1,407 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
Counterpart semantics for a second-order mu-calculus
We propose a novel approach to the semantics of quantified μ-calculi, considering models where states are algebras; the evolution relation is given by a counterpart relation (a family of partial homomorphisms), allowing for the creation, deletion, and merging of components; and formulas are interpreted over sets of state assignments (families of substitutions, associating formula variables to state components). Our proposal avoids the limitations of existing approaches, usually enforcing restrictions of the evolution relation: the resulting semantics is a streamlined and intuitively appealing one, yet it is general enough to cover most of the alternative proposals we are aware of
Quantified CTL: Expressiveness and Complexity
While it was defined long ago, the extension of CTL with quantification over
atomic propositions has never been studied extensively. Considering two
different semantics (depending whether propositional quantification refers to
the Kripke structure or to its unwinding tree), we study its expressiveness
(showing in particular that QCTL coincides with Monadic Second-Order Logic for
both semantics) and characterise the complexity of its model-checking and
satisfiability problems, depending on the number of nested propositional
quantifiers (showing that the structure semantics populates the polynomial
hierarchy while the tree semantics populates the exponential hierarchy)
Comparing and evaluating extended Lambek calculi
Lambeks Syntactic Calculus, commonly referred to as the Lambek calculus, was
innovative in many ways, notably as a precursor of linear logic. But it also
showed that we could treat our grammatical framework as a logic (as opposed to
a logical theory). However, though it was successful in giving at least a basic
treatment of many linguistic phenomena, it was also clear that a slightly more
expressive logical calculus was needed for many other cases. Therefore, many
extensions and variants of the Lambek calculus have been proposed, since the
eighties and up until the present day. As a result, there is now a large class
of calculi, each with its own empirical successes and theoretical results, but
also each with its own logical primitives. This raises the question: how do we
compare and evaluate these different logical formalisms? To answer this
question, I present two unifying frameworks for these extended Lambek calculi.
Both are proof net calculi with graph contraction criteria. The first calculus
is a very general system: you specify the structure of your sequents and it
gives you the connectives and contractions which correspond to it. The calculus
can be extended with structural rules, which translate directly into graph
rewrite rules. The second calculus is first-order (multiplicative
intuitionistic) linear logic, which turns out to have several other,
independently proposed extensions of the Lambek calculus as fragments. I will
illustrate the use of each calculus in building bridges between analyses
proposed in different frameworks, in highlighting differences and in helping to
identify problems.Comment: Empirical advances in categorial grammars, Aug 2015, Barcelona,
Spain. 201
Representation and duality of the untyped lambda-calculus in nominal lattice and topological semantics, with a proof of topological completeness
We give a semantics for the lambda-calculus based on a topological duality
theorem in nominal sets. A novel interpretation of lambda is given in terms of
adjoints, and lambda-terms are interpreted absolutely as sets (no valuation is
necessary)
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