This paper is an overview of existing applications of Linear Logic (LL) to issues of computation. After a substantial introduction to LL, it discusses the implications of LL to functional programming, logic programming, concurrent and object-oriented programming and some other applications of LL, like semantics of negation in LP, non-monotonic issues in AI planning, etc. Although the overview covers pretty much the state-of-the-art in this area, by necessity many of the works are only mentioned and referenced, but not discussed in any considerable detail. The paper does not presuppose any previous exposition to LL, and is addressed more to computer scientists (probably with a theoretical inclination) than to logicians. The paper contains over 140 references, of which some 80 are about applications of LL. 1 Linear Logic Linear Logic (LL) was introduced in 1987 by Girard [62]. From the very beginning it was recognized as relevant to issues of computation (especially concurrency and stat...

The paper describes a transition logic, TL, and a deductive formalism for it. It shows how various important aspects (such as ramification, qualification, specificity, simultaneity, indeterminism etc.) involved in planning (or in reasoning about action and causality for that matter) can be modelled in TL in a rather natural way. (The deductive formalism for) TL extends the linear connection method proposed earlier by the author by embedding the latter into classical logic, so that classical and resource-sensitive reasoning coexist within TL. The attraction of a logical and deductive approach to planning is emphasized and the state of automated deduction briefly described. 1 Introduction Artificial Intelligence (AI, or Intellectics [Bib92a]) aims at creating artificial (or computational [PMG98]) intelligence. Were there no natural intelligence, the sentence would be meaningless to us. Hence understanding natural intelligence by necessity has always been among the goals of Intel...

In this paper we present a logical framework for defining consistent axiomatizations of planning domains. A language to define basic actions and structured plans is embedded in a logic. This allows general properties of a whole planning scenario to be proved as well as plans to be formed deductively. In particular, frame assertions and domain constraints as invariants of the basic actions can be formulated and proved. Even for complex plans most frame assertions are obtained by purely syntactic analysis. In such cases the formal proof can be generated in a uniform way. The formalism we introduce is especially useful when treating recursive plans. A tactical theorem prover, the Karlsruhe Interactive Verifier KIV is used to implement this logical framework. 1 Introduction In this paper we present a logical framework for defining consistent axiomatizations of planning domains. An effective mechanism for defining the basic operations in a constructive way is embedded in a logic that allo...

We review some logic formalisms which were designed in order to reason about situations, goals, and actions. In particular, we focuss on the so-called frame problem, i.e. the technical problem of how to formalize the assumption that unless an action explicitly causes a certain fact to hold or not to hold, the facts are preserved by the action. It is shown that there is no need to explicitly state frame axioms, ie. axioms which deal with the frame problem, if the logic formalism treats facts as resources which are produced and consumed. The linear connection method, the linear logic as well as a particular equational logic are such formalisms. Moreover, we demonstrate that these three formalisms are equivalent for a large class of planning problems.

The idea that to perceive an object is to perceive its affordances---that is, the interactions of the perceiver with the world that the object supports or affords---is attractive from the point of view of theories in cognitive science that emphasize the fundamental role of actions in representing an agent's knowledge about the world. However, in this general form, the notion has so far lacked a formal expression. This paper offers a representation for objects in terms of their affordances using Linear Dynamic Event Calculus, a formalism for reasoning about causal relations over events. It argues that a representation of this kind, linking objects to the events which they are characteristically involved in, underlies some universal operations of natural language syntactic and semantic composition that are postulated in Combinatory Categorial Grammar (CCG). These observations imply that the language faculty is more directly related to prelinguistic cognitive apparatus used for planning action than formal theories in either domain have previously seemed to allow.

Recently, three approaches to deductive planning were developed, which solve the technical frame problem without the need to state frame axioms explicitly. These approaches are based on the linear connection method, an equational Horn logic, and linear logic. At first glance these approaches seem to be very different. In the linear connection method a syntactical condition -- each literal is connected at most once -- is imposed on proofs. In the equational logic approach situations and plans are represented as terms and SLDE-resolution is applied as an inference rule. The linear logic approach is a Gentzen style proof system without weakening and contraction rules. On second glance, however, and as a consequence of the results rigourously proved in this paper, it will turn out that the three approaches are equivalent. They are based on the very same idea that facts about a situation are taken as resources which can be consumed and produced.

l purpose resolution theorem prover, several extensions and modifications have been developed. They led to efficiency gains and increased the expressive power of the underlying logic; most prominent examples being Kowalski's version of the situational calculus [21] and Manna and Waldinger's fluent theory [22], respectively. Kowalski's approach provides a representational solution of the frame problem by drastically reducing the number of required frame axioms. Plans are generated using a Horn clause resolution procedure guided by several strategies. Fluent theory was the first approach which allowed for a domain axiomatisation using flexible function symbols and with that formed the basis for the generation of recursive plans. These were produced by Manna and Waldinger's resolution-based tableau calculus which also provides induction rules. The paradigm, which underlies these approaches as well as most of the more recent ones, is that of

The idea that natural language grammar and planned action are related systems has been implicit in psychological theory for more than a century. However, formal theories in the two domains have have tended to look very different. This article argues that both faculties share the formal character of applicative systems based on operations corresponding to the same two combinatory operations, namely functional composition and type-raising. Viewing them in this way suggests simpler and more cognitively plausible accounts of both systems, and suggests that the language faculty evolved in the species and develops in children by a rather direct adaptation of a more primitive apparatus for planning purposive action in the world by composing affordances of objects or tools. The knowledge representation that underlies such planning is also reflected in the natural language semantics of tense, mood, and aspect, which the paper begins by arguing provides the key to understanding both systems.

The problem of planning dialog moves can be viewed as an instance of the more general AI problem of planning with incomplete information and sensing. Sensing actions complicate the planning process since such actions engender potentially infinite state spaces. We adapt the Linear Dynamic Event Calculus (LDEC) to the representation of dialog acts using insights from the PKS planner, and show how this formalism can be applied to the problem of planning mixedinitiative collaborative discourse. 1

Recently three approaches for solving planning problems deductively were proposed each of which does not require to state frame axioms explicitly. These approaches are based on the linear connection method, an equational logic programming language, and on linear logic. In this paper, we briefly review these approaches and show that they are equivalent. Moreover, we illustrate that these approaches are not only restricted to deductive planning, but can be applied whenever actions are to be modelled in logic. We show that the approaches essentially amount on building predicates over the data structure multiset. Such multisets are interpreted as resources, which are consumed and produced by actions. We give a minimal and complete unification algorithm for the equational theory which defines the multisets. Finally, we discuss possible extensions of the equational logic programming approach.