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

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How do I study combinatory logic? [closed]

They told me about the book "To mock a mockingbird" by Raymond Smullyan, I'm liking it, but I would like to solve the questions made in the book in a mathematical way, and it's hard to find anything ...
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2answers
65 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 ...
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128 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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38 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 ...
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1answer
171 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 ...
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31 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 ...
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1answer
164 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 ...
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1answer
230 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 ...
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1answer
72 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?
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152 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 ...
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1answer
148 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 ...
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57 views

Is the number of reduction orders a computable function?

Let $e$ be any expression in the SKI combinator calculus. Let $R(e)$ denote the number of possible reduction orders for $e$. Is $R$ a computable function? I believe that $R$ is not computable. But ...
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769 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).
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1answer
514 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 ...
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1k 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 ...
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94 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 ...