Complexity of syntactic constructions - Revision history

RVulanovic at 17:26, 11 December 2006

2006-12-11T17:26:17Z

Rkoehler at 18:05, 9 December 2006

2006-12-09T18:05:25Z

Rkoehler at 18:26, 4 December 2006

2006-12-04T18:26:14Z

Altmann at 16:34, 4 December 2006

2006-12-04T16:34:42Z

RVulanovic at 21:50, 14 November 2006

2006-11-14T21:50:58Z

RVulanovic at 18:54, 14 November 2006

2006-11-14T18:54:45Z

Ahans at 13:14, 27 June 2006

2006-06-27T13:14:28Z

Ahans at 08:42, 17 June 2006

2006-06-17T08:42:31Z

Ahans at 15:07, 20 May 2006

2006-05-20T15:07:40Z

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'''1. Problem and history'''

The complexity of a syntactic construct is measured in terms of the number of its immediate constituents.The partitioning in immediate constituents can be performed on the basis of any grammar.
The first model seems to be that of Köhler and Altmann (2000).

'''2. Hypothesis'''

''The complexity of syntactic constructions follows the hyper-Pascal distribution''.

'''3. Derivation'''

The complexity depends on following quantities (Köhler, Altmann 2000:192):

minX – the requirement of minimization of the complexity of a syntactic construction in order to decrease memory effort in processing the construction;

maxH – the requirement of maximazing compactness. This enables us diminishing the complexity of the subordinated level of embedding by embedding constituents into the given level… minX on the level m corresponds to the requirement maxH on the level m+1;

E – a variable representing the average degree of fullness, the default value of complexity;

I(K) – the size of inventory of constructions.

Assumptions: The number of constructions with complexity x is proportional to that with complexity x-1. maxH increases the probability of a higher complexity, minX decreases it. E nad I(K) are inverse to each other: the greater the inventory, the less complexity is needed.
Putting these assumptions together, we obtain

(1)<math> P_{x+1}= \frac{maxH + x}{min X + x} \frac{E}{I(K)}P_x, \quad x= 1, 2, ...</math>

Setting maxH = k-1, minX = m-1 and E/I(K) = q yields

(2)<math> P_{x+1} = \frac{k+x-1}{m+x-1}qP_x, \quad x=1, 2, ...</math>

resulting in

(3)<math> P_X = \frac{{k+x-2 \choose x-1}}{{m+x-2 \choose x-1}}q^{x-1}P_1
, \quad x=1,2,3...</math>

where<math> P_1^{-1}</math>.

'''Example''': Complexity of syntactic constructions in the Negra corpus (Brants 1999)
Köhler and Altmann (2000) fitted (3) to the complexity of syntactic constructions in the Negra corpus. The result is presented in Table 1 and Fig. 1.

<div align="center">[[Image:Tabelle1_CoSC.jpg]]</div>

Since the number of observations is too great, the use of the chi-square is problematic. The authors use the contingency coefficient <math>C = X^2/N</math> which is acceptable. It would be advisable to use single texts instead of corpora.

<div align="center">[[Image:Grafik1_CoSC.jpg]]</div>
<div align="center">Fig. 1. Distribution of syntactic complexity in the Negra corpus</div>

'''4. Authors: G. Altmann'''

'''5. References'''

'''Brants, T'''. (1999). ''Tagging and parsing with cascaded Markov models. Automation of corpus annotation''. Saarbrücken: Universität der Saarlandes.

'''Köhler, R., Altmann, G.''' (2000). Probability distributions of syntactic units and properties. ''J. of Qwuantitative Linguistics 7, 189-200''.

@@ Line 20: / Line 20: @@
 I(K) –	the size of inventory of constructions.
-Assumptions: The number of constructions with complexity x+1 is proportional to that with complexity x. maxH increases the probability of a higher complexity, minX decreases it. The greater E the more complexity is needed to code the individual messages. On the other hand, the greater the inventory I(K) the less complexity is needed. Within a given period of time, the relation E/I(K) can be considered as a constant, say q.
+Assumptions: The number of constructions with complexity x+1 is proportional to that with complexity x. maxH increases the probability of a higher complexity, minX decreases it. The greater E, the more complexity is needed to code the individual messages. On the other hand, the greater the inventory size I(K), the less complexity is needed. With these assumptions, we obtain
 (1)<math> P_{x+1}= \frac{maxH + x}{minX + x} \frac{E}{I(K)}P_x, \quad x= 1, 2, ...</math>
-Setting maxH = k-1, minX = m-1 and E/I(K) = q yields
+Within a given period of time, the relation E/I(K) can be considered as a constant, say q. Setting E/I(K) = q, maxH = k-1, and minX = m-1, from (1) we get
 (2)<math> P_{x+1} = \frac{k+x-1}{m+x-1}qP_x, \quad x=1, 2, ...</math>

