604 research outputs found
Dual-Context Calculi for Modal Logic
We present natural deduction systems and associated modal lambda calculi for
the necessity fragments of the normal modal logics K, T, K4, GL and S4. These
systems are in the dual-context style: they feature two distinct zones of
assumptions, one of which can be thought as modal, and the other as
intuitionistic. We show that these calculi have their roots in in sequent
calculi. We then investigate their metatheory, equip them with a confluent and
strongly normalizing notion of reduction, and show that they coincide with the
usual Hilbert systems up to provability. Finally, we investigate a categorical
semantics which interprets the modality as a product-preserving functor.Comment: Full version of article previously presented at LICS 2017 (see
arXiv:1602.04860v4 or doi: 10.1109/LICS.2017.8005089
Mendler-style Iso-(Co)inductive predicates: a strongly normalizing approach
We present an extension of the second-order logic AF2 with iso-style
inductive and coinductive definitions specifically designed to extract programs
from proofs a la Krivine-Parigot by means of primitive (co)recursion
principles. Our logic includes primitive constructors of least and greatest
fixed points of predicate transformers, but contrary to the common approach, we
do not restrict ourselves to positive operators to ensure monotonicity, instead
we use the Mendler-style, motivated here by the concept of monotonization of an
arbitrary operator on a complete lattice. We prove an adequacy theorem with
respect to a realizability semantics based on saturated sets and
saturated-valued functions and as a consequence we obtain the strong
normalization property for the proof-term reduction, an important feature which
is absent in previous related work.Comment: In Proceedings LSFA 2011, arXiv:1203.542
Physics, Topology, Logic and Computation: A Rosetta Stone
In physics, Feynman diagrams are used to reason about quantum processes. In
the 1980s, it became clear that underlying these diagrams is a powerful analogy
between quantum physics and topology: namely, a linear operator behaves very
much like a "cobordism". Similar diagrams can be used to reason about logic,
where they represent proofs, and computation, where they represent programs.
With the rise of interest in quantum cryptography and quantum computation, it
became clear that there is extensive network of analogies between physics,
topology, logic and computation. In this expository paper, we make some of
these analogies precise using the concept of "closed symmetric monoidal
category". We assume no prior knowledge of category theory, proof theory or
computer science.Comment: 73 pages, 8 encapsulated postscript figure
A principled approach to programming with nested types in Haskell
Initial algebra semantics is one of the cornerstones of the theory of modern functional programming languages. For each inductive data type, it provides a Church encoding for that type, a build combinator which constructs data of that type, a fold combinator which encapsulates structured recursion over data of that type, and a fold/build rule which optimises modular programs by eliminating from them data constructed using the buildcombinator, and immediately consumed using the foldcombinator, for that type. It has long been thought that initial algebra semantics is not expressive enough to provide a similar foundation for programming with nested types in Haskell. Specifically, the standard folds derived from initial algebra semantics have been considered too weak to capture commonly occurring patterns of recursion over data of nested types in Haskell, and no build combinators or fold/build rules have until now been defined for nested types. This paper shows that standard folds are, in fact, sufficiently expressive for programming with nested types in Haskell. It also defines buildcombinators and fold/build fusion rules for nested types. It thus shows how initial algebra semantics provides a principled, expressive, and elegant foundation for programming with nested types in Haskell
Binary Lambda Calculus and Combinatory Logic
We introduce binary representations of both lambda calculus
and combinatory logic terms, and demonstrate their simplicity
by providing very compact parser-interpreters for these binary
languages.
We demonstrate their application to Algorithmic Information Theory
with several concrete upper bounds on program-size complexity,
including an elegant self-delimiting code for binary strings
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Programming and proving with classical types
The propositions-as-types correspondence is ordinarily presen-
ted as linking the metatheory of typed Ī»-calculi and the proof theory
of intuitionistic logic. Griffin observed that this correspondence could
be extended to classical logic through the use of control operators. This
observation set off a flurry of further research, leading to the development
of Parigotās Ī»Ī¼-calculus. In this work, we use the Ī»Ī¼-calculus as the
foundation for a system of proof terms for classical first-order logic. In
particular, we define an extended call-by-value Ī»Ī¼-calculus with a type
system in correspondence with full classical logic. We extend the language
with polymorphic types, add a host of data types in ādirect styleā, and
prove several metatheoretical properties. All of our proofs and definitions
are mechanised in Isabelle/HOL, and we automatically obtain an inter-
preter for a system of proof terms cum programming languageācalled
Ī¼MLāusing Isabelleās code generation mechanism. Atop our proof terms,
we build a prototype LCF-style interactive theorem proverācalled Ī¼TPā
for classical first-order logic, capable of synthesising Ī¼ML programs from
completed tactic-driven proofs. We present example closed Ī¼ML programs
with classical tautologies for types, including some inexpressible as closed
programs in the original Ī»Ī¼-calculus, and some example tactic-driven
Ī¼TP proofs of classical tautologies
Categorical Realizability for Non-symmetric Closed Structures
In categorical realizability, it is common to construct categories of
assemblies and categories of modest sets from applicative structures. These
categories have structures corresponding to the structures of applicative
structures. In the literature, classes of applicative structures inducing
categorical structures such as Cartesian closed categories and symmetric
monoidal closed categories have been widely studied. In this paper, we expand
these correspondences between categories with structure and applicative
structures by identifying the classes of applicative structures giving rise to
closed multicategories, closed categories, monoidal bi-closed categories as
well as (non-symmetric) monoidal closed categories. These applicative
structures are planar in that they correspond to appropriate planar lambda
calculi by combinatory completeness. These new correspondences are tight: we
show that, when a category of assemblies has one of the structures listed
above, the based applicative structure is in the corresponding class. In
addition, we introduce planar linear combinatory algebras by adopting linear
combinatory algebras of Abramsky, Hagjverdi and Scott to our planar setting,
that give rise to categorical models of the linear exponential modality and the
exchange modality on the non-symmetric multiplicative intuitionistic linear
logic
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