23 votes

Using lambda calculus to derive time complexity?

A recent developpement on this topic: U. dal Lago and B. Accatoli proved that the length of the leftmost-outermost reduction (LOr) of a $\lambda$-term is an invariant (time) cost model for $\lambda$-...
  • 686
22 votes
Accepted

Type-based memory safety without manual memory manage or runtime garbage collection?

Roughly speaking, there are two main strategies for safe manual memory management. The first approach is to use some substructural logic like linear logic to control resource usage. This idea has ...
20 votes

Why do functional programming languages require garbage collection?

All of the following comments are premised on the choice of a standard implementation strategy using closures to represent function values and a call-by-value evaluation order: For the pure lambda ...
17 votes

Status quo of category theory and monads in theoretical computer science research?

There have been a number of developments with regards to the use of monads in the theory of computation since Eugenio Moggi's work. I am not able to give a comprehensive account, but here are some ...
16 votes

What is the "question" that programming language theory is trying to answer?

The overall purpose of PLT is to make industrial software engineering (in a general sense) cheaper (also in a general sense), through optimising the most important tool (programming languages) and ...
15 votes
Accepted

Program reasoning about own source code

I think you are asking about two different things. The ability of a programming language to represent all its programs as data. Reasoning about programs as data. For analytical purposes it's ...
14 votes
Accepted

Logical Reations for an Impredicative System in a Predicative MetaTheory

In general, what we usually call the logical relations argument isn't really linked to impredicativity: the main idea is simply to interpret terms in some abstract algebra $\cal A$, and to represent ...
  • 13.4k
13 votes

Why/when do we ever need transfinite loop variants?

You are correct when you observe that for any particular terminating loop $L$ we may simply define the invariant "we're getting one step closer to termination". But proving that this is indeed a valid ...
12 votes
Accepted

A bicartesian closed category of strict complete partial orders (Hask)

Yes, it's impossible to have a nondegenerate CCC with general recursion and categorical coproducts. The standard reference for this is: H. Huwig and A. Poigne. A note on inconsistencies caused by ...
12 votes
Accepted

What kind of theoretical object corresponds to a C++ concept?

From a programming language theory perspective, as opposed to the computability perspective other answers and comments have offered, C++ templates combined with concepts correspond to bounded ...
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12 votes
Accepted

Applications of algebraic geometry in type theory/programming language theory

To my knowledge (which is definitely incomplete), there has been relatively little work on this, presumably because it requires assimilating two relatively intricate bodies of knowledge. However, ...
11 votes

What's the relation between OOP and category theory?

There are absolutely some relationships between the semantics and practice of OOP and category theory. This is somewhat unsurprising since both fields attempt to give a principled generic account of ...
  • 13.4k
11 votes
Accepted

Reference for the fact that (0=1) implies false requires a universe in MLTT

I know of: Jan M. Smith, The independence of Peano's fourth axiom from Martin-Löf's type theory without universes, The Journal of Symbolic Logic 53(3), p. 840-845, 1988.
11 votes

An example where smallest normal lambda term is not fastest

Blum’s speedup theorem is usually stated in the language of partially recursive functions, but up to trivial differences in notation, it works just the same in the language of $\lambda$-calculus. It ...
11 votes

Has the semantics of TeX (as a programming language) ever been formalized?

No, to my knowledge there has been no work on formalizing TeX of the kind you are interested in. (What follows is a subjective and personal commentary). I think it is an intriguing and well-posed ...
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11 votes

Representing bound variables with a function from uses to binders

Andrej's and Łukasz's answers make good points, but I wanted to add additional comments. To echo what Damiano said, this way of representing binding using pointers is the one suggested by proof-nets, ...
10 votes
Accepted

Why is the multi-step reduction of semantics reflexive?

The practical reason is that it is very convenient to include also the case "zero steps" in the definition of "many steps" (millennia of mathematical experience have taught us that it is usually a ...
10 votes

Representing bound variables with a function from uses to binders

I'm not sure how your variable-to-binder-function would be represented and for what purpose you'd like to use it. If you are using back-pointers then as Andrej noted the computational complexity of ...
  • 1,157
10 votes

Preservation under Substitution with Telescopes

The most general form of substitution theorems speaks about arbitrary contexts: Define what it means to have a substitution $\sigma : \Gamma \to \Delta$ from a context $\Gamma$ to a context $\Delta$ (...
10 votes
Accepted

How to tell if an effect is algebraic?

The general answer which you do not want to hear is: an effect is algebraic if it can be described using operations and equations. The question is a bit open-ended and gives no hint as to what you're ...
9 votes

What are some good introductory books on type theory?

Robert Harper's book Practical Foundations for Programming Languages (available as a draft online: http://www.cs.cmu.edu/~rwh/pfpl/2nded.pdf) is a somewhat more intense alternative to Types and ...
9 votes
Accepted

Has the semantics of TeX (as a programming language) ever been formalized?

(With apologies for a long answer that goes in a direction different from the scope of the site: frankly I was surprised to see the question here in the first place….) TeX was designed for ...
9 votes
Accepted

Question on subtyping of handlers in "An Effect System for Algebraic Effects and Handlers"

I'm surprised we don't get this question more often, for Andrej and I have considered adding this rule for quite some time and believed to have proven its correctness. But in the end, it turned out to ...
9 votes

Why/when do we ever need transfinite loop variants?

I would like to add the following to Andrej's response (not enough rep for a comment). Indeed, we cannot avoid ordinals but we may hide them. One approach is to use some modal logic that takes ...
9 votes
Accepted

Proof techniques for showing that dependent type checking is decidable

There is indeed a subtlety here, though things work out nicely in the case of type checking. I'll write down the issue here, since it seems to come up in many related threads, and try to explain why ...
  • 13.4k
9 votes

Hereditary substitution with a universe hierarchy

As of November 2018, how to do this for dependent type theories with large eliminations is an open question. Establishing that the recursion is well-founded is not too bad; you can use Pataraia's ...
9 votes
Accepted

Extending Hindley-Milner to type mutable references

To get behaviour similar to Ocaml, simply avoid generalizing the type of mutable variables. With ordinary let-bindings, you generalize if you bind a value, and don't generalize otherwise. With ...
9 votes
Accepted

Intuition Behind Strict Positivity?

It sounds like you want an overview of normalization arguments for type systems with positive datatypes. I'd recommend Nax Mendler's PhD dissertation: http://www.nuprl.org/documents/Mendler/...
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9 votes
Accepted

Similarities and differences between Pie and popular languages with dependent types

I'd say that Pie is a much smaller version of the core languages of those systems. It's a bit closer to Lean's core than to Coq, Agda, or Idris, but the differences are not very large there. When I ...
8 votes

Maximum computational power of a C implementation

As noted in the question, standard C requires that there exists a value UCHAR_MAX such that every variable of type unsigned char will always hold a value between 0 ...
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