On Peters and Ritchie's Definition of an n-Term Transformational MappingAuthor(s): Philip MillerSource: Linguistic Inquiry, Vol. 15, No. 4 (Autumn, 1984), pp. 716-718Published by: The MIT PressStable URL: http://www.jstor.org/stable/4178414 .

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716 SQUIBS AND DISCUSSION

ON PETERS AND RITCHIE'S Peters and Ritchie (1973, 60) give the following definition of an DEFINITION OF AN n-TERM n-term transformational mapping: TRANSFORMATIONAL MAPPING

Philip Miller, Definition 2.10. The set {lo , atp} defines an n-term trans- Universite Libre de formational mapping if each otq, 1 - q S p, is either a pair [Td, Bruxelles (hq, iq)] or one of the triples [Ts, (hq, iq), (jq, kq)], [T,, (hq, iq),

(jq, kq)], or [Tr, (hq, iq), (jq, kq)], and if 1 S hi S il < h2 S i2 < . . . < hp , ip - n. In this case, the value of the mapping of n- term factorizations of well-formed labeled bracketings to well- formed labeled bracketings, on the factorization (4'1, . . . ,) , is defined, if and only if there are a nonterminal A and a well-formed labeled bracketing ep such that iil. . . kPn = [A4C]A, to be:

(i) p(w I ... wA2p+l) if each wm is defined, 1 s m 6 2p+1, and if there is a well-formed labeled bracketing X such that p(w1 ... ()2p+I) = [AXIA,

and

(ii) [AW]A otherwise, where

,,,+l = iq+I . . . khq+1-1 if hq+i - 1 i 1q + 1 ~'~2q+1 (e otherwise, where we set io

l and hp+I = nJ

forq =0,. p

Td(thq ... Piq) if OXq = [Td, (hq, iq)] ]

W2q = Tp(lhq * * * Piq, 4jq ... 4'kq) if 1 is s, (l or r and Qlq = [Tp, (hq, iq), (jq, kq)]J

forq= 1,...,p.

Where p (defined by Peters and Ritchie in def. 2.3., p. 56, and on pp. 59-60) is a function from labeled bracketings to well-formed labeled bracketings that eliminates labeled bracketings in the following configurations: X1[A ]AX2 and X1[AU[AW]ATIAX2 where A E VN, cr L* and T E R* (L = {[A: A E VN}, R= {]A: A ( VN}) and where X and cr[AWIAT are well-formed labeled bracketings, and Xi, X2 are labeled bracketings. In the first case p eliminates the string [A ]A; in the second it eliminates the inner pair of A-labeled brackets. This is a formal definition that corresponds to a part of Ross's (1967) rule of tree pruning, the idea being to eliminate redundant brackets.

Not only is the whole definition difficult to read but, worse, it does not give correct results. It can easily be seen that io and hp+ I are not correctly defined. As they are defined, they give incorrect results if hi # 1 or ip # n-that is, if the first or last term of the factorization is not affected by a transformation.

Here is a short example:

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SQUIBS AND DISCUSSION 717

(1) 4114123414 = [Aa[BbIB[cC[Dd]DIcIA

n =4 p= 1

{lo,} = {[Ts, (2, 2), (3, 3)]} h= 2

ii = 2 11 = 3 k= 3

Let us calculate the w(? using the definition (2):

(2) forq = 0 w2q+I = I 1 = esincehq -1 =+ h, - 1 = 2 - 1 = I is not greater than or equal to iq + 1 io + 1 = 1 + 1 =2

for q = 1 W2q = =2 = Ts(412, 43) (W2q+1 = (i3 = 3 since hq+i - 1 = n - I

= 4 - 1 = 3 is equal to iq + 1 =2 + 1 =3

The result of the mapping, following the definition, is conse- quently (3):

(3) p(W1W2W3) = p(Ts(t42, 3)4'3) = Ts(42, 1)413

But the result should actually be (4):

(4) +j Ts(tP2, t3)+3 t+4

41p and +4 should stay, being unaffected by any transformation. Notice that it is impossible to define an (t)4 equal to 414 since this would amount to assigning the value 2 to index q.

The same problem occurs with the examples given by Pe- ters and Ritchie on pp. 62-64. The results they give are not those that follow from a strict application of their definition. For instance, in the first example, the first and last factors [s[PreQ]Pre and [Timebefore John left] Time]PP]S should disappear.

The correct result is obtained by assigning io = 0 and hp+ - n + 1, as can easily be verified.

But it seems to me that the whole definition can be con- siderably clarified by introducing an identity transformation that does not change anything (this was suggested to me by Prof. Dominicy).

Let us define an elementary identity transformation. The elementary identity transformation is the function Ti from sub- strings of well-formed labeled bracketings to substrings of well- formed labeled bracketings, defined by TX41) = 41.

We can now give a simpler and more compact definition of an n-term transformational mapping:

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718 SQUIBS AND DISCUSSION

(5) Definition 2.10.' The set {a1, ... , a, } defines an n- term transformational mapping if each aq, 1 - q - p, is either one of the pairs [Td, (hq, iq)] or [Ti, (hq, iq)I or one of the triples [Ta, (hq, iq), (jq, kq)], [T,, (hq, iq), (jq, kq)], or [Tr, (hq, iq), (jq, kq)], and if hI = 1, hq = iq-I +1 iq, for q = 2, . . ., p and ip = n. In this case the value of the mapping of n-term fac- torizations of well-formed labeled bracketings to well- formed labeled bracketings, on the factorization (4", * . . , 4,), is defined (if and only if there are a non- terminal A and a well-formed labeled bracketing p such that 4,' ... 'n = [dA(A) to be P(wi ... wp where

TO(Ihq... t4iq) if I is i or d (oq _) {and coq = [T3, (hq, iq)]

Ty(*hq . . . 'iq, jq . . . kq) if y is s, 1, or r and oq = [Ty, (hq, iq), (jq, kq)]

if each wq is defined, and 4j . .,.n otherwise.

The conditions imposed on hq and iq ensure that every term will be affected by a transformation (possibly the identity trans- formation). This solves the preceding problem and makes it un- necessary to define two types of wi, simplifying index calcu- lation.

References

Peters, P. S. and R. W. Ritchie (1973) "On the Generative Power of Transformational Grammars," Information Sciences 6, 49-83.

Ross, J. R. (1967) Constraints on Variables in Syntax, Doctoral dissertation, MIT, Cambridge, Massachusetts.

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