conditions of law equations as communicable knowledge

Conditions of Law Equations as Conditions of Law Equations as Communicable KnowledgeCommunicable Knowledge

Takashi WashioHiroshi Motoda

I.S.I.R., Osaka Univ.

Informal Workshop on Communicable KnowledgeDec., 6th, 2000

What are the conditions of What are the conditions of communicablecommunicable law equations? law equations?

(1) Generic conditions of law equations(2) Domain dependent conditions for

communicable law equations

Generic conditions of law equationsWhat are law equations?

Are objectiveness and generality of equations sufficient to represent laws?

Heat transfer between fluid and the wall of a round pipe under enforced turbulence flow

　 Dittus-Boelter Equation Nu = 0.023 Re0.8 Pr0.4

(Nu,Re,Pr ： defined from heat conductivity, density and flow velocity of the fluid.)　 Law Equation of Gravity Force　 F=G M1M2/R2

What are the generic What are the generic conditions of law equations?conditions of law equations?

“Law equation” is a emprical terminology. Its axiomatization without any exception may be difficult.

Its axiomatic analysis is important for the basis of the science. R.Descartes: distinctness and clearness of reasoning, divide and 　

　　　　　　　 conquer method, soundness, consistency I.Newton: 　 removal of non-natural causes (objectiveness), 　　

　　　　　　　 minimum causal assumptions (simplicity, 　　　　　　　　　　　　 parsimony), validity in wide phenomena 　　　　　　　　　　　　　　 (generality), no exception (soundness)

H.A.Simon: parsimony of description R.P.Feynman: mathematical constraints (admissibility)

Generic conditions of law equationsA Scientific Region: T=<S,A,L,D>where S={s is a syntax rule.}, 　　　　　　　　 A={a is an axiom.}, 　　　　　　 L={l is a postulate}, 　　 D={o is an objective phenomenon.} ．S: definitions of coordinate system, physical quantity 　　 and some algebraic operatorsA: axioms on distance and etc.L: empirical laws and empirical strong believesD: a domain on which the scientific region 　　　　 concentrates its analysis.

Generic conditions of law equationsEx.) Law of Gravity Force is not always required 　　　　　　　 for the objective phenomena of classical physics. 　　 → A law l is used to understand or model 　　　　　　　　　　 phenomena in the subset of D.

Objective domain of an equation eAn objective phenomenon of an equation e is a phenomenon where all quantities in e are required to describe the phenomenon.

A domain of e, De ( D),⊆ is a subset of objective phenomena of e in D.

Generic conditions of law equationsSatisfaction and Consistency of an equation e• An equation e is “satisfactory” for its objective 　　　　　　 phenomenon when e explains the phenomenon.

• An equation e is “consistent” with its objective 　　　　　　 phenomenon when e does not show any contradictory 　　 relation with the phenomenon.

Ex.) Collision of two mass points

The law of gravity force is considered to be satisfactory under the sufficiently heavy mass of the two points, otherwise it is ignored. In any case, the law of gravity force is consistent with this collision phenomenon.

Generic conditions of law equations

Objectiveness （ All quantities in e is observable.)Generality (e is satisfactory in wide phenomena.)Reproducibility (an identical result on e is obtaine

d under an identical condition.)Soundness (e is consistent with the measurement

under a certain condition.)Parsimony (e consists of minimum number of qua

ntities.)Mathematical Admissibility (e follows S and A.)

In the objective domain of e, De


Heat transfer between fluid and the wall of a round pipe under enforced turbulence flow

　 Dittus-Boelter Equation Nu = 0.023 Re0.8 Pr0.4 is satisfactory only in the region of 104<Re<105,

　 1<Pr<10. It does not satisfactory over entire De.

　 → It does not satisfy the soundness.

　 Law of gravity force　　 F=G M1M2/R2

　 → It is satisfactory over De.


Parsimony (e consists of minimum number of 　　　　　　　　 quantities)

Mathematical Admissibility (e follows S and A)

Conditions being confirmed through experiments and/or observationsObjectiveness （ All quantities in e is observable)Generality (e is satisfactory in wide phenomenaReproducibility (identical result on e is obtained 　　　　　　　　　　　 under identical condition)Soundness (e is consistent with the measurement 　　　　　　　　 under a certain condition)

Conditions on law equation formulae MDL, AIC, S-value

unit dimension and scale-types

What are the conditions of What are the conditions of communicablecommunicable law equations? law equations?



