Logical Axioms of a Combinator

Question

I am doing some experiments in proving a theorem with combinatorial logic, which looks promising, but there is one stumbling block: it was pointed out that in combinatorial logic it is true, for example, I = SKK, but this is not a theorem, it must be added as an axiom. Does anyone know a complete list of axioms to add?

Edit: Of course, you can prove that I = SKK, but if I do not miss something, this is not a theorem in the combinatorial logic system with equality. That was said, you can just expand the macro I to SKK ... but I still don't see anything important. Accepting a lot of sentences p (X) and ~ p (X), which easily resolve the contradiction in ordinary first-order logic and convert them to SK, performing a substitution and evaluating all the calls S and K, my program generates the following (where I use "for the opposite Unlambda ")

'' eq '' '' ks '' 'ks' 'kk' k eq '' 'ks' kk 'kk' 'kk' k false 'k true' k true

It seems that, perhaps, I need an appropriate set of rules for processing partial calls “k” and “s”, I just don’t see what these rules should do, and all the literature that I can find in this area has been written for the target audience mathematicians, not programmers. I suspect the answer is probably pretty simple if you understand this.

Answer 1

I ((S K) K). ( ) . ( ), , , :

• `` E = E '' ( , , E)

, , : I (( SK) K), , ​​

• `` (I E) = E '' ( I-)

,

( ) , I ((SK) K), ?

, , . , , ( ), . , .

, , langauge. , "" . , . , . , .

(, , , ), extensions . , .

Csörnyei 2007: 157-158, . , .

:

, . E, F, G. .

, : . , ( , ), . , , .

:

. : K, S, I.

I , , I / term S K K, .

.

: @` '' ( arity 2). : ( ) .

. , .., . , , . , . ( ) , x, y, z...

:

• -
• -
• E - , F , (E F)

, .

E F G H

(((E F) G) H).

:

• `` K E F = E '' ( K-axiom)
• `` S F G H = F H (G H) '' (S-)
• `` I E = E '' ( I-axiom)

(I) , , I /macro S K K.

• `` E = E '' ( )
• "E = F" , "F = E" ( Symmetry )
• "E = F" , "F = G", "E = G" ( Transitive)
• "E = F" , "E G = F G" ( I)
• "E = F" , "G E = G F" ( II)

. , , , , .

" I= S K K" ?

, . , , . , . , , . , ! , . -

E F = G F , E = G

: , , . , ,

`` I E = S K K E ''

E-, . ( ), :

• E: = K
• E: = S
• E: = K K
• .
• .
• .

...

, ! , .

. , . , , , , ***** *****. , . , :

• x E, F, (E x) = (F x) , E = F ( )

, : x - object, , E G meta​​strong > , , .

(. , , x meta​​strong > object x, y, z..., meta​​strong > E term. , .)

, ` I= S K K ''. p >

:

I x = x '' - I- [E: = x]

:

" S K K x = K x ( K x )" S- [E: = K, F: = K, G: = x], " K x ( K x) = x" K-axiom [E: = x, F: = K x],

:

" S K K x = K x ( K x )" , " K x ( K x) = x" . : [E: = S K K x, F: = K x ( K x), G = x]. , : E = G. , " S K K x = x", , .

:

S K K x = x ", :" x = S K K x "

:

I x = x "" x = S K K x ", infer" I x = S K K x"

:

" I x = S K K x" : (E x) = (F x), [E: = I, F: = S K K]. , , .. "E = F" ([E: = I, F: = S K K]), " I= S K K", quod erat.

Csörnyei, Zoltán (2007): Lambda-kalkulus. . : Typotex. ISBN-978-963-9664-46-3.

Answer 2

. :

I.x = x
K.x y = x
S.x y z = x z (y z)

SKanything = anything, SKanything , I.

, I = SKK I = SKS. , , SKK.

S K - .

Answer 3

-, eta. , - . Hindley and Seldin. -.

, , , - , , . , .