Questions tagged [combinatory-logic]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
26
votes
2answers
4k views

Are lambda calculus and combinatory logic the same?

I am currently reading "Lambda-Calculus and Combinators" by Hindley and Seldin. I'm not an expert, but have always taken an interest in lambda calculus because of involvement with functional ...
20
votes
2answers
1k views

Succinct circuit representation of graphs

The complexity class PPAD (e.g. computing various Nash equilibria) can be defined as the set of total search problems polytime reducible to END OF THE LINE: END OF THE LINE: Given circuits S and P ...
19
votes
2answers
1k views

Smallest possible universal combinator

I am looking for the smallest possible universal combinator, measured by the number of abstractions and applications required to specify such a combinator in the lambda calculus. Examples of universal ...
14
votes
1answer
691 views

How can non-terminating $\lambda$-terms be turned into fixed-point combinators?

I've been thinking about these questions: Is there a typed lambda calculus which is consistent and Turing complete? https://cs.stackexchange.com/questions/65003/if-%CE%BB-x-x-x-has-a-type-then-is-...
11
votes
1answer
977 views

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

I've been interested in various topics like Combinatory Logic, Lambda Calculus, Functional Programming for a while and have been studying them. However, unlike the "Theory of Computation" which ...
10
votes
1answer
286 views

Incomplete basis of combinators

This is inspired by this question. Let $\mathcal{C}$ be the collection of all combinators which only have two bound variables. Is $\mathcal{C}$ combinatorially complete? I believe the answer is ...
8
votes
1answer
284 views

Is there a normalizing (or perpetual) reduction strategy for untyped combinators?

Inspired by this question, I was curious whether there was a reduction strategy for untyped SKI combinators that was known to be either normalizing or perpetual. As described here (Twelfed here), ...
8
votes
0answers
404 views

Complete combinator basis for System F-omega

The S and K combinators form a complete (and Turing complete) basis when untyped. Within the Hindley-Milner type-system, and I believe within system $F$ as well, S and K can encode any well-typed ...
7
votes
3answers
954 views

What is the relationship between intuitionistic logic, combinatory logic and lambda calculus?

I've been reading Lectures on the Curry-Howard Isomorphism and it talks about intuitionistic/constructive logic (IL) , combinatory logic (CL) and lambda calculus ($\lambda$c) before moving on to the ...
7
votes
1answer
151 views

Upper bound on Chaitin's constant for lambda calculus and SKI combinatory logic

I'd like to have proof that Chaitin's constant for lambda calculus and/or SKI combinatory logic is pretty small. I've found some approximations (accurate to about 63 binary digits) for truing machines ...
7
votes
1answer
274 views

Church-Rosser equivalent for concatenative languages?

Looking at the striking parallels between combinatory logic and concatenative languages makes me wonder how many theorems of the former hold in the latter. The Church-Rosser theorem is particularly ...
7
votes
1answer
320 views

Concatenative binary lambda calculus/combinatory logic

John Tromp defines a version of the lambda calculus that is encoded in binary: https://tromp.github.io/cl/cl.html a) Does there exist a concatenative version of this language (or its combinatory ...
5
votes
1answer
164 views

Is combinatory strong reduction equivalent to lambda beta-eta reduction?

I'm familiar with the correspondence between combinatory logic and $\lambda$-calculus at equality level, ie $=_{\beta\eta}$ corresponds to $=_s$ (the equivalence relation generated by strong reduction)...
5
votes
0answers
83 views

Characterisation of the BCIW definable functions

Given a full model of the simply typed lambda calculus, it's possible to characterise the lambda definable functions as those that are invariant under every "Kripke logical relation". (See here.) I ...
4
votes
0answers
75 views

Can any Calculus of Construction term be built up from application of a finite number of terms?

Can we form a finite set of well typed calculus of construction terms such that any closed term can be built up from them (plus the type of large types) using only application? I conjecture that the ...
2
votes
1answer
161 views

Algorithm for extensional equality in combinator calculus

I'm dealing with combinator calculus, using the $S$ and $K$ combinators as a basis. Sometimes my code generates expressions that define equivalent functions, such as $$ (S\, K\, K) \qquad\text{and}\...
2
votes
2answers
528 views

Size of decision tree for f is polynomial in the DNF size of f and CNF size of f

I've been having hard time with proving the following claim: Let $f:\{T,F\}^n\rightarrow \{T,F\}$ be a boolean function. Let $size_{DT}(f)$ denote the number of leaves in the smallest (w.r.t the ...
1
vote
0answers
178 views

Is Combinatory Logic (CL) still relevant for programming language theory?

I've been reading up on R. Smullyan's "To Mock a Mockingbird" and Hindley's "Lambda-Calculus and Combinators: An Introduction". I've even read Schonfinkel's 1924 paper introducing the idea of ...
1
vote
0answers
134 views

translation of first-order logic to combinatory logic

I am new to combinatory logic, and I am interested in its relationship to first-order logic. In particular, I am wondering if there is a universal translation of first-order logic to combinatory logic ...
-1
votes
1answer
49 views

Algorithm for calculating substitution combination with ordering

I need to calculate the combinations of elements with a substitution element. For example for elements [A,B] if the substitution is X the results should be ...