Domain dependent conditions for Domain dependent conditions for communicable law equationscommunicable law equations

(1) Consistency of terms (quantities) with background knowledge

　　 A Scientific Region: T=<S,A,L,D> BK=A (axioms) and L (postulates): 　　　　　

　　　 quantities in other law equations, extensionally measurable quantities, intentional definitions of quantities having clear physical meaning

Ex.1) d = M/L3 ≡ V=L3, d=M/VEx.2) f=Gm1m2/r2 ? A=m1m2, f=GA/r2

physically unclear


(2) Consistency of relation with Background Knowledge 　　 A Scientific Region: T=<S,A,L,D> BK=A (axioms) and L (postulates): 　　　　　　　　　　 other law equations, empirical fact and 　　　 empirically strong evidence

Ex.1) f=m2a ≠ dv/dt=a, mdv=fdtEx.2) f=Gm1m2/r2 – k/Dα ← space term　　　　 Universe should be static. ≠ Red shift of light spectrum + Doppler effect


(3) Relation on relevant and/or interested phenomena 　 A Scientific Region: T=<S,A,L,D> 　　　　　　　　 where D={o is an objective phenomenon.} ．

　　 D should be relevant to the interest of scientists.

Ex.) f=ma is relevant to physicists’ interest. sp=f(cb,t,fb) is relevant to the interest 　　　　　　　　　　　 of stock fund managers.


(4) Relation on relevant and/or interested view　　 A Scientific Region: T=<S,A,L,D> BK=A (axioms), L (postulates), D (domain): 　　

　　 selection of quantities, 　　　　　　　　　　　　　　　　　 selection of equation class

Ex.1) Model equation of ideal gass PV=nRT : macroscopic veiw f = 2mv : microscopic viewEx.2) Model equation of air friction force　　　 f = - c v2 – k v : global view f = - k v : local view　　　　


(5) Appropriate simplicity and complexity for understanding Is the optimum simplicity in terms of the principle of parsimony real

ly appropriate for understanding? The most of the law equations in physics involves 3 – 7 quantities.

A complicated model is decomposed into multiple law equations in appropriate granule.

( R 3 h fe2

R 3 h fe2 + h ie2

R 2 h fe1

R 2 h fe1 + h ie1 r L 2

r L 2 + R 1

) ( V 1 - V 2 ) - Q C

- K h ie3 X B h fe3

= 0

V=IRIEC=hfeIBC

I0=I1+I2

(5) Appropriate simplicity and complexity for understanding (Continued)

Decision Tree (ID3,C4.5)Decision tree pruned in a comprehensive level

Depth5

A financial application:As far as the accuracy is sufficient for the object, the depth is set to 5.I-Ent. is used only to select features.

Domain dependent conditions for Domain dependent conditions for communicable law equationscommunicable law equationsIn case of the discovery of a new paradigm:(1) Terms (quantities) become inconsistent with background knowledge(2) Relations become inconsistent with Background Knowledge

A Scientific Region: T=<S,A,L,D> ⇒ T’=<S’,A’,L’,D’>Ex.) Classical Mechanics Quantum Mechanics ⇒

mdxdV

dtxd /2

2

Vxmt

i

2

22

2

Quantities and relations are different.

A model of A model of communicablecommunicable knowledge discoveryknowledge discovery



Is the communicable knowledge discovery really learning and/or mining?The most of the learning and data mining do not use generic and domain dependent conditions for communicable knowledge discovery!

A model of A model of communicablecommunicable knowledge discoveryknowledge discovery

Proposing framework:Data set features class explaining quantities objective quantity

HypothesisModel

Background Knowledge and Empirical Knowledge

-Anomaly?Confirmation of

current BK and EKno

yes

model composition and learning

belief revision and learning

SummarySummary(1) Conditions of Law Equations

as Communicable Knowledge1. Generic conditions of law equations2. Domain dependent conditions 　　　　　　　　　　　 for communicable law equations

(2) Proposal of a model of communicable knowledge discovery

Discovery is not the matter of only learning and data mining but also model composition, belief revision, consistency checking, model diagnosis, knowledge representation and reasoning of BK and EK and computer-human collaboration.

