124 research outputs found

### Introduction to linear logic and ludics, part II

This paper is the second part of an introduction to linear logic and ludics,
both due to Girard. It is devoted to proof nets, in the limited, yet central,
framework of multiplicative linear logic and to ludics, which has been recently
developped in an aim of further unveiling the fundamental interactive nature of
computation and logic. We hope to offer a few computer science insights into
this new theory

### Realizability algebras II : new models of ZF + DC

Using the proof-program (Curry-Howard) correspondence, we give a new method
to obtain models of ZF and relative consistency results in set theory. We show
the relative consistency of ZF + DC + there exists a sequence of subsets of R
the cardinals of which are strictly decreasing + other similar properties of R.
These results seem not to have been previously obtained by forcing.Comment: 28

### Categorified cyclic operads

In this paper, we introduce a notion of categorified cyclic operad for
set-based cyclic operads with symmetries. Our categorification is obtained by
relaxing defining axioms of cyclic operads to isomorphisms and by formulating
coherence conditions for these isomorphisms. The coherence theorem that we
prove has the form "all diagrams of canonical isomorphisms commute". Our
coherence results come in two flavours, corresponding to the "entries-only" and
"exchangeable-output" definitions of cyclic operads. Our proof of coherence in
the entries-only style is of syntactic nature and relies on the coherence of
categorified non-symmetric operads established by Do\v{s}en and Petri\'c. We
obtain the coherence in the exchangeable-output style by "lifting" the
equivalence between entries-only and exchangeable-output cyclic operads, set up
by the second author. Finally, we show that a generalisation of the structure
of profunctors of B\' enabou provides an example of categorified cyclic operad,
and we exploit the coherence of categorified cyclic operads in proving that the
Feynman category for cyclic operads, due to Kaufmann and Ward, admits an odd
version.Comment: 57 page

### Coherent Presentations of Monoidal Categories

Presentations of categories are a well-known algebraic tool to provide
descriptions of categories by means of generators, for objects and morphisms,
and relations on morphisms. We generalize here this notion, in order to
consider situations where the objects are considered modulo an equivalence
relation, which is described by equational generators. When those form a
convergent (abstract) rewriting system on objects, there are three very natural
constructions that can be used to define the category which is described by the
presentation: one consists in turning equational generators into identities
(i.e. considering a quotient category), one consists in formally adding
inverses to equational generators (i.e. localizing the category), and one
consists in restricting to objects which are normal forms. We show that, under
suitable coherence conditions on the presentation, the three constructions
coincide, thus generalizing celebrated results on presentations of groups, and
we extend those conditions to presentations of monoidal categories

### Realizability algebras: a program to well order R

The theory of classical realizability is a framework in which we can develop
the proof-program correspondence. Using this framework, we show how to
transform into programs the proofs in classical analysis with dependent choice
and the existence of a well ordering of the real line. The principal tools are:
The notion of realizability algebra, which is a three-sorted variant of the
well known combinatory algebra of Curry. An adaptation of the method of forcing
used in set theory to prove consistency results. Here, it is used in another
way, to obtain programs associated with a well ordering of R and the existence
of a non trivial ultrafilter on N

### Derivation Lengths Classification of G\"odel's T Extending Howard's Assignment

Let T be Goedel's system of primitive recursive functionals of finite type in
the lambda formulation. We define by constructive means using recursion on
nested multisets a multivalued function I from the set of terms of T into the
set of natural numbers such that if a term a reduces to a term b and if a
natural number I(a) is assigned to a then a natural number I(b) can be assigned
to b such that I(a) is greater than I(b). The construction of I is based on
Howard's 1970 ordinal assignment for T and Weiermann's 1996 treatment of T in
the combinatory logic version. As a corollary we obtain an optimal derivation
length classification for the lambda formulation of T and its fragments.
Compared with Weiermann's 1996 exposition this article yields solutions to
several non-trivial problems arising from dealing with lambda terms instead of
combinatory logic terms. It is expected that the methods developed here can be
applied to other higher order rewrite systems resulting in new powerful
termination orderings since T is a paradigm for such systems

### Syntactic aspects of hypergraph polytopes

This paper introduces an inductively defined tree notation for all the faces
of polytopes arising from a simplex by truncations. This notation allows us to
view inclusion of faces as the process of contracting tree edges. Our notation
instantiates to the well-known notations for the faces of associahedra and
permutohedra. Various authors have independently introduced combinatorial tools
for describing such polytopes. We build on the particular approach developed by
Dosen and Petric, who used the formalism of hypergraphs to describe the
interval of polytopes from the simplex to the permutohedron. This interval was
further stretched by Petric to allow truncations of faces that are themselves
obtained by truncations, and iteratively so. Our notation applies to all these
polytopes. We illustrate this by showing that it instantiates to a notation for
the faces of the permutohedron-based associahedra, that consists of
parenthesised words with holes. Dosen and Petric have exhibited some families
of hypergraph polytopes (associahedra, permutohedra, and hemiassociahedra)
describing the coherences, and the coherences between coherences etc., arising
by weakening sequential and parallel associativity of operadic composition. We
complement their work with a criterion allowing us to recover the information
whether edges of these "operadic polytopes" come from sequential, or from
parallel associativity. We also give alternative proofs for some of the
original results of Dosen and Petric.Comment: 42 pages, 4 figure

### A Logical Foundation for Environment Classifiers

Taha and Nielsen have developed a multi-stage calculus {\lambda}{\alpha} with
a sound type system using the notion of environment classifiers. They are
special identifiers, with which code fragments and variable declarations are
annotated, and their scoping mechanism is used to ensure statically that
certain code fragments are closed and safely runnable. In this paper, we
investigate the Curry-Howard isomorphism for environment classifiers by
developing a typed {\lambda}-calculus {\lambda}|>. It corresponds to
multi-modal logic that allows quantification by transition variables---a
counterpart of classifiers---which range over (possibly empty) sequences of
labeled transitions between possible worlds. This interpretation will reduce
the "run" construct---which has a special typing rule in
{\lambda}{\alpha}---and embedding of closed code into other code fragments of
different stages---which would be only realized by the cross-stage persistence
operator in {\lambda}{\alpha}---to merely a special case of classifier
application. {\lambda}|> enjoys not only basic properties including subject
reduction, confluence, and strong normalization but also an important property
as a multi-stage calculus: time-ordered normalization of full reduction. Then,
we develop a big-step evaluation semantics for an ML-like language based on
{\lambda}|> with its type system and prove that the evaluation of a well-typed
{\lambda}|> program is properly staged. We also identify a fragment of the
language, where erasure evaluation is possible. Finally, we show that the proof
system augmented with a classical axiom is sound and complete with respect to a
Kripke semantics of the logic

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