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 AxT, a set of possible actions.

Certain ‘Finite Structure Conditions’ on a geometric theory are shown to be sufficient for its classifying topos to be a presheaf topos. The conditions assert that every homomorphism from a finite structure of the theory to a model factors via a finite model, and they hold in cases where the finitely presentable models are all finite.

The conditions are shown to hold for the theory of strongly algebraic (or SFP) information systems and some variants, as well as for some other theories already known to be classified by presheaf toposes.

The work adheres to geometric constructivism throughout, and in consequence provides ‘topical’ categories of domains (internal in the category of toposes and geometric morphisms) with an analogue of Plotkin's double characterization of strongly algebraic domains, by sets of minimal upper bounds and by sequences of finite posets.

We define the class of divisibility monoids that arise as quotients of the free monoid Σ* modulo certain equations of the form ab = cd. These form a much larger class than free partially commutative monoids, and we show, under certain assumptions, that the recognizable languages in these divisibility monoids coincide with c-rational languages. The proofs rely on Ramsey's theorem, distributive lattice theory and on Hashigushi's rank function generalized to these monoids. We obtain Ochmański's theorem on recognizable languages in free partially commutative monoids as a consequence.

This paper concerns the elimination of higher type quantifiers and gives two theorems. The first theorem shows that quantifiers in formulae of a specific form can be eliminated. The second theorem shows that quantifiers in formulae of a similar form cannot be eliminated, that is, such formulae do not have an equivalent first-order formula. The proof is based on the Ehrenfeucht game. These theorems are important for design of an interpreter of a ν act, which is a representation of mathematical action. Moreover, even if the universe is assumed to be finite, these theorems hold.