Notes on computational linguistics.pdf - UCLA Department of ...

Stabler - Lx 185/209 2003
That is, to compute ηθ, first apply θ to the terms of η and then add θ itself, and finally remove any of the
θ variables that are also η variables and remove any substitutions of variables for themselves. Clearly, with
this definition, every composition of substitutions will itself satisfy the conditions for being a substitution.
Furthermore, since the composition ηθ just applies θ to η, A(ηθ) = (Aη)θ for any expression A.
Example 1. Let
Then
And, on the other hand,
η ={X1 ↦ Y1,X2 ↦ Y2}
θ ={Y1 ↦ a1,Y2 ↦ a2}.
ηθ ={X1 ↦ a1,X2 ↦ a2,Y1 ↦ a1,Y2 ↦ a2}.
θη ={Y1 ↦ a1,Y2 ↦ a2,X1 ↦ Y1,X2 ↦ Y2}
Since ηθ = θη, we see that composition is thus not commutative, although it is associative.
Example 2. Let
η ={X1 ↦ Y1}
θ ={Y1 ↦ X1}.
Then although neither η nor θ is empty, ηθ = θ and θη = η.
Example 3. Let
η ={}
θ ={Y1 ↦ X1}.
Then ηθ = θη = θ.
This empty substitution η ={}is called an “identity element” for the composition operation. 3
Now we are ready to present a procedure for unifying two expressions E and F to produce a (most general)
unifier mgu(E, F).
Unification algorithm:
1. Put k = 0andσ0 ={}.
2. If Eσk = Fσk, stop. σk is a mgu of S. Otherwise, compare Eσk and Fσk from left to right to find the
first symbol at which they differ. Select the subexpression E ′ of E that begins with that symbol, and the
subexpression F ′ of F that begins with that symbol.
3. If one of E ′ ,F ′ is a variable V and one is a term t, andifVdoes not occur as a (strict) subconstituent of
t, putσk+1 = σk{V ↦ t}, increment k to k + 1, and return to step 2. Otherwise stop, S is not unifiable.
The algorithm produces a most general unifier which is unique up to a renaming of variables, otherwise it
terminates and returns the judgment that the expressions are not unifiable. 4
Now we are ready to define predicate prolog. All clauses and goals are universally closed, so the language,
inference method, and semantics are fairly simple.
3 In algebra, when we have a binary associative operation on a set with an identity element, we say we have a monoid. Sothesetof
substitutions, the operation of composition, and the empty substitution form a monoid. Another monoid we will discuss below is given
by the set of sequences of words, the operation of appending these sequences, and the empty sequence.
4 Our presentation of the unification algorithm is based on Lloyd (1987), and this result about the algorithm is established as Lloyd’s
Theorem 4.3. The result also appears in the classic source, Robinson (1965).
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