Extended Calculus Of Constructions Research Articles

ANF preserves dependent types up to extensional equality

AbstractMany programmers use dependently typed languages such as Coq to machine-verify high-assurance software. However, existing compilers for these languages provide no guarantees after compiling, nor when linking after compilation.Type-preserving compilerspreserve guarantees encoded in types and then use type checking to verify compiled code and ensure safe linking with external code. Unfortunately, standard compiler passes do not preserve the dependent typing of commonly used (intensional) type theories. This is because assumptions valid in simpler type systems no longer hold, and intensional dependent type systems are highly sensitive to syntactic changes, including compilation. We develop an A-normal form (ANF) translation with join-point optimization—a standard translation for making control flow explicit in functional languages—from the Extended Calculus of Constructions (ECC) with dependent elimination of booleans and natural numbers (a representative subset of Coq). Our dependently typed target language has equality reflection, allowing the type system to encode semantic equality of terms. This is key to proving type preservation and correctness of separate compilation for this translation. This is the first ANF translation for dependent types. Unlike related translations, it supports the universe hierarchy, and does not rely on parametricity or impredicativity.

Formal foundations for situation awareness based on dependent type theory

Cognitive situation awareness has recently caught the attention of the information fusion community. Some approaches have developed formalizations that are both ontology-based and underpinned with Situation Theory. While the semantics of Situation Theory is very attractive from the cognitive point of view, the languages that are used to express knowledge and to reason with suffer from a number of limitations concerning both expressiveness and reasoning capabilities. In this paper we propose a more general formal foundation denoted S-DTT (Situation-based Dependent Type Theory) that is expressed with the language of the Extended Calculus of Constructions (ECC), a widely used theory in mathematical formalization and in software validation. Situation awareness relies on small blocks of knowledge called situation fragment types whose composition leads to a very expressive and unifying theory. The semantic part is provided by an ontology that is rooted in the S-DTT theory and, on which higher-order reasoning can be performed. The basis of the theory is summarized and its expressing power is illustrated with numerous examples. A scenario in the healthcare context for patient safety issues is detailed and a comparison with well-known approaches is discussed.

The Extended Calculus of Constructions (ECC) with inductive types

Luo's Extended Calculus of Constructions (ECC) is a higher order functional calculus based on Coquand's and Huet's Calculus of Constructions, but has in addition strong sums and a predicative cumulative type hierarchy. In this paper I introduce inductive types on the predicative type levels of ECC. I also show how the ω- Set model for ECC can be extended to a model for this augmented calculus.