8 research outputs found
The Doxastic Interpretation of Team Semantics
We advance a doxastic interpretation for many of the logical connectives
considered in Dependence Logic and in its extensions, and we argue that Team
Semantics is a natural framework for reasoning about beliefs and belief
updates
A Generalised Quantifier Theory of Natural Language in Categorical Compositional Distributional Semantics with Bialgebras
Categorical compositional distributional semantics is a model of natural
language; it combines the statistical vector space models of words with the
compositional models of grammar. We formalise in this model the generalised
quantifier theory of natural language, due to Barwise and Cooper. The
underlying setting is a compact closed category with bialgebras. We start from
a generative grammar formalisation and develop an abstract categorical
compositional semantics for it, then instantiate the abstract setting to sets
and relations and to finite dimensional vector spaces and linear maps. We prove
the equivalence of the relational instantiation to the truth theoretic
semantics of generalised quantifiers. The vector space instantiation formalises
the statistical usages of words and enables us to, for the first time, reason
about quantified phrases and sentences compositionally in distributional
semantics
An analysis of innocent interaction
We present two abstract machines for innocent interaction. The first, a rather complicated machine, operates directly on innocent strategies. The second, a far simpler machine, requires a “compilation” of the innocent strategies into “cellular” strategies before use. Given two innocent strategies, we get the same final result if we make them interact using the first machine or if we first cellularize them then use the other machine
Imperative programs as proofs via game semantics
Game semantics extends the Curry-Howard isomorphism to a three-way
correspondence: proofs, programs, strategies. But the universe of strategies
goes beyond intuitionistic logics and lambda calculus, to capture stateful
programs. In this paper we describe a logical counterpart to this extension, in
which proofs denote such strategies. The system is expressive: it contains all
of the connectives of Intuitionistic Linear Logic, and first-order
quantification. Use of Laird's sequoid operator allows proofs with imperative
behaviour to be expressed. Thus, we can embed first-order Intuitionistic Linear
Logic into this system, Polarized Linear Logic, and an imperative total
programming language.
The proof system has a tight connection with a simple game model, where games
are forests of plays. Formulas are modelled as games, and proofs as
history-sensitive winning strategies. We provide a strong full completeness
result with respect to this model: each finitary strategy is the denotation of
a unique analytic (cut-free) proof. Infinite strategies correspond to analytic
proofs that are infinitely deep. Thus, we can normalise proofs, via the
semantics