Questions tagged [combinatory-logic]
Combinatory logic, combinatorial calculi, and other questions about combinators and variable-free variants of the $\lambda$-calculus.
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Interpreting higher-order intuitionistic logic in typed variable-free combinatory logic
Consider the following typed variable-free combinatory logic:
$\iota$ i the type of individuals.
$o$ is the type of propositions.
If $\tau_1, \tau_2$ are types, then $\tau_1 \to \tau_2$ is the type ...
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Missing parentheses in $s(k (s I I))(s(\lambda y. s(k y))(\lambda y. s I I)$ leads to interesting error in an nLab page. Need a double check.
I think I found an error in the nLab page on partial combinatory algebra in the Example combinators section:
Finally, consider the classical construction of the fixed-point combinator, $Y = \lambda y....
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Beta reducing $SS(SK)$ using SKI calculus
I have an expression to $\beta$-reduce and I managed to brute force it using the $\lambda$-calculus. I was wondering though, if I could make it in less steps, than what I did, using the $SKI$-calculus....
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Doesn't lambda K-calculus include lambda I-calculus?
To mock a mockingbird, chapter 18:
From just S and K you can derive any combinatorial bird whatsoever!
Same book, chapter 19
[…] with just the two birds J and I, we would ultimately get the same ...
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Nondeterminism In Modeling Computation with SKI-combinators
Consider two terms of the SKI-combinator calculus $\alpha$ and $\beta$ such that the following derivations are valid.
$\alpha \rightarrow \alpha_1$
$\beta \rightarrow \beta_1$
Then we have two valid ...
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SKI Calculus prefix notation of odd/even number
I'm working on a homework with SKI calculus. I saw the hints in this very useful post.
We basically defined SKI functions as:
...
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Proof for $\frac{b}{h}=\frac{c(c-1)}{k(k-1)}$, a combinatorial identity
Imagine we have $b$ teachers and $c$ students in a school such that
each teacher teaches $k$ students,
every two students have $h$ similar teachers.
I'd like to prove that
$$\frac{b}{h}\:=\:\frac{c(...
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What axioms can be added to $S,K$ combinator algebra without making it collapse into triviality?
My understanding is that if you start with the free magma on two generators (call them $S$ and $K$) and then take a quotient with respect to the usual $S$ and $K$ equivalence rules ($Sfgx = fx(gx)$ ...
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An equation involving multisets
For multisets $A, B, C, A', B', C'$, if $A \uplus B \uplus \{B \uplus C\} \uplus \{A \uplus \{C\}\}$ = $A' \uplus B' \uplus \{B' \uplus C'\} \uplus \{A' \uplus \{C'\}\}$, must $A=A',B=B',C=C'$, where $...
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Relationship between the semantics of simply typed lambda calculus and combinatory logic
The simply typed lambda calculus has a class of extremely intuitive models where each basic type $\sigma$ is modeled by some set $[\![\sigma]\!]$, and a complex type $\sigma\rightarrow\tau$ is then ...
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number of surjective functions from X to Y where card(Y)=k < n =card(X) [duplicate]
Let $X=\{1,...,n\}$ and $Y=\{1,...,k\}$. If $n\gt\ k$ then the number of surjective functions $f:X\to Y$ $\;\;$is $\;\sum_{j=0}^{k-1} (-1)^j\binom{k}{j}(k-j)^n$
Here I show what I have done. I can'...
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Hindley and Seldin "Introduction to Combinators and Lambda calculus": Question about the solution to exercise 3.5.
The question is concerning the solution to exercise 3.5. In the exercise, they ask us to prove that any finite set of simultaneous equations $x_iy_1...y_n=Z_i$, where $1\leq i\leq k$, can be solved ...
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Could we define an arity of a term in combinatory logic and consider some inference rule?
In computer science, we know a function has an arity. And we noticed the similarity between function and a term in combinatory logic, so could we define the arity of a term in combinatory logic? And ...
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How to define halting status in combinatory logic
We all know that combinatory logic can be used to express programs, for example:
$S(K(SI))K\alpha\beta \rightarrow K(SI)\alpha(K\alpha)\beta
\rightarrow SI(K\alpha)\beta
...
