3,760 research outputs found
Unifying Functional Interpretations: Past and Future
This article surveys work done in the last six years on the unification of
various functional interpretations including G\"odel's dialectica
interpretation, its Diller-Nahm variant, Kreisel modified realizability,
Stein's family of functional interpretations, functional interpretations "with
truth", and bounded functional interpretations. Our goal in the present paper
is twofold: (1) to look back and single out the main lessons learnt so far, and
(2) to look forward and list several open questions and possible directions for
further research.Comment: 18 page
From IF to BI: a tale of dependence and separation
We take a fresh look at the logics of informational dependence and
independence of Hintikka and Sandu and Vaananen, and their compositional
semantics due to Hodges. We show how Hodges' semantics can be seen as a special
case of a general construction, which provides a context for a useful
completeness theorem with respect to a wider class of models. We shed some new
light on each aspect of the logic. We show that the natural propositional logic
carried by the semantics is the logic of Bunched Implications due to Pym and
O'Hearn, which combines intuitionistic and multiplicative connectives. This
introduces several new connectives not previously considered in logics of
informational dependence, but which we show play a very natural role, most
notably intuitionistic implication. As regards the quantifiers, we show that
their interpretation in the Hodges semantics is forced, in that they are the
image under the general construction of the usual Tarski semantics; this
implies that they are adjoints to substitution, and hence uniquely determined.
As for the dependence predicate, we show that this is definable from a simpler
predicate, of constancy or dependence on nothing. This makes essential use of
the intuitionistic implication. The Armstrong axioms for functional dependence
are then recovered as a standard set of axioms for intuitionistic implication.
We also prove a full abstraction result in the style of Hodges, in which the
intuitionistic implication plays a very natural r\^ole.Comment: 28 pages, journal versio
On Affine Logic and {\L}ukasiewicz Logic
The multi-valued logic of {\L}ukasiewicz is a substructural logic that has
been widely studied and has many interesting properties. It is classical, in
the sense that it admits the axiom schema of double negation, [DNE]. However,
our understanding of {\L}ukasiewicz logic can be improved by separating its
classical and intuitionistic aspects. The intuitionistic aspect of
{\L}ukasiewicz logic is captured in an axiom schema, [CWC], which asserts the
commutativity of a weak form of conjunction. This is equivalent to a very
restricted form of contraction. We show how {\L}ukasiewicz Logic can be viewed
both as an extension of classical affine logic with [CWC], or as an extension
of what we call \emph{intuitionistic} {\L}ukasiewicz logic with [DNE],
intuitionistic {\L}ukasiewicz logic being the extension of intuitionistic
affine logic by the schema [CWC]. At first glance, intuitionistic affine logic
seems very weak, but, in fact, [CWC] is surprisingly powerful, implying results
such as intuitionistic analogues of De Morgan's laws. However the proofs can be
very intricate. We present these results using derived connectives to clarify
and motivate the proofs and give several applications. We give an analysis of
the applicability to these logics of the well-known methods that use negation
to translate classical logic into intuitionistic logic. The usual proofs of
correctness for these translations make much use of contraction. Nonetheless,
we show that all the usual negative translations are already correct for
intuitionistic {\L}ukasiewicz logic, where only the limited amount of
contraction given by [CWC] is allowed. This is in contrast with affine logic
for which we show, by appeal to results on semantics proved in a companion
paper, that both the Gentzen and the Glivenko translations fail.Comment: 28 page
Focusing and Polarization in Intuitionistic Logic
A focused proof system provides a normal form to cut-free proofs that
structures the application of invertible and non-invertible inference rules.
The focused proof system of Andreoli for linear logic has been applied to both
the proof search and the proof normalization approaches to computation. Various
proof systems in literature exhibit characteristics of focusing to one degree
or another. We present a new, focused proof system for intuitionistic logic,
called LJF, and show how other proof systems can be mapped into the new system
by inserting logical connectives that prematurely stop focusing. We also use
LJF to design a focused proof system for classical logic. Our approach to the
design and analysis of these systems is based on the completeness of focusing
in linear logic and on the notion of polarity that appears in Girard's LC and
LU proof systems
Reasoning about Knowledge in Linear Logic: Modalities and Complexity
In a recent paper, Jean-Yves Girard commented that âit has been a long time since philosophy has stopped intereacting with logicâ[17]. Actually, it has no
Dialectica Categories for the Lambek Calculus
We revisit the old work of de Paiva on the models of the Lambek Calculus in
dialectica models making sure that the syntactic details that were sketchy on
the first version got completed and verified. We extend the Lambek Calculus
with a \kappa modality, inspired by Yetter's work, which makes the calculus
commutative. Then we add the of-course modality !, as Girard did, to
re-introduce weakening and contraction for all formulas and get back the full
power of intuitionistic and classical logic. We also present the categorical
semantics, proved sound and complete. Finally we show the traditional
properties of type systems, like subject reduction, the Church-Rosser theorem
and normalization for the calculi of extended modalities, which we did not have
before
The intuitionistic temporal logic of dynamical systems
A dynamical system is a pair , where is a topological space and
is continuous. Kremer observed that the language of
propositional linear temporal logic can be interpreted over the class of
dynamical systems, giving rise to a natural intuitionistic temporal logic. We
introduce a variant of Kremer's logic, which we denote , and show
that it is decidable. We also show that minimality and Poincar\'e recurrence
are both expressible in the language of , thus providing a
decidable logic expressive enough to reason about non-trivial asymptotic
behavior in dynamical systems
- âŚ