1,629 research outputs found
Distilling Abstract Machines (Long Version)
It is well-known that many environment-based abstract machines can be seen as
strategies in lambda calculi with explicit substitutions (ES). Recently,
graphical syntaxes and linear logic led to the linear substitution calculus
(LSC), a new approach to ES that is halfway between big-step calculi and
traditional calculi with ES. This paper studies the relationship between the
LSC and environment-based abstract machines. While traditional calculi with ES
simulate abstract machines, the LSC rather distills them: some transitions are
simulated while others vanish, as they map to a notion of structural
congruence. The distillation process unveils that abstract machines in fact
implement weak linear head reduction, a notion of evaluation having a central
role in the theory of linear logic. We show that such a pattern applies
uniformly in call-by-name, call-by-value, and call-by-need, catching many
machines in the literature. We start by distilling the KAM, the CEK, and the
ZINC, and then provide simplified versions of the SECD, the lazy KAM, and
Sestoft's machine. Along the way we also introduce some new machines with
global environments. Moreover, we show that distillation preserves the time
complexity of the executions, i.e. the LSC is a complexity-preserving
abstraction of abstract machines.Comment: 63 page
Cocycle Twists and Extensions of Braided Doubles
It is well known that central extensions of a group G correspond to
2-cocycles on G. Cocycles can be used to construct extensions of G-graded
algebras via a version of the Drinfeld twist introduced by Majid. We show how
2-cocycles can be defined for an abstract monoidal category C, following
Panaite, Staic and Van Oystaeyen. A braiding on C leads to analogues of Nichols
algebras in C, and we explain how the recent work on twists of Nichols algebras
by Andruskiewitsch, Fantino, Garcia and Vendramin fits in this context.
Furthermore, we propose an approach to twisting the multiplication in braided
doubles, which are a class of algebras with triangular decomposition over G.
Braided doubles are not G-graded, but may be embedded in a double of a Nichols
algebra, where a twist may be carried out if careful choices are made. This is
a source of new algebras with triangular decomposition. As an example, we show
how to twist the rational Cherednik algebra of the symmetric group by the
cocycle arising from the Schur covering group, obtaining the spin Cherednik
algebra introduced by Wang.Comment: 60 pages, LaTeX; v2: references added, misprints correcte
A Sequent Calculus for Modelling Interferences
A logic calculus is presented that is a conservative extension of linear
logic. The motivation beneath this work concerns lazy evaluation, true
concurrency and interferences in proof search. The calculus includes two new
connectives to deal with multisequent structures and has the cut-elimination
property. Extensions are proposed that give first results concerning our
objectives
A Strong Distillery
Abstract machines for the strong evaluation of lambda-terms (that is, under
abstractions) are a mostly neglected topic, despite their use in the
implementation of proof assistants and higher-order logic programming
languages. This paper introduces a machine for the simplest form of strong
evaluation, leftmost-outermost (call-by-name) evaluation to normal form,
proving it correct, complete, and bounding its overhead. Such a machine, deemed
Strong Milner Abstract Machine, is a variant of the KAM computing normal forms
and using just one global environment. Its properties are studied via a special
form of decoding, called a distillation, into the Linear Substitution Calculus,
neatly reformulating the machine as a standard micro-step strategy for explicit
substitutions, namely linear leftmost-outermost reduction, i.e., the extension
to normal form of linear head reduction. Additionally, the overhead of the
machine is shown to be linear both in the number of steps and in the size of
the initial term, validating its design. The study highlights two distinguished
features of strong machines, namely backtracking phases and their interactions
with abstractions and environments.Comment: Accepted at APLAS 201
Call-by-value non-determinism in a linear logic type discipline
We consider the call-by-value lambda-calculus extended with a may-convergent
non-deterministic choice and a must-convergent parallel composition. Inspired
by recent works on the relational semantics of linear logic and non-idempotent
intersection types, we endow this calculus with a type system based on the
so-called Girard's second translation of intuitionistic logic into linear
logic. We prove that a term is typable if and only if it is converging, and
that its typing tree carries enough information to give a bound on the length
of its lazy call-by-value reduction. Moreover, when the typing tree is minimal,
such a bound becomes the exact length of the reduction
Towards sharing in lazy computation systems
Work on proving congruence of bisimulation in functional programming languages often refers to [How89,How96], where Howe gave a highly general account on this topic in terms of so-called lazy computation systems . Particularly in implementations of lazy functional languages, sharing plays an eminent role. In this paper we will show how the original work of Howe can be extended to cope with sharing. Moreover, we will demonstrate the application of our approach to the call-by-need lambda-calculus lambda-ND which provides an erratic non-deterministic operator pick and a non-recursive let. A definition of a bisimulation is given, which has to be based on a further calculus named lambda-~, since the na1ve bisimulation definition is useless. The main result is that this bisimulation is a congruence and contained in the contextual equivalence. This might be a step towards defining useful bisimulation relations and proving them to be congruences in calculi that extend the lambda-ND-calculus
On some classes of lazy cocycles and categorical structures
We study some classes of lazy cocycles, called pure (respectively neat),
together with their categorical counterparts, entwined (respectively strongly
entwined) monoidal categories.Comment: Final version; 18 pages; the title has been changed; the order of the
sections has been changed; some things have been removed and appear now in
arXiv:0801.205
The Brauer group of modified supergroup algebras
The computation of the Brauer group BM of modified supergroup algebras is
perfomed, yielding, in particular, the computation of the Brauer group of all
finite-dimensional triangular Hopf algebras when the base field is
algebraically closed and of characteristic zero. The results are compared with
the computation of lazy cohomology and with Yinhuo Zhang's exact sequence. As
an example, we compute explicitely the Brauer group and lazy cohomology for
modified supergroup algebras with (extensions of) Weyl groups of irreducible
root systems as a group datum and their standard representation as a
representation datum.Comment: 47 pages, submitte
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