A Focused Sequent Calculus Framework for Proof Search in Pure Type Systems

Basic proof-search tactics in logic and type theory can be seen as the root-first applications of rules in an appropriate sequent calculus, preferably without the redundancies generated by permutation of rules. This paper addresses the issues of defining such sequent calculi for Pure Type Systems (PTS, which were originally presented in natural deduction style) and then organizing their rules for effective proof-search. We introduce the idea of Pure Type Sequent Calculus with meta-variables (PTSCalpha), by enriching the syntax of a permutation-free sequent calculus for propositional logic due to Herbelin, which is strongly related to natural deduction and already well adapted to proof-search. The operational semantics is adapted from Herbelin's and is defined by a system of local rewrite rules as in cut-elimination, using explicit substitutions. We prove confluence for this system. Restricting our attention to PTSC, a type system for the ground terms of this system, we obtain the Subject Reduction property and show that each PTSC is logically equivalent to its corresponding PTS, and the former is strongly normalising iff the latter is. We show how to make the logical rules of PTSC into a syntax-directed system PS for proof-search, by incorporating the conversion rules as in syntax-directed presentations of the PTS rules for type-checking. Finally, we consider how to use the explicitly scoped meta-variables of PTSCalpha to represent partial proof-terms, and use them to analyse interactive proof construction. This sets up a framework PE in which we are able to study proof-search strategies, type inhabitant enumeration and (higher-order) unification

Sequent Calculus in the Topos of Trees

Nakano's "later" modality, inspired by G\"{o}del-L\"{o}b provability logic, has been applied in type systems and program logics to capture guarded recursion. Birkedal et al modelled this modality via the internal logic of the topos of trees. We show that the semantics of the propositional fragment of this logic can be given by linear converse-well-founded intuitionistic Kripke frames, so this logic is a marriage of the intuitionistic modal logic KM and the intermediate logic LC. We therefore call this logic $\mathrm{KM}_{\mathrm{lin}}$. We give a sound and cut-free complete sequent calculus for $\mathrm{KM}_{\mathrm{lin}}$ via a strategy that decomposes implication into its static and irreflexive components. Our calculus provides deterministic and terminating backward proof-search, yields decidability of the logic and the coNP-completeness of its validity problem. Our calculus and decision procedure can be restricted to drop linearity and hence capture KM.Comment: Extended version, with full proof details, of a paper accepted to FoSSaCS 2015 (this version edited to fix some minor typos

On Deriving Nested Calculi for Intuitionistic Logics from Semantic Systems

This paper shows how to derive nested calculi from labelled calculi for propositional intuitionistic logic and first-order intuitionistic logic with constant domains, thus connecting the general results for labelled calculi with the more refined formalism of nested sequents. The extraction of nested calculi from labelled calculi obtains via considerations pertaining to the elimination of structural rules in labelled derivations. Each aspect of the extraction process is motivated and detailed, showing that each nested calculus inherits favorable proof-theoretic properties from its associated labelled calculus

Cisplatin-induced ototoxicity: the current state of ototoxicity monitoring in New Zealand.

Background: Many well-known pharmacologic agents have been shown to have toxic effects to the cochleo-vestibular system. Examples of such ototoxic agents include cisplatin and aminoglycoside antibiotics. Ototoxicity monitoring consists of a comprehensive pattern of audiological assessments designed to detect the onset of any hearing loss. Three main methods have emerged over the past decade, and include the basic audiological assessment, extended high frequency (EHF) audiometry, and otoacoustic emission (OAE) measurement. These measures can be used separately or in combination, depending on clinical purpose and patient considerations. It is suggested by the American Academy of Audiology Position Statement and Clinical Practice Guidelines: Ototoxicity Monitoring, that baseline testing be done in a fairly comprehensive manner, including pure-tone thresholds in both the conventional- and extended high frequency ranges, tympanometry, speech audiometry, and the testing of OAEs (AAA, 2009). Anecdotal evidence suggests that New Zealand Audiologists do not currently follow a national ototoxicity monitoring protocol. Therefore the main aim of this study was to explore the current status of ototoxicity monitoring within New Zealand. Hypothesis: It was hypothesized that hospital based Audiology departments across New Zealand each followed their own internal ototoxicity monitoring protocol based, to a large extent, on the guidelines proposed by the American Academy of Audiology and by the American Speech-Language-Hearing Association. Method: Through the use of a Telephone Interview Questionnaire, 16 charge Audiologists were interviewed to establish their current state of knowledge regarding ototoxicity monitoring at 16 out of 20 district health boards in New Zealand. Enquiries about the current systems and procedures in place at their departments together with any suggestions and recommendations to improve on these systems were made. Results: This study found that only 9 of the 16 DHBs interviewed currently follow an ototoxicity monitoring protocol. Furthermore, other than initially hypothesized the origin of the protocols followed by the remaining 7 departments were reported to have ranged from independently developed protocols to historically adopted protocols. One department implemented an adapted version of a protocol by Fausti et al. (Ear and Hearing 1999; 20(6):497-505). This diversity in origin however, does confirm our initial suspicion that no universal and standardized monitoring protocol is currently being followed by Audiologists working in the public health sector of New Zealand

