SKI Calculus: Difference between revisions
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A '''SKI calculus''' program is a binary tree where the leaves are combinators, the three symbols <code>S</code>, <code>K</code>, <code>I</code>. Using parentheses to notate the tree, a simple example of a SKI program is <code>(((SK)S)((KI)S))</code>. We can omit parentheses by assuming they are left-binding by default, so we simplify our program to <code>SKS(KIS)</code>. | |||
A SKI calculus program is a binary tree where the leaves are combinators, the three symbols <code>S</code>, <code>K</code>, <code>I</code>. Using parentheses to notate the tree, a simple example of a SKI program is <code>(((SK)S)((KI)S))</code>. We can omit parentheses by assuming they are left-binding by default, so we simplify our program to <code>SKS(KIS)</code>. | |||
Like lambda calculus, SKI calculus has a process called beta-reduction. We change the tree according to any reducible redex. | Like lambda calculus, SKI calculus has a process called beta-reduction. We change the tree according to any reducible redex. | ||
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* <code>Sxyz -> Sxz(yz)</code> | * <code>Sxyz -> Sxz(yz)</code> | ||
Note that <code>xyz</code> represent any valid trees, not just single combinators. | Note that <code>xyz</code> represent any valid trees, not just single combinators. We repeat this process and we say it terminates if the combinator cannot be beta-reduced. | ||
Busy Beaver for SKI calculus ('''BB_SKI''') is a variation of the [[Busy Beaver for lambda calculus|Busy Beaver problem for lambda calculus]]. BB_SKI(n) is defined as the size of the largest output of a terminating program of size n. | |||
BB_SKI(n) is defined as the size of the largest output of a terminating program of size n. | |||
== Champions == | == Champions == | ||
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|S(SSK(S(S(S(SSS))I)I))I(S(S(KS)K)I)KK | |S(SSK(S(S(S(SSS))I)I))I(S(S(KS)K)I)KK | ||
|? | |? | ||
|- | |||
|25 | |||
|> Graham's Number | |||
|SII(SI(SI(K(S(K(S(K(SS(K(K(S(S(KS)K)I)))))(SI)))K)))) | |||
| 2014MELO03 | |||
|} | |} | ||
== SK calculus == | == SK calculus == | ||
We can remove the <code>I</code> combinator and replace it by <code>(SKS)</code>, <code>(SKK)</code> or any <code>(SKx)</code>. | We can remove the <code>I</code> combinator and replace it by <code>(SKS)</code>, <code>(SKK)</code> or any <code>(SKx)</code>. These terms have a straightforward binary encoding where (prefix) application is 1, K=00, and S=01. Since n combinators take n-1 applications to combine, their code length is 2n + n-1 = 3n-1 bits. | ||
=== Champions === | === Champions === | ||
Latest revision as of 23:07, 9 May 2026
A SKI calculus program is a binary tree where the leaves are combinators, the three symbols S, K, I. Using parentheses to notate the tree, a simple example of a SKI program is (((SK)S)((KI)S)). We can omit parentheses by assuming they are left-binding by default, so we simplify our program to SKS(KIS).
Like lambda calculus, SKI calculus has a process called beta-reduction. We change the tree according to any reducible redex.
Ix -> IKxy -> KxSxyz -> Sxz(yz)
Note that xyz represent any valid trees, not just single combinators. We repeat this process and we say it terminates if the combinator cannot be beta-reduced.
Busy Beaver for SKI calculus (BB_SKI) is a variation of the Busy Beaver problem for lambda calculus. BB_SKI(n) is defined as the size of the largest output of a terminating program of size n.
Champions
| n | Value | Champion | Discoverered by |
|---|---|---|---|
| 1 | = 1 | S | ? |
| 2 | = 2 | SS | ? |
| 3 | = 3 | SSS | ? |
| 4 | = 4 | SSSS | ? |
| 5 | = 6 | SSS(SS) | ? |
| 6 | ≥ 17 | SSS(SI)S | ? |
| 7 | ≥ 40 | S(SS)S(SS)S | ? |
| 8 | ≥ 41 | SII(S(S(SS)))S | ? |
| 9 | ≥ 79 | SII(SS(SSS))S | ? |
| 10 | ≥ 164 | SII(SS(SS(SS)))S | ? |
| 11 | ≥ 681 | SII(SS(SS(SSS)))S | ? |
| 12 | ≥ 1530 | SII(SS(SS(SS(SS))))S | ? |
| 13 | ≥ 65537 | S(S(SI))I(S(S(KS)K)I)KK | ? |
| 14 | ≥ 2^256+1 | S(S(S(SI)))I(S(S(KS)K)I)KK | ? |
| 15 | > 2^2^2^2^21 | S(S(SSS)I)I(S(S(KS)K)I)KK | ? |
| 16 | > 2^^19 | S(S(S(SSS))I)I(S(S(KS)K)I)KK | ? |
| 17 | > 2^^2^128 | SSK(S(S(KS)K)I)(S(SI(SI))I)KK | ? |
| 18 | > 2^^2^2^2^2^21 | SSK(S(S(KS)K)I)(S(S(SSS)I)I)KK | ? |
| 19 | > 2^^^2^128 | S(SSK(S(SI(SI))I))I(S(S(KS)K)I)KK | ? |
| 20 | > 2^^^2^2^2^2^21 | S(SSK(S(S(SSS)I)I))I(S(S(KS)K)I)KK | ? |
| 21 | > 2^^^2^^19 | S(SSK(S(S(S(SSS))I)I))I(S(S(KS)K)I)KK | ? |
| 25 | > Graham's Number | SII(SI(SI(K(S(K(S(K(SS(K(K(S(S(KS)K)I)))))(SI)))K)))) | 2014MELO03 |
SK calculus
We can remove the I combinator and replace it by (SKS), (SKK) or any (SKx). These terms have a straightforward binary encoding where (prefix) application is 1, K=00, and S=01. Since n combinators take n-1 applications to combine, their code length is 2n + n-1 = 3n-1 bits.
Champions
| n | bits | Value | Champion | Discoverered by |
|---|---|---|---|---|
| 1 | 2 | = 1 | S | ? |
| 2 | 5 | = 2 | SS | ? |
| 3 | 8 | = 3 | SSS | ? |
| 4 | 11 | = 4 | SSSS | ? |
| 5 | 14 | = 6 | SSS(SS) | ? |
| 6 | 17 | ≥ 10 | SSS(SS)S | ? |
| 7 | 20 | ≥ 40 | S(SS)S(SS)S | ? |
| 8 | 23 | ≥ 41 | S(S(SS)S(SS)S) | ? |
| 9 | 26 | ≥ 42 | S(S(S(SS)S(SS)S)) | ? |
| 10 | 29 | ≥ 66 | SS(SSS)(SS(SS))S | ? |
| 11 | 32 | ≥ 79 | SS(SSS)(SS(SSS))S | ? |
| 12 | 35 | ≥ 164 | SS(SKK)(SS)(SS(SS))S | ? |
| 13 | 38 | ≥ 681 | SS(SKK)(SS)(SS(SSS))S | ? |
| 14 | 41 | ≥ 1530 | SS(SKK)(SS)(SS(SS(SS)))S | ? |
| 15 | 44 | ≥ 7811 | SS(SKK)(SS)(SS(SS(SSS)))S | ? |
See Also
Lower bounds of this function (archived)