The framework Pure Type System (\(\mathcal{PTS}\)) offers a simple and general approach to designing and formalizing type systems. However, in the presence of dependent types, there often exist some acute problems that make it difficult for \(\mathcal{PTS}\) to accommodate many common realistic programming features such as general recursion, recursive types, effects (e.g., exceptions, references, input/output), etc. In this paper, we propose a new framework Applied Type System (\(\mathcal{ATS}\)) to allow for designing and formalizing type systems that can readily support common realistic programming features. The key salient feature of \(\mathcal{ATS}\) lies in a complete separation between statics, in which types are formed and reasoned about, and dynamics, in which programs are constructed and evaluated. With this separation, it is no longer possible for a program to occur in a type as is otherwise allowed in \(\mathcal{PTS}\). We present not only a formal development of \(\mathcal{ATS}\) but also mention some examples in support of using \(\mathcal{ATS}\) as a framework to form type systems for practical programming.

Linear logic

Applied Type System

Game semantics has almost achieved the status of a paradigm in computer science but philosophers are slow to take notice. One reason for this might be the lack of a convincing philosophical account of logical games, what it means to play them, for the proponent to win, etc., pointedly raised by Wilfrid Hodges as the ‘Dawkins question’. In this paper, I critically examine two available answers: after a brief discussion of an argument by Tennant against Hintikka games, I focus on Lorenzen's attempt at providing a direct foundation for his game rules in the life-world, showing some of the difficulties inherent to that project. I then propose an alternative based on the theory of assertions developed by Dummett and Brandom.

methods

Why Play Logical Games

We design an intuitionistic predicate logic that supports a limited amount of classical reasoning, just enough to prove a variant of Markov's principle suited for predicate logic. At the computational level, the extraction of an existential witness out of a proof of its double negation is done by using a form of statically-bound exception mechanism, what can be seen as a direct-style variant of Friedman's A-translation.

/pdf/an-intuitionistic-logic-that-proves-markov-s-principle-qob563oqdk.pdf

An Intuitionistic Logic that Proves Markov's Principle

Rigid E-unification is a restricted kind of unification modulo equational theories, or E-unification, that arises naturally in extending P. Andrews' (1981) theorem-proving method of mating to first-order languages with equality. It is shown that rigid E-unification is NP-complete and that finite complete sets of rigid E-unifiers always exist. As a consequence, deciding whether a family of mated sets is an equational mating is an NP-complete problem. Some implications of this result regarding the complexity of theorem proving in first-order logic with equality are discussed. >

Rigid E-unification is NP-complete

In this paper we present a theory of proof nets for full multiplicative linear logic, including the two units. It naturally extends the well-known theory of unit-free multiplicative proof nets. A linking is no longer a set of axiom links but a tree in which the axiom links are subtrees. These trees will be identified according to an equivalence relation based on a simple form of graph rewriting. We show the standard results of sequentialization and strong normalization of cut elimination. Furthermore, the identifications enforced on proofs are such that the proof nets, as they are presented here, form the arrows of the free (symmetric) *-autonomous category.

/pdf/on-proof-nets-for-multiplicative-linear-logic-with-units-2hhuucuvn1.pdf

Linear logic

Citations

Applied Type System

Why Play Logical Games

An Intuitionistic Logic that Proves Markov's Principle

Rigid E-unification is NP-complete

On Proof Nets for Multiplicative Linear Logic with Units

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