Redundancy Ratio: An Invariant Property of the Consonant Inventories of the World’s Languages

1. Redundancy Ratio: An Invariant Property of the Consonant Inventories of the World’s Languages Animesh Mukherjee, Monojit Choudhury, Anupam Basu and Niloy Ganguly Department of Computer Science & Engg. Indian Institute of Technology, Kharagpur

2. Redundancy in Natural Systems • Reduce the risk of information loss – fault tolerance • Examples of redundancy: • Biological systems – Codons, genes, proteins etc. • Linguistic systems – Synonymous words • Human Brain – Perhaps the biggest example of neuronal redundancy

3. Redundancy in Sound Systems • Like any other natural system, human speech sound systems are expected to show redundancy in the information they encode • In this work we attempt to • Mathematically formulate this redundancy, and, • Unravel the interesting patterns (if any) that results from this formulation

4. plosive voiceless voiced bilabial /b/ /p/ /t/ /d/ dental Feature Economy: An age-old Principle • Sounds, especially consonants, tend to occur in pairs that are highly correlated in terms of their features • Languages tend to maximize combinatorial possibilities of a few features to produce many consonants If a language has in its inventory then it will also tend to have

5. Mathematical Formulation • We use the concepts of information theory to quantify feature economy (assuming features are Boolean) • The basic idea is to compute the number of bits req-uired to pass the information of an inventory of size N over a transmission channel Ideal Scenario Inventory of Size N Info. Undistorted Noiseless Channel log2N bits are required for lossless transmission

6. Mathematical Formulation • We use the concepts of information theory to quantify feature economy (assuming features are Boolean) • The basic idea is to compute the number of bits req-uired to pass the information of an inventory of size N over a transmission channel General Scenario Inventory of Size N Info. Distorted Noisy Channel >log2N bits are required for lossless transmission

7. pf pf qf N N N Feature Entropy • The actual number of bits required can be estimated by calculating the binary entropy as follows • pf – number of consonants in the inventory in which feature f is present • qf –number of consonants in the inventory in which feature f is absent • The probability that a consonant chosen at random form the inventory has f is and that is does not have f is (=1- )

8. FE log2N pf pf qf qf N N N N Feature Entropy • If F denote the set of all features, FE= –∑fєF log2 + log2 • Redundancy Ratio (RR) RR = • The excess number of bits required to represent the inventory

9. Example

10. Experimentation • Data Source • UCLA Phonological Inventory Database • Samples data uniformly from almost all linguistic families • Hosts phonological systems of 317 languages • Number of Consonants: 541 • Number of Vowels: 151

11. RR: Consonant Inventories • The slope of the line fit is -0.0178 RR is almost invariant with respect to the inventory size • The result means that consonant inventories are organized to have similar redundancy irrespective of their size  important because no such explanation yet Redundancy Ratio Inventory Size

12. The Invariance is not “by chance” • The invariance in the distribution of RRs for consonant inventories did not emerge by chance • Can be validated by a standard test of hypothesis • Null Hypothesis:The invariance in the distribution of RRs observed across the real consonant inventories is also prevalent across the randomly generated inventories.

13. Generation of Random Inventories • Model I – Purely random model • The distribution of the consonant inventory size is assumed to be known a priori • Conceive of 317 bins corresponding to the languages in UPSID • Pick a bin and fill it by randomly choosing consonants (without repetition) from the pool of 541 available consonants • Repeat the above step until all the bins are packed Pool of phonemes /t/ /d/ /n/ /b/ /k/ /p/ /m/ ……………… Fill randomly Bin 1 Bin 2 Bin 317 /m/ /k/ /t/ /b/ /g/ /n/ …………………………………………….. /p/ /n/ /p/ /p/ /d/ /d/ 4 6 2

14. Generation of Random Inventories • Model II – Random model based on Occurrence Frequency • For each consonant c let the frequency of occurrence in UPSID be denoted by fc. • Let there be 317 bins each corresponding to a language in UPSID. • fc bins are then chosen uniformly at random and the consonant c is packed into these bins without repetition. Pool of phonemes /t/ (25) /n/ (12) /p/ (100) ……………………. Bin 1 Bin 2 Bin 317 Choose 25 bins randomly and fill with /t/ /m/ /k/ /t/ /t/ /b/ /g/ /n/ …………………………………………….. /p/ /n/ /p/ /p/ /d/ /d/

15. Results • Model I – t-test indicates that the null hypothesis can be rejected with (100 - 9.29e-15)% confidence • Model II – Once again in this case t-test shows that the null hypothesis can be rejected with (100–2.55e–3)% confidence • Occurrence frequency governs the organization of the consonant inventories at least to some extent Average Redundancy Ratio Model I Model II Real Inventory Size

16. The Case of Vowel Inventories • The slope of the line fit is -0.125 • For small inventories RR is not invariant while for Larger ones (size > 12) it is so • Smaller inventories  perceptual contrast and Larger inventories  feature economy • t-test shows that we can be 99.93% confident that the two inventories are different in terms of RR Vowels Redundancy Ratio Consonants Inventory Size

17. Error Correcting Capability • For most of the consonant inventories the average hamming distance between two consonants is 4  1 bit error correcting capability • Vowel inventories do not indicate any such fixed error correcting capability Average Hamming Distance Consonants Vowels Inventory Size

18. Conclusions • Redundancy ratio is almost an invariant property of the consonant inventories with respect to the inventory size, • This invariance is a direct consequence of the fixed error correcting capabilities of the consonant inventories, • Unlike the consonant inventories, the vowel inventories are not indicative (at least not all of them) of such an invariance.

19. Discussions • Cause of the origins of redundancy in a linguistic system • Fault tolerance: Redundancy acts as a failsafe mechanism against random distortion • Evolutionary Cause: Redundancy allows a speaker to successfully communicate with speakers of neighboring dialects – “Linguistic junk” as pointed out by Lass (Lass, 1997)

20. Děkuji