4,146 research outputs found
Labelled Lambda-calculi with Explicit Copy and Erase
We present two rewriting systems that define labelled explicit substitution
lambda-calculi. Our work is motivated by the close correspondence between
Levy's labelled lambda-calculus and paths in proof-nets, which played an
important role in the understanding of the Geometry of Interaction. The
structure of the labels in Levy's labelled lambda-calculus relates to the
multiplicative information of paths; the novelty of our work is that we design
labelled explicit substitution calculi that also keep track of exponential
information present in call-by-value and call-by-name translations of the
lambda-calculus into linear logic proof-nets
Beta Reduction is Invariant, Indeed (Long Version)
Slot and van Emde Boas' weak invariance thesis states that reasonable
machines can simulate each other within a polynomially overhead in time. Is
-calculus a reasonable machine? Is there a way to measure the
computational complexity of a -term? This paper presents the first
complete positive answer to this long-standing problem. Moreover, our answer is
completely machine-independent and based over a standard notion in the theory
of -calculus: the length of a leftmost-outermost derivation to normal
form is an invariant cost model. Such a theorem cannot be proved by directly
relating -calculus with Turing machines or random access machines,
because of the size explosion problem: there are terms that in a linear number
of steps produce an exponentially long output. The first step towards the
solution is to shift to a notion of evaluation for which the length and the
size of the output are linearly related. This is done by adopting the linear
substitution calculus (LSC), a calculus of explicit substitutions modelled
after linear logic and proof-nets and admitting a decomposition of
leftmost-outermost derivations with the desired property. Thus, the LSC is
invariant with respect to, say, random access machines. The second step is to
show that LSC is invariant with respect to the -calculus. The size
explosion problem seems to imply that this is not possible: having the same
notions of normal form, evaluation in the LSC is exponentially longer than in
the -calculus. We solve such an impasse by introducing a new form of
shared normal form and shared reduction, deemed useful. Useful evaluation
avoids those steps that only unshare the output without contributing to
-redexes, i.e., the steps that cause the blow-up in size.Comment: 29 page
(Leftmost-Outermost) Beta Reduction is Invariant, Indeed
Slot and van Emde Boas' weak invariance thesis states that reasonable
machines can simulate each other within a polynomially overhead in time. Is
lambda-calculus a reasonable machine? Is there a way to measure the
computational complexity of a lambda-term? This paper presents the first
complete positive answer to this long-standing problem. Moreover, our answer is
completely machine-independent and based over a standard notion in the theory
of lambda-calculus: the length of a leftmost-outermost derivation to normal
form is an invariant cost model. Such a theorem cannot be proved by directly
relating lambda-calculus with Turing machines or random access machines,
because of the size explosion problem: there are terms that in a linear number
of steps produce an exponentially long output. The first step towards the
solution is to shift to a notion of evaluation for which the length and the
size of the output are linearly related. This is done by adopting the linear
substitution calculus (LSC), a calculus of explicit substitutions modeled after
linear logic proof nets and admitting a decomposition of leftmost-outermost
derivations with the desired property. Thus, the LSC is invariant with respect
to, say, random access machines. The second step is to show that LSC is
invariant with respect to the lambda-calculus. The size explosion problem seems
to imply that this is not possible: having the same notions of normal form,
evaluation in the LSC is exponentially longer than in the lambda-calculus. We
solve such an impasse by introducing a new form of shared normal form and
shared reduction, deemed useful. Useful evaluation avoids those steps that only
unshare the output without contributing to beta-redexes, i.e. the steps that
cause the blow-up in size. The main technical contribution of the paper is
indeed the definition of useful reductions and the thorough analysis of their
properties.Comment: arXiv admin note: substantial text overlap with arXiv:1405.331
Acyclic Solos and Differential Interaction Nets
We present a restriction of the solos calculus which is stable under
reduction and expressive enough to contain an encoding of the pi-calculus. As a
consequence, it is shown that equalizing names that are already equal is not
required by the encoding of the pi-calculus. In particular, the induced solo
diagrams bear an acyclicity property that induces a faithful encoding into
differential interaction nets. This gives a (new) proof that differential
interaction nets are expressive enough to contain an encoding of the
pi-calculus. All this is worked out in the case of finitary (replication free)
systems without sum, match nor mismatch
A Polynomial Translation of pi-calculus FCPs to Safe Petri Nets
We develop a polynomial translation from finite control pi-calculus processes
to safe low-level Petri nets. To our knowledge, this is the first such
translation. It is natural in that there is a close correspondence between the
control flows, enjoys a bisimulation result, and is suitable for practical
model checking.Comment: To appear in special issue on best papers of CONCUR'12 of Logical
Methods in Computer Scienc
Expansion Trees with Cut
Herbrand's theorem is one of the most fundamental insights in logic. From the
syntactic point of view it suggests a compact representation of proofs in
classical first- and higher-order logic by recording the information which
instances have been chosen for which quantifiers, known in the literature as
expansion trees.
Such a representation is inherently analytic and hence corresponds to a
cut-free sequent calculus proof. Recently several extensions of such proof
representations to proofs with cut have been proposed. These extensions are
based on graphical formalisms similar to proof nets and are limited to prenex
formulas.
In this paper we present a new approach that directly extends expansion trees
by cuts and covers also non-prenex formulas. We describe a cut-elimination
procedure for our expansion trees with cut that is based on the natural
reduction steps. We prove that it is weakly normalizing using methods from the
epsilon-calculus
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