Do optional activities matter in virtual learning environments?

Virtual Learning Environments (VLEs) provide students with activi-ties to improve their learning (e.g., reading texts, watching videos or solving exercises). But VLEs usually also provide optional activities (e.g., changing an avatar profile or setting goals). Some of these have a connection with the learn-ing process, but are not directly devoted to learning concepts (e.g., setting goals). Few works have dealt with the use of optional activities and the relation-ships between these activities and other metrics in VLEs. This paper analyzes the use of optional activities at different levels in a specific case study with 291 students from three courses (physics, chemistry and mathematics) using the Khan Academy platform. The level of use of the different types of optional ac-tivities is analyzed and compared to that of learning activities. In addition, the relationship between the usage of optional activities and different student be-haviors and learning metrics is presented

Focalisation and Classical Realisability (version with appendices)

The original publication is avalaible at : www.springerlink.comInternational audienceWe develop a polarised variant of Curien and Herbelin's lambda-bar-mu-mu-tilde calculus suitable for sequent calculi that admit a focalising cut elimination (i.e. whose proofs are focalised when cut-free), such as Girard's classical logic LC or linear logic. This gives a setting in which Krivine's classical realisability extends naturally (in particular to call-by-value), with a presentation in terms of orthogonality. We give examples of applications to the theory of programming languages

Towards a canonical classical natural deduction system

This paper studies a new classical natural deduction system, presented as a typed calculus named \lml. It is designed to be isomorphic to Curien-Herbelin's calculus, both at the level of proofs and reduction, and the isomorphism is based on the correct correspondence between cut (resp. left-introduction) in sequent calculus, and substitution (resp. elimination) in natural deduction. It is a combination of Parigot's $\lambda\mu$-calculus with the idea of ``coercion calculus'' due to Cervesato-Pfenning, accommodating let-expressions in a surprising way: they expand Parigot's syntactic class of named terms. This calculus aims to be the simultaneous answer to three problems. The first problem is the lack of a canonical natural deduction system for classical logic. \lml is not yet another classical calculus, but rather a canonical reflection in natural deduction of the impeccable treatment of classical logic by sequent calculus. The second problem is the lack of a formalization of the usual semantics of Curien-Herbelin's calculus, that explains co-terms and cuts as, respectively, contexts and hole-filling instructions. The mentioned isomorphism is the required formalization, based on the precise notions of context and hole-expression offered by \lml. The third problem is the lack of a robust process of ``read-back'' into natural deduction syntax of calculi in the sequent calculus format, that affects mainly the recent proof-theoretic efforts of derivation of $\lambda$-calculi for call-by-value. An isomorphic counterpart to the $Q$-subsystem of Curien-Herbelin's-calculus is derived, obtaining a new $\lambda$-calculus for call-by-value, combining control and let-expressions.Fundação para a Ciência e a Tecnologia (FCT

A genetic algorithm for the one-dimensional cutting stock problem with setups

This paper investigates the one-dimensional cutting stock problem considering two conflicting objective functions: minimization of both the number of objects and the number of different cutting patterns used. A new heuristic method based on the concepts of genetic algorithms is proposed to solve the problem. This heuristic is empirically analyzed by solving randomly generated instances and also practical instances from a chemical-fiber company. The computational results show that the method is efficient and obtains positive results when compared to other methods from the literature. © 2014 Brazilian Operations Research Society