928 research outputs found
Zipper logic
Zipper logic is a graph rewrite system, consisting in only local rewrites on
a class of zipper graphs. Connections with the chemlambda artificial chemistry
and with knot diagrammatics based computation are explored in the article.Comment: 16 pages, 24 colour figure
A Structural Approach to Reversible Computation
Reversibility is a key issue in the interface between computation and
physics, and of growing importance as miniaturization progresses towards its
physical limits. Most foundational work on reversible computing to date has
focussed on simulations of low-level machine models. By contrast, we develop a
more structural approach. We show how high-level functional programs can be
mapped compositionally (i.e. in a syntax-directed fashion) into a simple kind
of automata which are immediately seen to be reversible. The size of the
automaton is linear in the size of the functional term. In mathematical terms,
we are building a concrete model of functional computation. This construction
stems directly from ideas arising in Geometry of Interaction and Linear
Logic---but can be understood without any knowledge of these topics. In fact,
it serves as an excellent introduction to them. At the same time, an
interesting logical delineation between reversible and irreversible forms of
computation emerges from our analysis.Comment: 30 pages, appeared in Theoretical Computer Scienc
The involutions-as-principal types/ application-as-unification analogy
In 2005, S. Abramsky introduced various universal models of computation based on Affine Combinatory Logic, consisting of partial involutions over a suitable formal language of moves, in order to discuss reversible computation in a game-theoretic setting. We investigate Abramsky\u2019s models from the point of view of the model theory of \u3bb-calculus, focusing on the purely linear and affine fragments of Abramsky\u2019s Combinatory Algebras. Our approach stems from realizing a structural analogy, which had not been hitherto pointed out in the literature, between the partial involution interpreting a combinator and the principal type of that term, with respect to a simple types discipline for \u3bb-calculus. This analogy allows for explaining as unification between principal types the somewhat awkward linear application of involutions arising from Geometry of Interaction (GoI). Our approach provides immediately an answer to the open problem, raised by Abramsky, of characterising those finitely describable partial involutions which are denotations of combinators, in the purely affine fragment. We prove also that the (purely) linear combinatory algebra of partial involutions is a (purely) linear \u3bb-algebra, albeit not a combinatory model, while the (purely) affine combinatory algebra is not. In order to check the complex equations involved in the definition of affine \u3bb-algebra, we implement in Erlang the compilation of \u3bb-terms as involutions, and their execution
Is game semantics necessary?
We discuss the extent to which game semantics is implicit in the formalism of
linear logic and in the intuitions underlying linear logic
Spurious ambiguity and focalization
Spurious ambiguity is the phenomenon whereby distinct derivations in grammar may assign the same structural reading, resulting in redundancy in the parse search space and inefficiency in parsing. Understanding the problem depends on identifying the essential mathematical structure of derivations. This is trivial in the case of context free grammar, where the parse structures are ordered trees; in the case of type logical categorial grammar, the parse structures are proof nets. However, with respect to multiplicatives, intrinsic proof nets have not yet been given for displacement calculus, and proof nets for additives, which have applications to polymorphism, are not easy to characterize. In this context we approach here multiplicative-additive spurious ambiguity by means of the proof-theoretic technique of focalization.Peer ReviewedPostprint (published version
Types and forgetfulness in categorical linguistics and quantum mechanics
The role of types in categorical models of meaning is investigated. A general
scheme for how typed models of meaning may be used to compare sentences,
regardless of their grammatical structure is described, and a toy example is
used as an illustration. Taking as a starting point the question of whether the
evaluation of such a type system 'loses information', we consider the
parametrized typing associated with connectives from this viewpoint.
The answer to this question implies that, within full categorical models of
meaning, the objects associated with types must exhibit a simple but subtle
categorical property known as self-similarity. We investigate the category
theory behind this, with explicit reference to typed systems, and their
monoidal closed structure. We then demonstrate close connections between such
self-similar structures and dagger Frobenius algebras. In particular, we
demonstrate that the categorical structures implied by the polymorphically
typed connectives give rise to a (lax unitless) form of the special forms of
Frobenius algebras known as classical structures, used heavily in abstract
categorical approaches to quantum mechanics.Comment: 37 pages, 4 figure
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