Abstract. We present an algebraic view on logic programming, related to proof theory and more specifically linear logic and geometry of interaction. Within this construction, a characterization of logspace (deterministic and non-deterministic) computation is given via a synctactic restriction, using an encoding of words that derives from proof theory. We show that the acceptance of a word by an observation (the counterpart of a program in the encoding) can be decided within logarithmic space, by reducing this problem to the acyclicity of a graph. We show moreover that observations are as expressive as two-ways multi-heads finite automata, a kind of pointer machines that is a standard model of logarithmic space computation.

We present an algebraic characterization of the complexity classes Logspace and NLogspace, using an algebra with a composition law based on unification. This new bridge between unification and complexity classes is inspired from proof theory and more specifically linear logic and Geometry of Interaction. We show how unification can be used to build a model of computation by means of specific subalgebras associated to finite permutations groups. We then prove that whether an observation (the algebraic counterpart of a program) accepts a word can be decided within logarithmic space. We also show that the construction can naturally represent pointer machines, an intuitive way of understanding logarithmic space computing

Abstract. Many correctness criteria have been proposed since linear logic was introduced and it is not clear how they relate to each other. In this paper, we study proof-nets and their correctness criteria from the perspective of dependency, as introduced by Mogbil and Jacobé de Naurois. We introduce a new correctness criterion, called DepGraph, and show that together with Danos' contractibility criterion and Mogbil and Naurois criterion, they form the three faces of a notion of dependency which is crucial for correctness of proof-structures. Finally, we study the logical meaning of the dependency relation and show that it allows to recover and characterize some constraints on the ordering of inferences which are implicit in the proof-net.

No abstract

We give a characterization of deterministic polynomial time computation based on an algebraic structure called the resolution semiring, whose elements can be understood as logic programs or sets of rewriting rules over first-order terms. This construction stems from an interactive interpretation of the cut-elimination procedure of linear logic known as the geometry of interaction. This framework is restricted to terms (logic programs, rewriting rules) using only unary symbols, and this restriction is shown to be complete for polynomial time computation by encoding pushdown automata. Soundness w.r.t. Ptime is proven thanks to a saturation method similar to the one used for pushdown systems and inspired by the memoization technique. A Ptime-completeness result for a class of logic programming queries that uses only unary function symbols comes as a direct consequence.

No abstract

No abstract

No abstract