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Relation between Lambda calculus and SKI combinator calculus
I have played around with evaluation of Lambda terms in Lambda calculus and their counterparts using SKI combinators. While the results are extensionally equal (as they should be), there are striking ...
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Would this be considered a combinator
A combinator is a function that doesn't utilize free variables.
E.g.
$$
\lambda a. \lambda b.a
$$
However all of this works in an untyped environment. When working with types, let's say $nat$. can'...
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Expressing I combinator and numerals via X combinator
So I understand that the X Combinator is defined as X = λx (x S) K
I am also aware of definitions of S and K and I: S = XK <...
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Why can't we formalize the lambda calculus in first order logic?
I'm reading through Hindley and Seldin's book about the lambda calculus and combinatory logic. In the book, the authors express that, though combinatory logic can be expressed as an equational theory ...
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The definition of the Church numerals in combinatory logic
Hindley & Seldin define ([1] Definition 4.2, p. 48) the Church numerals as follows: (I'm paraphrasing to save space. Here's the original page.)
For every $n \in \{0,1,\dots\}$, the Church ...
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How to prove the Church-Rosser theorem for combinatory logic
I need help proving the Church-Rosser theorem for combinatory logic. I will break down my post in three parts: part I will establish the notation required to state the Church-Rosser theorem as well as ...
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Number of unique items in a big set from small sample
We have a box with $m=1\,000\,000$ cards. Each card contains one word. The words are repeated so there is a relatively small number of $n$ unique words. $n$ is unknown.
If we get a sample of $k=5000$ ...
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Equality in the untyped lambda calculus without extensionality
Raymond Turner (in p.66 of "Truth and Modality for Knowledge Representation") elaborates a combinatory logic, $PT$, whose language $L2$ is the following language of terms (together with a language of ...
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Combinatory logic - Evaluation exercise (abstraction and weak reduction)
I am going through the book "Lambda-Calculus and Combinators: An Introduction".
I am trying to solve the following exercise: evaluation of $[x,y,z].xzy$
The result should be, according to solutions: ...
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Can all computable numeric functions on church numerals in ski-combinator calculus be expressed using only completely evaluated terms?
Let a term in ski-combinator calculus be called "complete" if every primitive is partially applied (so all S's are applied to at most two arguments, all K's to at most 1, and all I's are not applied).
...
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Rolling Dice Probability
A fair dice is rolled 3 times, The probability of the product of the three outcomes is a prime number is?
The products which give a prime number I found out to be only 4.
However for the total ...
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Consistency of the SKI calculus as unprovability of S = K
The exercise I'm dealing with asks me to show that by adding $S = K$ to the usual reduction rules for the SKI-calculus, one obtains an inconsistent equivalence. This must be done without using Böhm's ...
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An exercise on combinatory logic
Can somebody help me with the following exercise?
(1) Find a combinator $X$ such that $Xy = X$; (2) Find a combinator in normal form with the same property. Rules for reduction are
$Ix > x$
$Kxy &...
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Hindley's "Introduction to combinatory logic", exercise 6 chapter 2.
Can somebody help me with the following exercise?
Find a combinator X such that X = S(KK)(XS). Reduction rules are usual:
IX reduces to X (identity combinator)
KXY reduces to X
SXYZ reduces to XZ(...
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What is this P4 correspond to in proposition as types?
I was reading "Proofs and Types", so there came across that any proposition can be converted to lambda form. So was trying out with Hilbert system's axioms
P1. $A \rightarrow A $
P2. $A \rightarrow (...
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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?
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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 ...
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Fixed point combinator (Y) and fixed point equation
In Hindley (Lambda-Calculus and Combinators, an Introduction), Corollary 3.3.1 to the fixed-point theorem states:
In $\lambda$ and CL: for every $Z$ and $n \ge 0$ the equation $$xy_1..y_n = Z$$
can ...
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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 $m = n$ and ...
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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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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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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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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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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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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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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 this ...
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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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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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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 ...
user17137
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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 ...
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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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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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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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