Interaction Systems I: The theory of optimal reductions

Andrea Asperti, Cosimo Laneve
1994 Mathematical Structures in Computer Science  
A new class of higher order rewriting systems, called Interaction Systems, is introduced. >From one side, Interaction Systems provide the intuitionistic generalization of Lafont's Interaction Nets Laf90] (recall that Interaction Nets are linear). In particular, we keep the idea of binary interaction, and the syntactical bipartition of operators into constructors and destructors. From the other side, Interaction Systems are the subsystem of Klop's Combinatory Reduction Systems Kl80, Ac78] where
more » ... he Curry-Howard analogy still \makes sense". This means that we can associate with every IS a suitable logical (intuitionistic) system: constructors and destructors respectively correspond to right and left introduction rules, interaction is cut, and computation is cut-elimination. Interaction Systems have been primarily motivated by the necessity of extending Lamping's optimal graph reduction technique for the -calculus Lam90, GAL92] to other computational constructs than just -reduction. This implementation style can be smootly generalized to arbitrary IS's, providing in this way a uniform description of essential rules such as conditionals and recursion. The optimal implementation of IS's will be only sketched here (it will eventually be the subject of the forthcoming Part II). The main aim of this paper is to introduce this new class of Systems, to discuss the motivations behind its de nition, and to investigate the theoretical aspect of optimal reductions (in particular, the notion of redex-family). On introduit une nouvelle classe de syst emes de r e ecriture d'ordre sup erieur: les Syst emes d'Interaction. D'une part, les Syst emes d'Interaction sont la g en eralisation intuitioniste des R eseaux d'Interaction de Lafont (les R eseaux d'Interaction sont lin eaires); en particulier, on reprend l'id ee d'interaction binaire, et la bipartition syntaxique des op erateurs en constructeurs et destructeurs. D'autre part, les Syst emes d'Interaction sont la sous-classe des Combinatory Reduction Systems de Klop o u on peut toujours appliquer l'analogie de Curry-Howard, c'est-a-dire qu'on peut d e nir, pour chaque IS, le syst eme logique intuitioniste correspondant: les constructeurs et les destructeurs correspondent respectivement a des r egles d'introduction a droite et a gauche, l'interaction a une coupure, et la r eduction est l' elimination des coupures. Les Syst emes d'Interaction ont leur motivation primordiale dans la n ecessit e d' etendre la technique d' evaluation optimale pour le lambda calcul de Lamping a d'autres op erations que la -r eduction. Ce style d'impl ementation peutêtre appliqu e sans probl emes a tous les Syst emes d'Interaction, ce qui donne une description uniforme de r egles de r e ecriture essentielles, comme les sauts conditionnels ou la r ecursion. L'impl ementation optimale des IS's sera seulement esquiss ee ici (il sera le sujet du deuxi eme partie). L'objectif principal de ce papier est d'introduire cette nouvelle classe des Syst emes, de motiver leur d e nition, et d' etudier les aspects th eoriques des r eductions optimales (en particulier, la notion de famille de radicaux). In every partition there is at most one negative port. If a negative port exists, we shall call it an input partition; otherwise it is an output partition. Every agent has exactly \one output" (functionality). In particular, if the agent is a constructor, the main port is already an output, and all the partitions must be input. Conversely, in the case of destructors, we have exactly one output partition among the auxiliary ports, and this partition has to be a singleton. Remark 2.1 1. As the reader has surely understood, the partitions of the auxiliary ports are meant to express a binding mechanism. In case all partitions are singletons (discrete case, in Laf90]) the net is acyclic, corresponding to a rst order rewriting system.
doi:10.1017/s0960129500000566 fatcat:kic7lxe57rbmfk3xyoupwiwj5y