Introduction to linear logic and ludics, part II

This paper is the second part of an introduction to linear logic and ludics, both due to Girard. It is devoted to proof nets, in the limited, yet central, framework of multiplicative linear logic and to ludics, which has been recently developped in an aim of further unveiling the fundamental interactive nature of computation and logic. We hope to offer a few computer science insights into this new theory

Interactive observability in Ludics: The geometry of tests

AbstractLudics [J.-Y. Girard, Locus solum, Math. Structures in Comput. Sci. 11 (2001) 301–506] is a recent proposal of analysis of interaction, developed by abstracting away from proof-theory. It provides an elegant, abstract setting in which interaction between agents (proofs/programs/processes) can be studied at a foundational level, together with a notion of equivalence from the point of view of the observer.An agent should be seen as some kind of black box. An interactive observation on an agent is obtained by testing it against other agents.In this paper we explore what can be observed interactively in this setting. In particular, we characterize the objects that can be observed in a single test: the primitive observables of the theory.Our approach builds on an analysis of the geometrical properties of the agents, and highlights a deep interleaving between two partial orders underlying the combinatorial structures: the spatial one and the temporal one

Preliminares al estudio de la huella en lingüística

The present paper constitutes a brief advance of much longer and more detailed ongoing work on the concept of “trace” in contemporary linguistic theory, particularly in syntax. It is commonly believed that the idea was coined by Noam Chomsky. However, we already detect its use, with a very accurate value, in the early work of Zellig Harris on mathematical linguistics or, to be more precise, on mathematical structures of language. In its origins, rather than being an index responsible for marking the location occupied by a unit previous to its syntactic movement (which always takes the form of fronting ), the trace was the result of a matrix product between n-adic functions. Thus, in Harris the trace is primarily a concept anchored in matrix calculus, or, put it differently, an algebraic notion. Chomsky’s notion, on its turn, is closely related with the LISP programming language. This text seeks to provide a preliminary analysis of the conceptual complexity implied in the concept of trace, which linguists should become aware of, for otherwise they will be doomed to be entangled in misunderstandings unfruitful to our discipline for decades to come.El presente documento constituye un breve avance de una obra en curso mucho más larga y más detallada sobre el concepto de “huella” en la teoría lingüística contemporánea, particularmente en la sintaxis. Se cree, por lo común, que la idea fue acuñada por Noam Chomsky. Sin embargo, ya detectamos su uso, con un valor muy preciso, en los primeros trabajos de Zellig Harris sobre lingüística matemática o, para ser más exactos, sobre estructuras 2matemáticas del lenguaje. En sus orígenes, en lugar de ser un índice responsable de marcar la ubicación de una unidad antes de su movimiento sintáctico (que siempre toma la forma de fronting), la traza o huella era el resultado de un producto matricial entre funciones n-ádicas. Por lo tanto, en Harris la huella es principalmente un concepto anclado en el cálculo matricial o, dicho de otro modo, una noción algebraica. La noción de Chomsky, por su parte, está estrechamente relacionada con el lenguaje de programación LISP. EL presente texto busca proporcionar un análisis preliminar de la complejidad conceptual implícita en el concepto de huella, del cual los lingüistas deben tomar conciencia, porque de lo contrario estarán condenados a enredarse en malentendidos infructuosos para nuestra disciplina durante las próximas décadas

On the meaning of focalization

Abstract In this paper, we use Girard's Ludics to analyze focalization, a fundamental property of linear logic. In particular, we show how this can be realized interactively thanks to section-retraction pairs (u αβ , f αβ ) between behaviours α ˆ(β Y ), X and αβ Y, X

DRAFT -Do Not Distribute Ludics Programming I: Interactive Proof Search

Abstract Proof theory and Computation are research areas which have very strong relationships: new concepts in logic and proof theory often apply to the theory of programming languages. The use of proofs to model computation led to the modelling of two main programming paradigms which are functional programming and logic programming. While functional programming is based on proof normalization, logic programming is based on proof search. This approach has shown to be very successful by being able to capture many programming primitives logically. Nevertheless, important parts of real logic programming languages are still hardly understood from the logical point of view and it has been found very difficult to give a logical semantics to control primitives. Girard introduced Ludics [12] as a new theory to study interaction. In Ludics, everything is built on interaction or in an interactive way. In this paper, which is the first of a series investigating a new computational model for logic programming based on Ludics, namely computation as interactive proof search, we introduce the interactive proof search procedure and study some of its properties

Preliminares al estudio de la huella en lingüística

The present paper constitutes a brief advance of much longer and more detailed ongoing work on the concept of “trace” in contemporary linguistic theory, particularly in syntax. It is commonly believed that the idea was coined by Noam Chomsky. However, we already detect its use, with a very accurate value, in the early work of Zellig Harris on mathematical linguistics or, to be more precise, on mathematical structures of language. In its origins, rather than being an index responsible for marking the location occupied by a unit previous to its syntactic movement (which always takes the form of fronting ), the trace was the result of a matrix product between n-adic functions. Thus, in Harris the trace is primarily a concept anchored in matrix calculus, or, put it differently, an algebraic notion. Chomsky’s notion, on its turn, is closely related with the LISP programming language. This text seeks to provide a preliminary analysis of the conceptual complexity implied in the concept of trace, which linguists should become aware of, for otherwise they will be doomed to be entangled in misunderstandings unfruitful to our discipline for decades to comeEl presente documento constituye un breve avance de una obra en curso mucho más larga y más detallada sobre el concepto de “huella” en la teoría lingüística contemporánea, particularmente en la sintaxis. Se cree, por lo común, que la idea fue acuñada por Noam Chomsky. Sin embargo, ya detectamos su uso, con un valor muy preciso, en los primeros trabajos de Zellig Harris sobre lingüística matemática o, para ser más exactos, sobre estructuras matemáticas del lenguaje. En sus orígenes, en lugar de ser un índice responsable de marcar la ubicación de una unidad antes de su movimiento sintáctico (que siempre toma la forma de fronting), la traza o huella era el resultado de un producto matricial entre funciones n-ádicas. Por lo tanto, en Harris la huella es principalmente un concepto anclado en el cálculo matricial o, dicho de otro modo, una noción algebraica. La noción de Chomsky, por su parte, está estrechamente relacionada con el lenguaje de programación LISP. EL presente texto busca proporcionar un análisis preliminar de la complejidad conceptual implícita en el concepto de huella, del cual los lingüistas deben tomar conciencia, porque de lo contrario estarán condenados a enredarse en malentendidos infructuosos para nuestra disciplina durante las próximas década

Travelling on Designs -- Ludics Dynamics

Proofs in Ludics are represented by designs. Designs (desseins) can be seen as an intermediate syntax between sequent calculus and proof nets, carrying advantages from both approaches, especially w.r.t. cut-elimination. To study interaction between designs and develop a geometrical intuition, we introduce an abstract machine which presents normalization as a token travelling along a net of designs. This allows a concrete approach, from which to carry on the study of issues such as: (i) which part of a design can be recognized interactively; (ii) how to reconstruct a design from the traces of its interactions in different tests