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
23
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 ...
- 32.4k
21
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 ...
- 32.4k
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 ...
- 28.3k
17
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 ...
- 11.4k
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 ...
- 13.7k
14
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, ...
- 32.4k
14
votes
Where is the model theory in programming language theory?
Let me amend Damiano's answer with more specific comments.
Terms of type bool are not logical statements. Expressions like ...
- 28.3k
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 ...
- 28.3k
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 ...
- 32.4k
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 ...
- 16.7k
12
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 ...
- 2,040
12
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, ...
- 4,431
12
votes
Accepted
Where is the model theory in programming language theory?
The model theory of programming languages is called denotational semantics. You can google the term to find out more about it, I'll give an extreme synthesis of it.
Denotational semantics is a ...
- 5,273
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 ...
- 13.7k
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 ...
- 5,273
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 ...
- 1,177
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 $\Delta$ (...
- 28.3k
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 ...
- 28.3k
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 ...
- 295
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 ...
- 106
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 ...
- 326
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 ...
- 13.7k
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 ...
- 32.4k
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 ...
- 32.4k
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/...
- 13.7k
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
How can you encode natural numbers operations on interaction combinators?
Contrarily to the $\lambda$-calculus, the interaction combinators have no underlying logical system (i.e., there is no Curry-Howard correspondence for them), it is therefore hard to say that a numeral ...
- 5,273
8
votes
What's the difference between reduction strategies and evaluation strategies?
A reduction strategy is a function on Lambda that picks one redex (reducible expression) from all possible redexes -- depending on what you define as a redex.
Informally, an evaluation strategy is ...
8
votes
Accepted
Do we care about confluence because of unique normal forms?
I don't know what you mean by "practical", but confluence is very useful from the semantic point of view. Hopefully other people will be able to give you other answers from other points of view (for ...
- 5,273
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