Jacqueline Vauzeilles

Generating plans in linear logic I: actions as proofs

Abstract There is an increasing interest in the relation between logic and the changes involved in reasoning and, specifically, in plan generation. Up to now, several attempts in this direction have been made, either by embedding actions into a classical framework or by using nonstandard formalisms. We think that these attempts, though promising, miss their objectives, for a lack of a suitable logic, and that the effort must be pursued. In this paper, we show how to obtain a strong and clean correspondence between proofs and sequences of actions by using only Girards linear logic, eliminating from the classical logic the structural rules which are not adapted to our purpose. A theorem is presented which expresses the new adequacy between proofs and actions.

Generating plans in linear logic

We dealt with a well-delineated domain of planning and, thanks to linear logic, we gave it an adequate logical characterization, so that we managed to solve the completeness problem. Although this domain may appear restricting at first glance, it seems that it includes most of the examples in the literature on plan analysis and it gives rise to important questions which are quite satisfactorily handled by the formalism we proposed. On the other hand, we hope that our work modestly contributed to a deeper intuitive understanding of linear logic. Besides, we think that integrating other features of linear logic, such as the ones attached to linear implication (--o) or the bounded modalities ?n and !n will enable to cope with more realistic situations.

The classical AI planning problems in the mirror of Horn linear logic: semantics, expressibility, complexity

We introduce Horn linear logic as a comprehensive logical system capable of handling the typical AI problem of making a plan of the actions to be performed by a robot so that he could get into a set of final situations, if he started with a certain initial situation. Contrary to undecidability of propositional Horn linear logic, the planning problem is proved to be decidable for a reasonably wide class of natural robot systems. The planning problem is proved to be EXPTIME -complete for the robot systems that allow actions with non-deterministic effects. Fixing a finite signature, that is a finite set of predicates and their finite domains, we get a polynomial time procedure of making plans for the robot system over this signature. The planning complexity is reduced to PSPACE for the robot systems with only pure deterministic actions. As honest numerical parameters in our algorithms we invoke the length of description of a planning task ‘from W to Z˜’ and the Kolmogorov descriptive complexity of Ax T, a set of possible actions.

Functors and Ordinal Notations. II: A Functorial Construction of the Bachmann Hierarchy

This paper is the first of several works: the matter is the systematic rewriting of usual results of ordinal notations, using Pi-1-2-logic. The purpose of these articles is already realized in Pi-1-2-logic, in a relatively abstract form. The paper presents a functorial construction of the Veblen hierarchy.

Functors and Ordinal Notations III - Dilators and Gardens

Publisher Summary This chapter proves that there is an isomorphism between the categories of gardens and dilators. This isomorphism enables the investigation of the relation of a functor introduced by Jean–Yves Girard in Π 1 2 -logic with the Bachmann hierarchy, using the result of the work done in collaboration with Jean–Yves Girard on the functorial construction of the Bachmann hierarchy. The chapter presents definitions and some properties of gardens and dilators. It also defines a functor DEC (decomposition) from the category of dilators to the category of gardens. The chapter associates to each garden J a dilator SYN(J) (synthesis of J) using the inverse functor of SEP—that is, UN (unification of variables) . It is also proved that the functors DEC and SYN are inverse to each other.

Coping Polynomially with Numerous but Identical Elements within Planning Problems

Since the typical AI problem of making a plan of the actions to be performed by a robot so that it could get into a set of final situations, if it started with a certain initial situation, is generally exponential (it is even EXPTIME-complete in the case of games ‘Robot against Nature’), the planners are very sensitive to the number of variables, the inherent symmetry of the problem, and the nature of the logic formalisms being used. The paper shows that linear logic provides a convenient tool for representing planning problems. In particular, the paper focuses on planning problems with an unbounded number of functionally identical objects. We show that for such problems linear logic is especially effective and leads to dramatic contraction of the search space (polynomial instead of exponential). The paper addresses the key issue: “How to automatically recognize functions similarity among objects and break the extreme combinatorial explosion caused by this symmetry,” by means of replacing the unbounded number of specific names of objects with one generic name and contracting thereby the exponential search space over ‘real’ objects to a small polynomial search space but over the ‘generic’ one, with providing a more abstract formulation whose solutions are proved to be directly translatable into (optimal) polytime solutions to the original planning problem.

