviour of functions. Bistructures form a categorical model of Girard's classical linear logic in which the involution of linear logic is modelled, roughly speaking, by a reversal of the roles of input and output. The comonad of the model has associated co-Kleisli category which is equivalent to a cartesian-closed full subcategory of Berry's bidomains.

Taming the Snake and the Mongoose of Indian History

2018 ◽  
Author(s):  
Nathan McGovern
Keyword(s):  
The Other ◽  
Indian History ◽  
Other Hand ◽  
Single Category ◽  
The One

This chapter shows that, as we “peer over the horizon” of Indian history, most of the evidence we have for the use of the categories Brahman and śramaṇa together in Indian sources actually does not portray them as mutually antagonistic. On the one hand, the two terms are often used together to describe a single category of “others” against which early Buddhists and Jains constructed their own identities. On the other hand, they are also used together in the early Buddhist texts and Aśokan inscriptions to refer to a single undifferentiated class of worthy recipients of offerings.

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