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Type structure is a syntactic discipline for enforcing levels of abstraction.

2
votes
Virtually every programming language employs types in some form or other. Some languages use types only at run-time or only at compile-time, but hybrids are also widespread. Program specification and …
answered Feb 16 '15 by Martin Berger
3
votes
Counter-question: why should every type be inhabited by a term? You could not have the Curry-Howard correspondence between typing systems and logic if every type was inhabited. Concrete answer: I don …
answered Jun 11 '15 by Martin Berger
6
votes
In my experience, a good and operational way to understand duality of types for $\lambda$-calculi is by going through $\pi$-calculus. When you translate (decompose) types into process calculus, duali …
answered Nov 27 '12 by Martin Berger
12
votes
Summary. A logical framework is a meta-language for the formalisation of deductive systems, where deductions become syntactic objects. Of course what counts as a meta-language is quite vague, and it …
answered Feb 18 '15 by Martin Berger
10
votes
One dimension is type inference. System F's type inference for example is not decidable, but some its predicative fragments have decidable (partial) type inference. Another dimension is consistency …
answered Mar 22 '15 by Martin Berger
6
votes
The definition of an object $X$ is impredicative, if the definition uses a collection $C$ in the construction of $X$, such that $X$ is a member of $C$. So impredicativity is a form of circularity. W …
answered Jun 25 '16 by Martin Berger
11
votes
Type theories have multiple uses, and with each kind of usage comes a different notion of correctness. They two key uses are As a foundation of mathematics. In this context correctness means primari …
answered Apr 4 '14 by Martin Berger
11
votes
I'm not sure this question is ideal for CSTheory, but given that it's already gathering upvotes, here is an answer somebody might have given had the question been posted on cs.stackexchange. In ord …
answered Jul 15 '17 by Martin Berger
6
votes
One approach to such questions is via encodings. Say you have a language $L_1$ and a language $L_2$ and you want to show that they are somehow "the same", you can do this by finding an encoding $$ …
answered Aug 8 '16 by Martin Berger
8
votes
Quite a bit is know about this. The concept of Pure Type Systems (PTS) is useful for showing Church-Rosser (CR) for large classes of typed $\lambda$-calculi. Paraphrasing (1): PTS with only β reduct …
answered Dec 15 '15 by Martin Berger
3
votes
"Types are the leaven of computer programming; they make it digestible." Robin Milner
answered Jul 11 '21 by Martin Berger
4
votes
1answer
Programming languages with dependent types and/or higher-kinded types feature what might be called compile-time computation at the type-level. This is usually defined as follows (I'm omitting some de …
asked Aug 3 '15 by Martin Berger
6
votes
Inductive types have been studied heavily and many variants exist. A well-known introduction to inductive definitions is P. Aczel, An Introduction to Inductive Definitions which was originally pub …
answered Mar 29 '14 by Martin Berger
4
votes
A practical example of an axiom behaving badly you ask, what about this? 0 = 1 The Coquand paper referred to might be [ 1 ], where he shows that dependent ITT (Martin-Löf's intuitionistic type the …
answered Dec 19 '15 by Martin Berger
1
vote
I suggest not to give up on the operational intuition. Operational is primary, all semantics are derived, and are but proof techniques for operational semantics. The key ideas are as follows. A prog …
answered Sep 13 '15 by Martin Berger

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