Functors and ordinal notations. IV: The Howard ordinal and the functor ∧.

In this paper, we prove a result that J. Y. Girard has conjectured for the last few years. That is, This result is part of a study of ordinal notations. For the proof, we use some concepts introduced in Parts I, II and III of this article (see [GV1], [GV2] and [V]); in particular, in Part III (see [V]) we defined the category GAR of gardens (of type Ω), and we have shown that there is an isomorphism between GAR and the category DIL Ω of dilators which send Ω into Ω. To describe this isomorphism, we defined a functor SYN from GAR to DIL Ω and a functor DEC from DIL Ω to GAR which are inverse to each other. The functor SYN is constructed by induction on the height x of the garden Jx, iterating, at each step of cofinality Ω, the operation UN (unification of variables; see [G1, Chapter 3]). Here, we define the functor S⊿ from GAR to DIL Ω: this construction is very close to that of SYN, but we iterate, at each step of cofinality Ω, the operation ⊿ (identification of variables; see [G, Chapter 3]). Technically, the definition of S⊿ is easier and more natural than that of SYN (as ⊿ is easier than UN) but is not inversible (as ⊿); for example, if we consider the (regular) garden G eΩ+1 introduced in [GV2],S⊿( G eΩ+1 ) is exactly the dilator N = ⋃ 0 n N n , with N 1 = 2 + Id, and , but we are incapable of expressing SYN( G eΩ+1 ).

Les premiers recursivement inaccessible et Mahlo et la theorie des dilatateurs

Dans cet article, on utilise la theorie des dilatateurs pour decrire les premiers recursivement inaccessible et Mahlo

Linear logic as a tool for planning under temporal uncertainty

The typical AI problem is that of making a plan of the actions to be performed by a controller so that it could get into a set of final situations, if it started with a certain initial situation.The plans, and related winning strategies, happen to be finite in the case of a finite number of states and a finite number of instant actions.The situation becomes much more complex when we deal with planning under temporal uncertainty caused by actions with delayed effects.Here we introduce a tree-based formalism to express plans, or winning strategies, in finite state systems in which actions may have quantitatively delayed effects. Since the delays are non-deterministic and continuous, we need an infinite branching to display all possible delays. Nevertheless, under reasonable assumptions, we show that infinite winning strategies which may arise in this context can be captured by finite plans.The above planning problem is specified in logical terms within a Horn fragment of affine logic. Among other things, the advantage of linear logic approach is that we can easily capture `preemptive/anticipative? plans (in which a new action s may be taken at some moment within the running time of an action ? being carried out, in order to be prepared before completion of action ?).In this paper we propose a comprehensive and adequate logical model of strong planning under temporal uncertainty which addresses infinity concerns. In particular, we establish a direct correspondence between linear logic proofs and plans, or winning strategies, for the actions with quantitative delayed effects.

Tree Adjoining Grammars in Noncommutative Linear Logic

This paper presents a logical formalization of Tree-Adjoining Grammar (TAG). TAG deals with lexicalized trees and two operations are available: substitution and adjunction. Adjunction is generally presented as an insertion of a tree inside another, surrounding the subtree at the adjunction node. This seems to be contradictory with standard logical ability. We prove that some logic, namely a fragment of non-commutative intuitionistic linear logic (N-ILL), can serve this purpose. Briefly speaking, linear logic is a logic considering facts as resources. NILL can then be considered either as an extension of Lambek calculus, or as a restriction of linear logic. We model the TAG formalism in four steps: trees (initial or derived) and the way they are constituted, the operations (substitution and adjunction), and the elementary trees, i.e. the grammar. The sequent calculus is a restriction of the standard sequent calculus for N-ILL. Trees (initial or derived) are then obtained as the closure of the calculus under two rules that mimic the grammatical ones. We then prove the equivalence between the language generated by a TAG grammar and the closure under substitution and adjunction of its logical representation. Besides this nice property, we relate parse trees to logical proofs, and to their geometric representation: proofnets. We briefly present them and give examples of parse trees as proofnets. This process can be interpreted as an assembling of blocks (proofnets corresponding to elementary trees of the grammar).