Marc Bagnol

Logic Programming and Logarithmic Space

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 syntactic restriction, using an encoding of words that derives from proof theory.

Unification and Logarithmic Space

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.

Unary Resolution: Characterizing Ptime

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.

MALL Proof Equivalence is Logspace-Complete, via Binary Decision Diagrams

Proof equivalence in a logic is the problem of deciding whether two proofs are equivalent modulo a set of permutation of rules that reflects the commutative conversions of its cut-elimination procedure. As such, it is related to the question of proofnets: finding canonical representatives of equivalence classes of proofs that have good computational properties. It can also be seen as the word problem for the notion of free category corresponding to the logic. It has been recently shown that proof equivalence in MLL (the multiplicative with units fragment of linear logic) is PSPACE-complete, which rules out any low-complexity notion of proofnet for this particular logic. Since it is another fragment of linear logic for which attempts to define a fully satisfactory low-complexity notion of proofnet have not been successful so far, we study proof equivalence in MALL- (multiplicative-additive without units fragment of linear logic) and discover a situation that is totally different from the MLL case. Indeed, we show that proof equivalence in MALL- corresponds (under AC0 reductions) to equivalence of binary decision diagrams, a data structure widely used to represent and analyze Boolean functions efficiently. We show these two equivalent problems to be LOGSPACE-complete. If this technically leaves open the possibility for a complete solution to the question of proofnets for MALL-, the established relation with binary decision diagrams actually suggests a negative solution to this problem.

Representation of Partial Traces

The notion of trace in a monoidal category has been introduced to give a categorical account of a situation occurring in very different settings: linear algebra, topology, knot theory, proof theory... with the trace operation understood as a feedback operation. Partially traced categories were later introduced to account for cases where the trace is not always defined, and it was shown that partially traced category can always be seen as a subcategory of a totally traced one. We give a new proof of this representation theorem, using a construction that is different from the original one. However, since they satisfy the same universal property they are naturally isomorphic.

The Shuffle Quasimonad and Modules with Differentiation and Integration

Abstract Differential linear logic and the corresponding categorical structure, differential categories, introduced the idea of differential structure associated to a (co)monad. Typically in settings such as algebraic geometry, one expresses differential structure for an algebra by having a module with a derivation, i.e. a map satisfying the Leibniz rule. In the monadic approach, we are able to continue to work with algebras and derivations, but the additional structure allows us to define other rules of the differential calculus for such modules; in particular one can define a monadic version of the chain rule as well as other basic identities. In attempting to develop a similar theory of integral linear logic, we were led to consider the shuffle multiplication. This was shown by Guo and Keigher to be fundamental in the construction of the free Rota-Baxter algebra, the Rota-Baxter equation being the integral analogue of the Leibniz rule. This shuffle multiplication induces a quasimonad on the category of vector spaces. The notion of quasimonad, called r-unital monad by Wisbauer, is slightly weaker than that of monad, but is still sufficient to define a sensible notion of module with differentiation and integration. In this paper, we demonstrate this quasimonad structure, show that its free modules have both differential and integral operators satisfying the Leibniz and Rota-Baxter rules and satisfy the fundamental theorems of calculus.

On the Dependencies of Logical Rules

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 Jacobe 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.