Spring 2012
Juyeon Kang
qkang@fi.muni.cz
B410, Faculty of Informatics, Masaryk University,
Brno, Czech Rep.
IA165
Combinatory Logic for
Computational Semantics
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Lecture 3
● Continue about the theory of
combinators
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● Interesting book
”To mock a Mockingbird and Other LOgic Puzzles: including an Amazing
Adventure in Combinatory Logic” by Raymond Smullyan (1985)
→ combinator Birds
B from Bluebird, C from Cardinal, I from Identity Bird, K from
Kestrel, S from Starling, and so on.
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Theory of combinators
● Combinatory base: S and K
All combinators can be defined from the two combinators S and K.
● Combinators, called elemantary : I, K, B, W, C, S, Φ, Ψ
● All combinators work with the elimination (e.) and the introduction (i.)
rules.
● All combiantors are defined by β-reduction: Xfgx... →β fgx....
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Introduction and elimination rules (in Gentzen style)
● Rules analogues to the rules of the natural deduction of Gentzen
● Rules introduce or eliminate the logical constants
: => of implication, & of conjonction, v of disjonction
Rule of elimination Rule of introduction
p p=>q q
-------- [e.=>] -------------[i.=>]
q p=>q
p&q p&q p, q
----[e-&] -----[e-&] ----[i-&]
p q p&q
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Comment:
a. With each of the simpler combinators we shall associate a rule of reduction, designated by
the symbol for the combinator between parentheses.
b. This rule states that when the combinator in question is applied sucessively to a (finite )
series of variables, the resulting combination reduces to a certain combination of those
variables.
c. This reduction will follow from the definition of the combinator by (β-) rule.
d. And we shall use lower case italic letters for unspecified variables. The letters x, y, z will
often be used, when in the usual application, the variable is thought of as a function or an
argument.
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● The combinator B takes two functiors f and g and composes the
function g with the argument x. → composition
The introduction and elimination
rule of the combinator B
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● The combinator W takes one functor f and applies the functor f to the
argument x by duplicating the argument x. → duplication
The introduction and elimination
rule of the combiantor W
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● The combinator C takes one functor f and two arguments x and y. The
elimination of the combinator C by β-reduction allows to converse the
position of the arguement x with y. → conversion
The introduction and elimination
rule of the combinator C
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● The combinator Φ takes three functors f, g, and h, and the functor g
and h become intricate with the argument x. → intrication
The introduction and elimination
rule of the combinator Φ
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● The combinator C* takes an argument x (operand) and transforms it in
functor(operator). → transformation of operand to operator
The introduction and elimination
rule of the combinator C∗
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● Ψ fgxy →β f(gx)(gy)
the combinator Ψ takes two functions f and g and composes with the
arguments x and y by distributing the second function g to each of
them. → distribution
(Ψ f g) x y f (g x) (g y)
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β-reduction
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● Examples of β-reduction : X ≥β Y
By the scheme of β-reduction [B]
B(fx)gx ≥β fx(gx)
X = xy(Bxyz)UT ≥β xy(x(yz))UT =Y
By the scheme of β-reduction [W]
X= B(Wxy)U ≥β B(xyy)U =Y
We use here the infix ≤ to denote the relation converse to ≥.
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Interdefinability of simple combinators
● Certain of the combinators can be defined by the others.
=> by the way that the reduction rule for the derived combinator will
follow from its definition and the reduction rule for the basic
combinators.
Important point
We shall show that (1) W, S, Φ and ψ can be so defined in terms of
B, C and either W or S (see the classwork n°3); (2) that I can be
defined in terms of W and K; and (3) that all the combinators on
the list can be expressed in terms of S and K (refer to the
classwork n°2: completeness of the S-K base).
Important point
We shall show that (1) W, S, Φ and ψ can be so defined in terms of
B, C and either W or S (see the classwork n°3); (2) that I can be
defined in terms of W and K; and (3) that all the combinators on
the list can be expressed in terms of S and K (refer to the
classwork n°2: completeness of the S-K base).
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Properties of B-Theorems
Theorem 1. If x is a regular combinaotr and Y a combinator; then X·Y
applied to a sequence of arguments performs upon them first the
transoformation X, then the transformation Y.
Example:
C performs a permutation, W performs a duplication, C·W performs a
permutation, then a duplication on the result:
(C·W)fxy ≥ C(Wf)xy ≥ Wfyx ≥ fyyx
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Theorem 2. The product is associative, i.e.,
X·(Y ·Z)=(X· Y)·Z
Proof
a. ((X· Y)·Z)f ≥ B(BXY)Zf
≥ BXY(Zf)
≥ X(Y(Zf))
b. (X (Y Z))f ≥ BX(BYZ)f
≥ X(BYZf)
≥ X(Y(Zf))
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Theorem 3. The product is distributive with respect to pre-application
of B, i.e.,
B(X ·Y)= BX · BY
Proof
B(X ·Y)fx ≥(X ·Y)(fx)
≥X(Y(fx)
≤ X(BYfx)
≤ BX(BYf)x
≤ (BX · BY)fx
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Power of combinators
● Definition: We define the powers of a combinatory composite by natural
induction thus:
X0
=def
I
X1
=def
X
Xn+1
=def
X·Xn
● From these definitions, it is possible to deduce the following X, X2
, X3
,
that is, X, X·X, X·X·X, …
● Application to the combinators
C2
fxy ≥ C(Cf)xy ≥ Cfyx ≥ fxy
W2
fx ≥ W(Wfx) ≥ Wfxx ≥ fxxx
K2
fxy ≥ K(Kf)xy ≥ Kfy ≥ f
given the combinatorial expression X:
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● B2
fxyz ≡ B(Bf)xyz ≥ Bf(xy)z ≥ f((xy)z) ≡ f(xyz)
Theorem: For any expressions U, X, Y, Z1
,....,Zn
,
Bn
UXZ1
...Zn
≥ U(XZ1
...Zn
),
Φn
UXYZ1
...Zn
≥ U(XZ1
...Zn
)(YZ1
...Zn
)
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Summing up-combinators
combinators β-reduction
B for functional composition Bxyz ≥β x(yz)
I for identity Ix ≥β x
C for conversion Cxyz ≥β x(zy)
W for duplication Wxy ≥β xyy
S for distribution Sxyz ≥β xz(yz)
K for effacement Kxy ≥β x
C* for transposition C*xy ≥β yx
Φ for intrication Φxyzu ≥β x(yu)(zu)
Ψ for distribution ψ xyzu ≥β
x(yz)(yu)
Summing up-combinators
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Summing up-theorems
S K K x → x
S (K S) K x y z → x(yz)
S (K (S I)) K x y → (y x)
S (K (S I)) (S (K K ) I) x y (y x)→
**Completeness of the S-K base
[S K K =I]
[S (K S) K=B]
[S (K (S I)) K=C]
[S (K (S I)) (S (K K ) I)=C]
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Next week...
● About the Combinators vs λ-conversion and the application of
the combinators to natural language analysis