Dynamically typed languages such as Python and Ruby have experienced a rapid grown in popularity in recent times. However, there is much confusion as to what makes these languages interesting relative to statically typed languages, and little knowledge of their rich history. In this chapter I explore the general topic of dynamically typed languages, how they differ from statically typed languag...

We present a typed intermediate language λ for optimizing compilers for function-oriented and polymorphically typed programming languages (e.g., ML). The language λ is a typed lambda calculus with product, sum, intersection, and union types as well as function types annotated with flow labels. A novel formulation of intersection and union types supports encoding flow information in the typed pr...

Dynamically typed languages such as JavaScript present a challenge to just-in-time compilers. In contrast to statically typed languages such as JVML, in which there are specific opcodes for common operations on primitive types (such as iadd for integer addition), all operations in dynamically typed language such as JavaScript are late-bound. Often enough, types cannot be inferred with certainty...

We present the type rules for a dependently typed core calculus together with a straightforward implementation in Haskell. We explicitly highlight the changes necessary to shift from a simply-typed lambda calculus to the dependently typed lambda calculus. We also describe how to extend our core language with data types and write several small example programs. The article is accompanied by an e...

We introduce a new kind of tree automaton, a dependency tree automaton, that is suitable for deciding properties of classes of terms with binding. Two kinds of such automaton are defined, nondeterministic and alternating. We show that the nondeterministic automata have a decidable nonemptiness problem and leave as an open question whether this is true for the alternating version. The families o...

Functional Kan simplicial sets are simplicial sets in which the horn-fillers required by the Kan extension condition are given explicitly by functions. We show the non-constructivity of the following basic result: if B and A are functional Kan simplicial sets, then A is a Kan simplicial set. This strengthens a similar result for the case of non-functional Kan simplicial sets shown by Bezem, Coq...

2 Natural Deduction and Sequent Calculus 1 2.1 Natural Deduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2.2 Sequent Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.3 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...