112 research outputs found
De Morgan Dual Nominal Quantifiers Modelling Private Names in Non-Commutative Logic
This paper explores the proof theory necessary for recommending an expressive
but decidable first-order system, named MAV1, featuring a de Morgan dual pair
of nominal quantifiers. These nominal quantifiers called `new' and `wen' are
distinct from the self-dual Gabbay-Pitts and Miller-Tiu nominal quantifiers.
The novelty of these nominal quantifiers is they are polarised in the sense
that `new' distributes over positive operators while `wen' distributes over
negative operators. This greater control of bookkeeping enables private names
to be modelled in processes embedded as formulae in MAV1. The technical
challenge is to establish a cut elimination result, from which essential
properties including the transitivity of implication follow. Since the system
is defined using the calculus of structures, a generalisation of the sequent
calculus, novel techniques are employed. The proof relies on an intricately
designed multiset-based measure of the size of a proof, which is used to guide
a normalisation technique called splitting. The presence of equivariance, which
swaps successive quantifiers, induces complex inter-dependencies between
nominal quantifiers, additive conjunction and multiplicative operators in the
proof of splitting. Every rule is justified by an example demonstrating why the
rule is necessary for soundly embedding processes and ensuring that cut
elimination holds.Comment: Submitted for review 18/2/2016; accepted CONCUR 2016; extended
version submitted to journal 27/11/201
Private Names in Non-Commutative Logic
We present an expressive but decidable first-order system (named MAV1) defined by using the calculus of structures, a generalisation of the sequent calculus. In addition to first-order universal and existential quantifiers the system incorporates a de Morgan dual pair of nominal quantifiers called `new\u27 and `wen\u27, distinct from the self-dual Gabbay-Pitts and Miller-Tiu nominal quantifiers. The novelty of the operators `new\u27 and `wen\u27 is they are polarised in the sense that `new\u27 distributes over positive operators while `wen\u27 distributes over negative operators. This greater control of bookkeeping enables private names to be modelled in processes embedded as predicates in MAV1. Modelling processes as predicates in MAV1 has the advantage that linear implication defines a precongruence over processes that fully respects causality and branching. The transitivity of this precongruence is established by novel techniques for handling first-order quantifiers in the cut elimination proof
The Sub-Additives: A Proof Theory for Probabilistic Choice extending Linear Logic
Probabilistic choice, where each branch of a choice is weighted according to a probability distribution, is an established approach for modelling processes, quantifying uncertainty in the environment and other sources of randomness. This paper uncovers new insight showing probabilistic choice has a purely logical interpretation as an operator in an extension of linear logic. By forbidding projection and injection, we reveal additive operators between the standard with and plus operators of linear logic. We call these operators the sub-additives. The attention of the reader is drawn to two sub-additive operators: the first being sound with respect to probabilistic choice; while the second arises due to the fact that probabilistic choice cannot be self-dual, hence has a de Morgan dual counterpart. The proof theoretic justification for the sub-additives is a cut elimination result, employing a technique called decomposition. The justification from the perspective of modelling probabilistic concurrent processes is that implication is sound with respect to established notions of probabilistic refinement, and is fully compositional
A System of Interaction and Structure III: The Complexity of BV and Pomset Logic
Pomset logic and BV are both logics that extend multiplicative linear logic
(with Mix) with a third connective that is self-dual and non-commutative.
Whereas pomset logic originates from the study of coherence spaces and proof
nets, BV originates from the study of series-parallel orders, cographs, and
proof systems. Both logics enjoy a cut-admissibility result, but for neither
logic can this be done in the sequent calculus. Provability in pomset logic can
be checked via a proof net correctness criterion and in BV via a deep inference
proof system. It has long been conjectured that these two logics are the same.
In this paper we show that this conjecture is false. We also investigate the
complexity of the two logics, exhibiting a huge gap between the two. Whereas
provability in BV is NP-complete, provability in pomset logic is
-complete. We also make some observations with respect to possible
sequent systems for the two logics
An Analytic Propositional Proof System on Graphs
In this paper we present a proof system that operates on graphs instead of
formulas. Starting from the well-known relationship between formulas and
cographs, we drop the cograph-conditions and look at arbitrary undirected)
graphs. This means that we lose the tree structure of the formulas
corresponding to the cographs, and we can no longer use standard proof
theoretical methods that depend on that tree structure. In order to overcome
this difficulty, we use a modular decomposition of graphs and some techniques
from deep inference where inference rules do not rely on the main connective of
a formula. For our proof system we show the admissibility of cut and a
generalisation of the splitting property. Finally, we show that our system is a
conservative extension of multiplicative linear logic with mix, and we argue
that our graphs form a notion of generalised connective
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