Colloquium Nijmegen, May 24, Arjen P. Versloot Crazy Rules: Self-Organization and Chaos Theory Crazy Rules Self-Organization and Chaotic behaviour in a

Colloquium Nijmegen, May 24, Arjen P. Versloot

Crazy Rules: Self-Organization and Chaos Theory

Crazy Rules

Self-Organization and Chaotic behaviour in a Deterministic System



Is language just a dialect with an army?

Icelandic:takaheiðiaugu



Speaker population and sounds

p (2 tailed) = 0.014



Archaic dialects in the Swiss Alps

snîdan ‘to cut’heftan ‘to attatch’machôn ‘to make’losên ‘to losen’

..\..\..\..\All Users\Bureaublad\ILWIS 3.3 Academic.lnk

CH.xlsx



Components of Human Language• Data Storage and retrieval in the brain

(Neural Darwinism ⇒ Frequencies)

• Acoustics

• Semantics(from idiosyncratic to systematic:“two dogs”)



The Bidirectional Table

• Speakers are probabilistic learners, guided by:– Memory and Common Practice– Perceptual Reliability / Avoiding Semantic

Biases

• Large scale individual behaviour can be modelled with stochastic approximations “Ideal Speaker”



The working of the bidirectional table:

Middle Frisian hab, habbe & habbaMiddle Frisian hab, habbe & habba ‘to have’, 1430-1460

ø <e> <a> production

sg 50 30 2 61% 37% 2%

inf 0 1 39 0% 3% 97%

100% 97% 5%61%

(63%)36%

(37%) 0%

0% 3% 95% 0% 0%93%

(100%)

perception



• If language is deterministic and developing towards equilibria: where do the changes come from?

• Example: Old English:- 3 vowel system in unstressed syllables- Stable for 200 years, followed by change- Learning not based on unlimited dataset

Stability and Instability



Balancing PowersGeneral phonetic centralization tendency

/u/ 28%

/a/ 41%

[ə]31% /e/

10%

8%

9%

Distributional proportions based on Old English text

Noise levels based on modern vowel duration contrast

Equilibrium /a/ /e/ /u/

Full 85% 87%

86%

Schwa 15% 13%

14%



..\Language Fractal\modelcalc.xls

..\Proefskrift\Afr-a\matrixcalcul.xls



/e/ and the equilibrium

/e/ > /ə /

/a/ > /ə / equil.equil.

The equilibrium is only preserved in a narrow strip in the proportion – noise area!



/a/ and the equilibrium

/a/ > /ə /

/u/>/ə /

equil.equil.

The threshold of /u/ > /ə/ showsCHAOTICbehaviour!



Chance and resolution (I)• Large scale individual behaviour can be

modelled with stochastic behaviour• Only true for sample size (n) ⇒ ∞

p = observed proportion for phenomenon xP = statistical chance for x to appearnx = number of observations with x

nt = total number of observations

nx/nt = p ≈ P



Chance and resolution (II)

• Throw a die 10 times• Chance for ‘6’ = 1/6

• 1/6 * 10 = 1,67• ‘best match’ = 2 times ‘6’

• (2-1,67)/1,67 = 20% deviation



Chance and resolution (III)

drunkards walk.xlsx



Consequences for system stability

/a/ > /ə /

/u/>/ə /

equil.equil.

/a/ > /ə /

/u/>/ə /

equil.equil.



Consequences for system stability

• Smaller sample sets broaden the equilibrium zone: robust for chance variation.

• Smaller sample sets increase unpredictable threshold behaviour: chaotic character.

/a/ > /ə /

/u/>/ə /

equil.equil.



Conclusion• The Bidirectional Table models the

interaction of memory (frequency), acoustics (noise levels) and semantics (functional contrasts)

• It predicts and explains e.g.:• stability of sound or lexical contrasts• structural differences depending on

population size• both equilibrium situations and

chaotic disturbances

Tank foar jo omtinken!

Arjen VerslootUvA/Fryske Akademy: [email protected]://www.fryske-akademy.nl/Members/arjen/arjen-p-versloot

Frisian Language Database: www.fryske-akademy.nl/tdb



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Colloquium Nijmegen, May 24, Arjen P. Versloot Crazy Rules: Self-Organization and Chaos Theory Crazy Rules Self-Organization and Chaotic behaviour in a