Combinatory logic, combinatorial calculi, and other questions about combinators and variable-free variants of the $\lambda$-calculus.

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2
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1answer
23 views

How to find the next higher combination out of a fixed group of digits?

I have a group of contiguous digits ordered from smallest to highest: 1234. I want a formula (in case it exists) to find the next closer higher combination of the same digits. In this example the next ...
1
vote
1answer
22 views

Proof of B, C, K, W system

There is a B,C,K,W system. In particular, there is presented the following identity: $B = S (K S) K$ How to prove this statement?
2
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1answer
76 views

Intuitively speaking, why was there a need to “eliminate” quantified variables in mathematical logic?

I'm trying to wrap my head around the understanding of lambda-calculus, from a math/computing/logic standpoint and am reading more about its very genesis. This has taken me to 1924 - Schonfinkel's ...
2
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0answers
53 views

Fixed point combinator (Y) and fixed point equation

In Hindley (Lambda-Calculus and Combinators, an Introduction), Corollary 3.3.1 on fixed point combinator. In $\lambda$ and CL: for every $Z$ and $n \ge 0$ the equation $$xy_1..y_n = Z$$ can be ...
2
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1answer
28 views

On a corollary of the Church-Rosser Theorem

In the proof of Corollary 1.41.5 from Hindley-Seldin, $\lambda$-Calculus and Combinators - An Introduction, If $a$ and $b$ are atoms and $aM_1...M_m =_\beta bN_1...N_n$ then $a = b$ and $m = n$ ...
3
votes
2answers
102 views

In what sense is the S-combinator “substitution”?

According to the Wikipedia page on SKI-combinator calculus, I is the identity function, K is the constant function, and S is "substitution". I understand the first two, but I don't see what S has to ...
3
votes
1answer
160 views

Is there a proof of (non)existence of a proper universal combinator?

It is a well-known fact that all combinators can be derived from the two fundamental combinators K and S. It seems only natural to also ask whether there is a single universal combinator, but I can’t ...
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0answers
69 views

What is the precise statement of Craig's theorem?

I'm interested writing a proof of Craig's theorem. After several attempts I realized that there are several possible ways to state the theorem, each with subtle but important differences. Here's one ...
0
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1answer
175 views

Proof completion: if $Y$ is a closed term in strong nf, then $Yx$ weakly reduces to a strong nf $Z$

I am self-studying Hindley & Seldin's Lambda-Calculus and Combinators. I would appreciate some help with filling in a final detail for a proof for the following statement regarding combinatory ...
2
votes
0answers
42 views

Chaitin's constant for lambda calculus and combinatory logic

I have found some approximations of Chaitin's Constant for turing machines but I have not found approximations for others. I'd like to have a rough estimate or upper bound on it for lambda calculus ...
5
votes
1answer
197 views

Smullyan-To-Mock-a-Mockingbird, Find egocentric bird in L

Question (29, p. 81). Let me tell you the most surprising thing I know about larks: Suppose we are given that the forest contains a lark $L$ and we are not given any other information. From just ...
3
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1answer
268 views

How to represent Smullyan's “Mockingbird” puzzles in (Homotopy) Type Theory?

(If you're unfamiliar with the puzzles from To Mock a Mockingbird, three pages tell you everything you should need.) Is it possible to solve the riddles in To Mock a Mockingbird in a "propositions as ...
1
vote
1answer
82 views

Why SKI when SK is complete

Why people talk about SKI calculus when S and K combinators can be used to create any other combinator including I?
8
votes
0answers
165 views

Where to go after _To Mock A Mockingbird_?

So long ago I read Raymond Smullyan's delightful To Mock A Mockingbird, a gentle introduction to combinatory logic (representing combinators as 'birds' singing back and forth to each other). I fell ...
4
votes
1answer
156 views

Prove that all combinators must fulfill A x = x for some x, given that M x = x x and composability of any two combinators

I'm working through Raymond Smullyan's "To Mock a Mockingbird" and I'm stuck on the first problem in the combinatory logic section. I'd appreciate hints, but no spoilers please. The problem is ...
1
vote
2answers
792 views

Proving a combinator is a fixed point

Show that the term ZZ where Z is λz.λx. x(z z x) satisfies the requirement for fixed point combinators that ZZM =β M(ZZM).
1
vote
1answer
807 views

Looping (ω) Combinator

Can someone explain this combinator? I understand $\lambda x. x$, but I don't understand $\lambda x. x x$ From what I've gathered, this means given x, return the application of x to x. I don't ...
12
votes
2answers
2k views

Can someone explain the Y Combinator?

The Y combinator is a concept in functional programming, borrowed from the lambda calculus. It is a fixed-point combinator. A fixed point combinator $G$ is a higher-order function (a functional, in ...
3
votes
1answer
104 views

combinatory basis for head reduction

Consider combinatory calculi that don't have tail reduction. So there may be combinators $x$, $y$ and $z$ such that $y\to z$ but $xy\nrightarrow xz$. We can still write every combinator as a ...