Taus Brock-Nannestad

Computation in focused intuitionistic logic

We investigate the control of evaluation strategies in a variant of the Λ-calculus derived through the Curry-Howard correspondence from LJF, a sequent calculus for intuitionistic logic implementing the focusing technique. The proof theory of focused intuitionistic logic yields a single calculus in which a number of known Λ-calculi appear as subsystems obtained by restricting types to a certain fragment of LJF. In particular, standard Λ-calculi as well as the call-by-push-value calculus are analysed using this framework, and we relate cut elimination for LJF to a new abstract machine subsuming well-known machines for these different strategies.

Disproving Using the Inverse Method by Iterative Refinement of Finite Approximations

In first-order logic, forward search using a complete strategy such as the inverse method can get stuck deriving larger and larger consequence sets when the goal query is unprovable. This is the case even in trivial theories where backward search strategies such as tableaux methods will fail finitely. We propose a general mechanism for bounding the consequence sets by means of finite approximations of infinite types. If the inverse method also implements forward subsumption and globalization, then the search space under this approximation is finite. We therefore obtain a type-directed iterative refinement algorithm for disproving queries. The method has been implemented for intuitionistic first-order logic, and we discuss its performance on a variety of problems.

Cut Elimination in Multifocused Linear Logic

We study cut elimination for a multifocused variant of full linear logic in the sequent calculus. The multifocused normal form of proofs yields problems that do not appear in a standard focused system, related to the constraints in grouping rule instances in focusing phases. We show that cut elimination can be performed in a sensible way even though the proof requires some specific lemmas to deal with multifocusing phases, and discuss the difficulties arising with cut elimination when considering normal forms of proofs in linear logic. The two most important results in the proof theory of linear logic [Gir87] are the admissibility of the cut rule, and the completeness of the focused normal form of proofs. The notion of focusing was originally developped by Andreoli [And92] with the purpose of improving proof search, but recently it has been considered more often as a normal form that can be obtained by reorganising inference steps in a given proof [MS07]: permutations can be used to group positive rule instances — and to move negative rule instances down. This viewpoint is particularly useful in the natural extension of focusing tomultifocusing [CMS08], where several positive formulas are selected to be decomposed in parallel. This stronger normal form is difficult to use for proof search, since not all positive formulas can be selected in a given sequent: there are complex dependencies. In the multiplicative fragment without units, when the selection of positives is done maximally, proofs are canonical in the sense that they are in bijection with proof-nets [Gir96]. For this reason, investigating the proof theory of multifocused linear logic is necessary to understand the notion of canonicity in sequent calculi, and possibly design normal forms of proofs that could be used as proof-nets, while retaining the usual syntax based on trees of rule instances.

Space-efficient acyclicity constraints a declarative pearl

Abstract Many constraints on graphs, e.g. the existence of a simple path between two vertices, or the connectedness of the subgraph induced by some selection of vertices, can be straightforwardly represented by means of a suitable acyclicity constraint. One method for encoding such a constraint in terms of simple, local constraints uses a 3-valued variable for each edge, and an ( N + 1 ) -valued variable for each vertex, where N is the number of vertices in the entire graph. For graphs with many vertices, this can be somewhat inefficient in terms of space usage. In this paper, we show how to refine this encoding into one that uses only a single bit of information, i.e. a 2-valued variable, per vertex, assuming the graph in question is planar. More generally, for graphs that are embeddable in genus g (i.e. on a torus with g handles), we show that 2 g + 1 bits per vertex suffices. We furthermore show how this same constraint can be used to encode connectedness constraints, and a variety of other graph-related constraints.

Space-Efficient Planar Acyclicity Constraints

Many constraints on graphs, e.g. the existence of a simple path between two vertices, or the connectedness of the subgraph induced by some selection of vertices, can be straightforwardly represented by means of a suitable acyclicity constraint. One method for encoding such a constraint in terms of simple, local constraints uses a 3-valued variable for each edge, and an (N+1)-valued variable for each vertex, where N is the number of vertices in the entire graph. For graphs with many vertices, this can be somewhat inefficient in terms of space usage. In this paper, we show how to refine this encoding into one that uses only a single bit of information, i.e. a 2-valued variable, per vertex, assuming the graph in question is planar. We furthermore show how this same constraint can be used to encode connectedness constraints, and a variety of other graph-related constraints.

Focused Linear Logic and the λ-calculus

Linear logic enjoys strong symmetries inherited from classical logic while providing a constructive framework comparable to intuitionistic logic. However, the computational interpretation of sequent calculus presentations of linear logic remains problematic, mostly because of the many rule permutations allowed in the sequent calculus. We address this problem by providing a simple interpretation of focused proofs, a complete subclass of linear sequent proofs known to have a much stronger structure than the standard sequent calculus for linear logic. Despite the classical setting, the interpretation relates proofs to a refined linear λ-calculus, and we investigate its properties and relation to other calculi, such as the usual λ-calculus, the λµ-calculus, and their variants based on sequent calculi.

Hybrid Extensions in a Logical Framework

We discuss the extension of the LF logical framework with operators for manipulating worlds, as found in hybrid logics or in the HLF framework. To overcome the restrictions of HLF, we present a more general approach to worlds in LF, where the structure of worlds can be described in an explicit way. We give a canonical presentation for this system and discuss the encoding of logical systems, beyond the limited scope of linear logic that formed the main goal of HLF.

Truthful monadic abstractions

In intuitionistic sequent calculi, detecting that a sequent is unprovable is often used to direct proof search. This is for instance seen in backward chaining, where an unprovable subgoal means that the proof search must backtrack. In undecidable logics, however, proof search may continue indefinitely, finding neither a proof nor a disproof of a given subgoal. In this paper we characterize a family of truth-preserving abstractions from intuitionistic first-order logic to the monadic fragment of classical first-order logic. Because they are truthful, these abstractions can be used to disprove sequents in intuitionistic first-order logic.