@@ Line 20: / Line 20: @@
 I(K) –	the size of inventory of constructions.
-Assumptions: The number of constructions with complexity x+1 is proportional to that with complexity x. maxH increases the probability of a higher complexity, minX decreases it. E and I(K) are inversely proportional to each other: the greater the inventory, the less complexity is needed. The relation E/I(K) can be considered as a constant, say q.
+Assumptions: The number of constructions with complexity x+1 is proportional to that with complexity x. maxH increases the probability of a higher complexity, minX decreases it. The greater E the more complexity is needed to code the individual messages. On the other hand, the greater the inventory I(K) the less complexity is needed. Within a given period of time, the relation E/I(K) can be considered as a constant, say q.
 With these assumptions, we obtain

@@ Line 20: / Line 20: @@
 I(K) –	the size of inventory of constructions.
-Assumptions: The number of constructions with complexity x+1 is proportional to that with complexity x. maxH increases the probability of a higher complexity, minX decreases it. E and I(K) are inversely proportional to each other: the greater the inventory, the less complexity is needed. The relation E/I(K) yield a constant, say q.
+Assumptions: The number of constructions with complexity x+1 is proportional to that with complexity x. maxH increases the probability of a higher complexity, minX decreases it. E and I(K) are inversely proportional to each other: the greater the inventory, the less complexity is needed. The relation E/I(K) can be considered as a constant, say q.
-Putting these assumptions together, we obtain
+With these assumptions, we obtain
 (1)<math> P_{x+1}= \frac{maxH + x}{minX + x} \frac{E}{I(K)}P_x, \quad x= 1, 2, ...</math>

@@ Line 20: / Line 20: @@
 I(K) –	the size of inventory of constructions.
-Assumptions: The number of constructions with complexity x+1 is proportional to that with complexity x. maxH increases the probability of a higher complexity, minX decreases it. E and I(K) are inversely proportional to each other: the greater the inventory, the less complexity is needed.
+Assumptions: The number of constructions with complexity x+1 is proportional to that with complexity x. maxH increases the probability of a higher complexity, minX decreases it. E and I(K) are inversely proportional to each other: the greater the inventory, the less complexity is needed. The relation E/I(K) yield a constant, say q.
 Putting these assumptions together, we obtain

@@ Line 1: / Line 1: @@
 '''1. Problem and history'''
-The complexity of a syntactic construct is measured in terms of the number of its immediate constituents.The partitioning in immediate constituents can be performed on the basis of any grammar.
+The complexity of a syntactic construct is measured in terms of the number of its immediate constituents. The partitioning in immediate constituents can be performed on the basis of any grammar.
 The first model seems to be that of Köhler and Altmann (2000).
@@ Line 20: / Line 20: @@
 I(K) –	the size of inventory of constructions.
-Assumptions: The number of constructions with complexity x is proportional to that with complexity x-1. maxH increases the probability of a higher complexity, minX decreases it. E nad I(K) are inversely proportional to each other: the greater the inventory, the less complexity is needed.
+Assumptions: The number of constructions with complexity x+1 is proportional to that with complexity x. maxH increases the probability of a higher complexity, minX decreases it. E and I(K) are inversely proportional to each other: the greater the inventory, the less complexity is needed.
 Putting these assumptions together, we obtain
@@ Line 34: / Line 34: @@
 , \quad x=1,2,3...</math>
-where<math> P_1^{-1}= _2 F_1 (k,1;m;q)</math>.
+where<math> P_1^{-1}= _2F_1 (k,1;m;q)</math>.
 '''Example''': Complexity of syntactic constructions in the Negra corpus (Brants 1999)
@@ Line 53: / Line 53: @@
 '''Brants, T'''. (1999). ''Tagging and parsing with cascaded Markov models. Automation of corpus annotation''. Saarbrücken: Universität der Saarlandes.
-'''Köhler, R., Altmann, G.''' (2000). Probability distributions of syntactic units and properties. ''J. of Qwuantitative Linguistics 7, 189-200''.
+'''Köhler, R., Altmann, G.''' (2000). Probability distributions of syntactic units and properties. ''J. of Quantitative Linguistics 7, 189-200''.

@@ Line 39: / Line 39: @@
 Köhler and Altmann (2000) fitted (3) to the complexity of syntactic constructions in the Negra corpus. The result is presented in Table 1 and Fig. 1.
-<div align="center">[[Image:Tabelle1_CoSC.jpg]]</div>
+<div align="center">[[Image:Tabelle11_CoSC.jpg]]</div>
 Since the number of observations is too great, the use of the chi-square is problematic. The authors use the contingency coefficient <math>C = X^2/N</math> which is acceptable. It would be advisable to use single texts instead of corpora.