Semi-continuous Sized Types and Termination

Some type-based approaches to termination use sized types: an ordinal bound for the size of a data structure is stored in its type. A recursive function over a sized type is accepted if it is visible in the type system that recursive calls occur just at a smaller size. This approach is only sound if the type of the recursive function is admissible, i.e., depends on the size index in a certain way. To explore the space of admissible functions in the presence of higher-kinded data types and impredicative polymorphism, a semantics is developed where sized types are interpreted as functions from ordinals into sets of strongly normalizing terms. It is shown that upper semi-continuity of such functions is a sufficient semantic criterion for admissibility. To provide a syntactical criterion, a calculus for semi-continuous functions is developed.Comment: 33 pages, extended version of CSL'0

Type-Based Termination, Inflationary Fixed-Points, and Mixed Inductive-Coinductive Types

Type systems certify program properties in a compositional way. From a bigger program one can abstract out a part and certify the properties of the resulting abstract program by just using the type of the part that was abstracted away. Termination and productivity are non-trivial yet desired program properties, and several type systems have been put forward that guarantee termination, compositionally. These type systems are intimately connected to the definition of least and greatest fixed-points by ordinal iteration. While most type systems use conventional iteration, we consider inflationary iteration in this article. We demonstrate how this leads to a more principled type system, with recursion based on well-founded induction. The type system has a prototypical implementation, MiniAgda, and we show in particular how it certifies productivity of corecursive and mixed recursive-corecursive functions.Comment: In Proceedings FICS 2012, arXiv:1202.317

Beauly Sub-Station, Beauly, Highland. Archive Report: the lithic assemblage (4004161)

An analysis of the lithic assemblage from the excavations undertaken by Northlight Heritage at Beauly Sub-station

Tower FT187, Dalnacardoch, Perth & Kinross Archive Report: the lithic assemblage (4016161)

An analysis of the lithic assemblage from the excavations undertaken by Northlight Heritage at Dalnacardoch, Perth & Kinross

Access Track 81 - Tower TD125, Pittentian, Crieff, Perth & Kinross: Archive Report: the lithic assemblage (4008161)

An analysis of the lithic assemblage form the excavations undertaken by Northlight Heritage at Pittentian, Crieff

Lair, Glenshee, Perth & Kinross Archive Report: the lithic assemblage (4268161; 4415161)

An analysis of the lithic assemblage from the excavations undertaken by the Tay Landscape Partnership in 2012 and 2014 at The Lair, Glenshee

Tower BF7, Fanellan, Beauly, Highland Archive Report: the lithic assemblage (4011161)

An analysis of the lithic assemblage from the excavations undertaken by Northlight Heritage at Famellan, Beauly

Refinement Types as Higher Order Dependency Pairs

Refinement types are a well-studied manner of performing in-depth analysis on functional programs. The dependency pair method is a very powerful method used to prove termination of rewrite systems; however its extension to higher order rewrite systems is still the object of active research. We observe that a variant of refinement types allow us to express a form of higher-order dependency pair criterion that only uses information at the type level, and we prove the correctness of this criterion