Spring 2012
Juyeon Kang
qkang@fi.muni.cz
B410, Faculty of Informatics, Masaryk University,
Brno, Czech Rep.
IA165
Combinatory Logic for
Computational Semantics
Interesting Readings
Curry, Haskell B. Combinatory Logic. Vol. 1 by Curry and R. Feys;
vol. 2 by Curry, J.R. Hindley, and J.P. Seldin. North-Holland,
1958, 1972.
Fitch, Frederic. Elements of Combinatory Logic. Yale University
Press, 1974.
Hindley, R., B. Lercher, J. Seldin. Introduction to Combinatory
Logic. Cambridge University Press, 1972.
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Lecture 2
● Introduction to the Combinatory Logic
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Historical background on CL
1) first invented by M.I Schönfinkel in 1920s
what for? Elimination of bound variables
example: see the table...
2) abstract operators: combinators
Z, T, I, C, S => B, C, I, K, S
K and S define the other three combinators.
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● Main idea
from K ans S, with a logical operator, one can generate all
formulas of predicate logic without the use of bound variables.
● Extra remark
multi-variable applications such as F(x,y) can be replaced by
(f(x))(y) where f is a function whose output-value f(x) is
also a function => Currying
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CL by Haskell Curry
1) a formal system of combinators and a proof of the combiantory
completeness of {B, C, K, W}
completeness proof => abstraction algorithm (coming next slides)
● Important remark 1
For Curry, as Schönfinkel,
every combinator was allowed to be applied to every other
combinator and even to itself.
Combinator
X
Combinator
X
Combinator
y
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● Important remark 2
All expressions of CL are applicative expressions where an operator is
applied to an operand. CL is generated from abstract operators, called
combinators, whose aim is to combine more elementary operators.
CL
Combinators
Combination operation
free variables
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● Combinatorial expression
Definition : The combinatorial expression will be represented by
X, Y, Z, U, V, T...; the variables by x, y, z, t...
(i) the atomes is the combinatorial expressions
(ii) If X and Y are the combinatory expression, then (XY) is a
combinatory expression.
Comment
we omit the most external parenthesis, where XY=det(XY).
Associativity
XYZ=det((XY)Z)≠X(YZ)
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Applications of CL
● In constructing the founcdations of mathematics
● In constructiuon methods and tools for implementing the programming
languages => Haskell
● Working on the Combinatory Logic:
Fitch (1974)
Klop (1992, 1993)
Shaumyan (1987); Universal Applicative Grammar :application to the NLP
Desclés (1999): study of the grammatical and lexical meanings
Steedman (2000): syntax-semantic interface
Terese (2003)
Bimbó (2011)
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Theory of combinators
● Combinatory base: S and K
All combinators can be defined from the combinators S and K.
● Combinators, called elemantary : I, K, B, W, C, S, Φ, Ψ
● A combinator is a combinatorial expression which contains only the
occurrences of combinators.
● Example: is combinators?
SKK
(S(Kx))((SK)K)
S(KS)K
S(SSKS)(KK)
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Combinatory base: S and K
 K is defined by the rule : Kxy:=x
the combinator K takes two arguments and returns the
first argument as result. → effacement
 S is defined by the rule : Sxyz:=xz(yz)
the combinator S composes the functions x (binary) and
y (unary) with the argument z. → composition
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● Sxyz-> xz(yz)
S x y z x z (y z)
S x y z
x z (y z)
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 I is defined by the rule : Ix:=x
the combiantor I takes one argument x and returns this
argument as result. → identification
K x y x→ I x x→
K x y x I x x
x xK x y I x
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● Combinators is composable between them.
● The combinators organize an algebraic structure, for some of them, we
have an algebraic tree.
● The action of combinators is intrinsic, that is, independent of the
domains of the compound operators.
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Normal form
● A normal form is a combinatoryal expression which can not be
reduced, that is, it contains any occurences of combinators
Definition
If a combinatorial expression is reduced to a combinatorial
expression which is in the normal form, then N is called the Normal
form of X.
Redex
Ux1....x2
Redex
Ux1....x2
Contratum
z1....zp
Contratum
z1....zp
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● Completeness of the S-K basis
S and K can be composed to produce combinators that are
extensionnally equal to any lambda term, and therefore, to any
computable function by Church's thesis. The proof is to present a
transformation, T[], which converts an arbitrary lambda term into an
equivalent combinator. operation of abstraction→
See the 2nd
question of the classwork N°2.
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Abstraction and substitution
● Two operations which construct combinatorial expressions from the
combinatorial expression already defined.
(1) operation of abstraction
The expression [λx].e is a combinatorial expression which is a result of a
calculus defined by the following conditions:
a. [λx].e=Ke (condition: e does not appear in x)
b. [λx].e=I
c. [λx].ex=e
d. [λx].e1e2= S([λx].e1)([λx].e2)
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(1.1) Abstraction algorithm
● An algorithm of abstraction aims to carry out the actual
calculus, by abstraction of the variable x, of the combinatorial
expression [λx].e.
● Abstraction algorithms are generally presented in the form of
algorithms of Markov (string rewriting system). The reasonning
of the algorithm is gouverned by the following 4 metarules:
i) we apply obligatorily one rule if possible, if not we pass to
the next step;
ii) we start always by trying the first step;
iii) since one rule was applied, we return to the first step;
iv) the result is obtained when any rule can be applied.
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→ The algorithm of Markov given by the set totally
ordered by the rules (a), (b), (c) and (d) is an
algorithm of abstraction. These rules function on the
combinatorial expressions.
● Example
[λx].xy = S([λx].x)([λx].y) rule (d)
= SI ([λx].y) rule (b)
= SI (Ky) rule (a)
= S (Ky) elimin. Of I
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(2) operation of substitution
λx.(e1 e2)
a function which takes an argument, say a, and substitutes it into the
lambda term (e1 e2) in place of x, yielding (e1 e2)[x := a].
(e1 e2)[x:=a]= (e1[x:=a] e2[x:=a])
(λx.(e1 e2) a) =((λx.e1 a)(λx.e2 a))
= (S λx.e1 λx.e2 a)
= ((S λx.e1 λx.e2) a)
By extensional equality,
λx.(e1 e2)= (S λx.e1 λx.e2)
● Example : [λxy].x = [λx.x] [λy.x]
= I (Kx) rule (a and b)
= Kx elimin. Of I
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Summing up
● The CL is a logic of the operating process by means of intrinsic
compositions of operators.
● The composition is intrinsic when it is independent from domains of the
operators (thus of their extensionnelles meanings).
● The CL allows to build operators and complex predicates from operators
and from more elementary predicates.
● The combinators of the CL is operators of " intrinsic composition ".
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Next week...
● More about the combinators: elementary and complex.