2,401 research outputs found
On noncommutative extensions of linear logic
Pomset logic introduced by Retor\'e is an extension of linear logic with a
self-dual noncommutative connective. The logic is defined by means of
proof-nets, rather than a sequent calculus. Later a deep inference system BV
was developed with an eye to capturing Pomset logic, but equivalence of system
has not been proven up to now. As for a sequent calculus formulation, it has
not been known for either of these logics, and there are convincing arguments
that such a sequent calculus in the usual sense simply does not exist for them.
In an on-going work on semantics we discovered a system similar to Pomset
logic, where a noncommutative connective is no longer self-dual. Pomset logic
appears as a degeneration, when the class of models is restricted. Motivated by
these semantic considerations, we define in the current work a semicommutative
multiplicative linear logic}, which is multiplicative linear logic extended
with two nonisomorphic noncommutative connectives (not to be confused with very
different Abrusci-Ruet noncommutative logic). We develop a syntax of proof-nets
and show how this logic degenerates to Pomset logic. However, a more
interesting problem than just finding yet another noncommutative logic is to
find a sequent calculus for this logic. We introduce decorated sequents, which
are sequents equipped with an extra structure of a binary relation of
reachability on formulas. We define a decorated sequent calculus for
semicommutative logic and prove that it is cut-free, sound and complete. This
is adapted to "degenerate" variations, including Pomset logic. Thus, in
particular, we give a variant of sequent calculus formulation for Pomset logic,
which is one of the key results of the paper
From Differential Linear Logic to Coherent Differentiation
In this survey, we present in a unified way the categorical and syntactical
settings of coherent differentiation introduced recently, which shows that the
basic ideas of differential linear logic and of the differential
lambda-calculus are compatible with determinism. Indeed, due to the Leibniz
rule of the differential calculus, differential linear logic and the
differential lambda-calculus feature an operation of addition of proofs or
terms operationally interpreted as a strong form of nondeterminism. The main
idea of coherent differentiation is that these sums can be controlled and kept
in the realm of determinism by means of a notion of summability, upon enforcing
summability restrictions on the derivatives which can be written in the models
and in the syntax
Applying quantitative semantics to higher-order quantum computing
Finding a denotational semantics for higher order quantum computation is a
long-standing problem in the semantics of quantum programming languages. Most
past approaches to this problem fell short in one way or another, either
limiting the language to an unusably small finitary fragment, or giving up
important features of quantum physics such as entanglement. In this paper, we
propose a denotational semantics for a quantum lambda calculus with recursion
and an infinite data type, using constructions from quantitative semantics of
linear logic
Tropical Mathematics and the Lambda Calculus I: Metric and Differential Analysis of Effectful Programs
We study the interpretation of the lambda-calculus in a framework based on
tropical mathematics, and we show that it provides a unifying framework for two
well-developed quantitative approaches to program semantics: on the one hand
program metrics, based on the analysis of program sensitivity via Lipschitz
conditions, on the other hand resource analysis, based on linear logic and
higher-order program differentiation. To do that we focus on the semantic
arising from the relational model weighted over the tropical semiring, and we
discuss its application to the study of "best case" program behavior for
languages with probabilistic and non-deterministic effects. Finally, we show
that a general foundation for this approach is provided by an abstract
correspondence between tropical algebra and Lawvere's theory of generalized
metric spaces
On the linear structure of cones
For encompassing the limitations of probabilistic coherence spaces which do not seem to provide natural interpretations of continuous data types such as the real line, Ehrhard and al. introduced a model of probabilistic higher order computation based on (positive) cones, and a class of totally monotone functions that they called "stable". Then Crubillé proved that this model is a conservative extension of the earlier probabilistic coherence space model. We continue these investigations by showing that the category of cones and linear and Scott-continuous functions is a model of intuitionistic linear logic. To define the tensor product, we use the special adjoint functor theorem, and we prove that this operation is and extension of the standard tensor product of probabilistic coherence spaces. We also show that these latter are dense in cones, thus allowing to lift the main properties of the tensor product of probabilistic coherence spaces to general cones. Last we define in the same way an exponential of cones and extend measurability to these new operations
Coinduction in Flow: The Later Modality in Fibrations
This paper provides a construction on fibrations that gives access to the so-called later modality, which allows for a controlled form of recursion in coinductive proofs and programs. The construction is essentially a generalisation of the topos of trees from the codomain fibration over sets to arbitrary fibrations. As a result, we obtain a framework that allows the addition of a recursion principle for coinduction to rather arbitrary logics and programming languages. The main interest of using recursion is that it allows one to write proofs and programs in a goal-oriented fashion. This enables easily understandable coinductive proofs and programs, and fosters automatic proof search.
Part of the framework are also various results that enable a wide range of applications: transportation of (co)limits, exponentials, fibred adjunctions and first-order connectives from the initial fibration to the one constructed through the framework. This means that the framework extends any first-order logic with the later modality. Moreover, we obtain soundness and completeness results, and can use up-to techniques as proof rules. Since the construction works for a wide variety of fibrations, we will be able to use the recursion offered by the later modality in various context. For instance, we will show how recursive proofs can be obtained for arbitrary (syntactic) first-order logics, for coinductive set-predicates, and for the probabilistic modal mu-calculus. Finally, we use the same construction to obtain a novel language for probabilistic productive coinductive programming. These examples demonstrate the flexibility of the framework and its accompanying results
Double Glueing over Free Exponential: with Measure Theoretic Applications
This paper provides a compact method to lift the free exponential
construction of Mellies-Tabareau-Tasson over the Hyland-Schalk double glueing
for orthogonality categories. A condition "reciprocity of orthogonality" is
presented simply enough to lift the free exponential over the double glueing in
terms of the orthogonality. Our general method applies to the monoidal category
TsK of the s-finite transition kernels with countable biproducts. We show (i)
TsK^{op} has the free exponential, which is shown to be describable in terms of
measure theory. (ii) The s-finite transition kernels have an orthogonality
between measures and measurable functions in terms of Lebesgue integrals. The
orthogonality is reciprocal, hence the free exponential of (i) lifts to the
orthogonality category O_I(TsK^{op}), which subsumes Ehrhard et al's
probabilistic coherent spaces as the full subcategory of countable measurable
spaces. To lift the free exponential, the measure-theoretic uniform convergence
theorem commuting Lebesgue integral and limit plays a crucial role. Our
measure-theoretic orthogonality is considered as a continuous version of the
orthogonality of the probabilistic coherent spaces for linear logic, and in
particular provides a two layered decomposition of Crubille et al's direct free
exponential for these spaces
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