Example: Trial of Communicable Knowledge Example: Trial of Communicable Knowledge Discovery using scale-type constraints and BKDiscovery using scale-type constraints and BK

Mathematical scale-type constraints [R.D.Luce 1959]

Ex. ） Fechner’ Law ：musical scale: s (order of piano’s keys)Sound frequency: f (Hz)

s:interval scale ， f:ratio scale s = a log f + b

Antigen=Antibody Reaction DataAntigen=Antibody Reaction DataJapanese domestic KDD challenge: KBS (Sep.,2000)Japanese domestic KDD challenge: KBS (Sep.,2000)Background Knowledge usedBackground Knowledge used

Ratio scale ： Ka, Cp ， interval scale ： G, H, TSG=αlog Ka + β G=αKaβ+δ

DG=αlog Ka + β’ DG=αKaβ+δ’

G=αH + β

DG=αDH + (β’)H=αlog Cp + β H=αCpβ+δ

DH=αlog Cp + (β’) DH=αCpβ+(δ’)

TS=αH + β

TDS=αDH + (β’)

G-G0=αlog Ka + β- αlog Ka0 - β

Background Knowledge usedBackground Knowledge usedChemical features of amino-acids ：21 natural amino-acids

Volume

Length

Solvable Unsolvable Aromatic

Result of AnalysisResult of AnalysisChange of H and G between before and after reaction (DH,DG)

-55 -50 -45 -40 -35 -30-130

-120

-110

-100

-90

-80

-70

-60

-50

-40

DH

DG

*:298K+:303Kx:308K

DH, DG:interval scale

Correlation coefficient: 0.690 ⇒ 　 Relation is unclear.

DG

DH

Result of Analysis: regression of EqResult of Analysis: regression of Eq..

-55 -50 -45 -40 -35 -30-130

-120

-110

-100

-90

-80

-70

-60

-50

-40

31-s-a

32-d-a

33-y-a 50-y-a

53-y-a

56-s-a

58-y-a

98-w-a

99-d-a 3299-dd-aa

142-n-a

143-n-a

161-y-a

164-q-a 202-s-a

203-n-a

31-s-a

32-d-a

33-y-a 50-y-a

53-y-a

56-s-a

58-y-a

98-w-a

99-d-a 3299-dd-aa

142-n-a

143-n-a

161-y-a 164-q-a

202-s-a

203-n-a

31-s-a

32-d-a

33-y-a 50-y-a

53-y-a

56-s-a

58-y-a

98-w-a

99-d-a 3299-dd-aa

142-n-a

143-n-a

161-y-a

164-q-a 202-s-a

203-n-a Change of H and G between before and after reaction (DH,DG)

To a （ solvable ， small ）

-55 -50 -45 -40 -35 -30-130

-120

-110

-100

-90

-80

-70

-60

-50

-40

142-n-d

143-n-d

203-n-d 142-n-d

143-n-d

203-n-d

142-n-d

143-n-d

203-n-d

To d （ solvable ， acid ， middle ）

-55 -50 -45 -40 -35 -30-130

-120

-110

-100

-90

-80

-70

-60

-50

-40

33-y-l

50-y-l

53-y-l

58-y-l

33-y-l 50-y-l

53-y-l

58-y-l

33-y-l

50-y-l

53-y-l

58-y-l

To l （ unsolvable ， middle ）

-55 -50 -45 -40 -35 -30-130

-120

-110

-100

-90

-80

-70

-60

-50

-40

32-d-e 164-q-e

32-d-e 164-q-e

32-d-e 164-q-e

To e （ solvable ， acid ， middle ）DH DH

DH DH

DG DG

DG DG

Summary of ResultSummary of ResultFor each type of amino-acid:

Relation (DH,DG)・ Clear linear relation for unsolvable amino-acid.

The gradient of the linear relation depends on the size of amino-acid.

・ Unclear relation for solvable amino-acid. Relation (DH,DCp)

・ Clear linear relation for unsolvable amino-acid.

・ Unclear relation for solvable amino-acid.

conditions of law equations as communicable knowledge

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