# Presentation: 000991

## Conditions of Law Equations as Communicable Knowledge

**Takashi Washio**
**Hiroshi Motoda**
**I.S.I.R., Osaka Univ.**

**Informal Workshop on Communicable Knowledge**
**Dec., 6****th****, 2000**

## What are the conditions of communicable law equations?

- Generic conditions of law equations
- Domain dependent conditions for communicable law equations

## What are law equations?

*What 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 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 equations

**A ****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 operators**
**A: axioms on distance and etc.**
**L: empirical laws and empirical strong believes**
**D: a domain on which the scientific region                ****    ****concentrates its analysis.**

## Generic conditions of law equations

**Ex.) 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 e**
**An 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 equations

**Satisfaction 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 obtained under an identical condition.)**
**Soundness**** (e is consistent with the measurement under a certain condition.)**
**Parsimony**** (e consists of minimum number of quantities.)**
**Mathematical Admissibility**** (e follows S and A.)**

**In the objective domain of e, De**

## Generic conditions of law equations

-    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.

## Generic conditions of law equations

**Parsimony**** (e consists of minimum number of ****         ****quantities)**
**Mathematical Admissibility**** (e follows S and A)**

- Conditions being confirmed
- through experiments and/or observations

**Objectiveness****(****All quantities in e is observable)**
**Generality**** (e is satisfactory in wide phenomena**
**Reproducibility**** (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 communicable law equations?

- Generic conditions of law equations
- Domain dependent conditions for communicable law equations

## Domain dependent conditions for communicable 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/L****3****  ****≡**** V=L****3****, d=M/V**
**Ex.2) ****f=Gm****1****m****2****/r****2**** ****?**** A=****m****1****m****2****,**** ****f=GA/r****2**** **

**physically unclear**

## Domain dependent conditions for communicable law equations

- (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=m****2****a  ****≠****  dv/dt=a, mdv=fdt**
**Ex.2)  f=Gm****1****m****2****/r****2**** – k/D****α ****← space term**
**                   ****Universe should be static.**
**           ****≠ ****Red shift of light spectrum**
**                 ****+ Doppler effect**

## Domain dependent conditions for communicable law equations

- (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.**

## Domain dependent conditions for communicable law equations

- (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 view**
**Ex.2) Model equation of air friction force**
**     ****f = - c v****2**** – k v : global view**
**          ****f = - k v            : local view**
**    **

## Domain dependent conditions for communicable law equations

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

**V=IR**
**I****EC****=h****fe****I****BC**
**I****0****=I****1****+I****2**

## (5) Appropriate simplicity and complexity for understanding

-      (Continued)

**Decision Tree (ID3,C4.5)**

**Decision tree pruned **
**in a comprehensive level**

**Depth**
**5**

**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 communicable law equations

- In 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

**Quantities and relations are different.**

## A model of communicable 

*knowledge discovery*

- Generic conditions of law equations
- Domain dependent conditions for communicable law equations

**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!**

## -

*knowledge discovery*

**Proposing framework:**

**Data set**
**    ****features**
**    ****class**
**    ****explaining quantities **
**    ****objective quantity**

**Hypothesis**
**Model**

**Background Knowledge and Empirical Knowledge**

**-**

**Anomaly?**

**Confirmation of **
**current BK and EK**

**no**

**yes**

**model composition and ****learning**

**belief revision and ****learning**

## Summary

- (1) Conditions of Law Equations                                         as Communicable Knowledge
- 1. Generic conditions of law equations
- 2. 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 Discovery 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 Data

*Japanese domestic KDD challenge: KBS (Sep.,2000)*
*Background Knowledge used*

**Ratio scale****:****Ka, Cp****,  ****interval scale****:****G, H, TS**

**G=αlog Ka + β**

**G=αKa****β****+δ**

**DG=αlog Ka + β’**

**DG=αKa****β****+δ’**

**G=αH + β**

**DG=αDH + (β’)**

**H=αlog Cp + β**

**H=αCp****β****+δ**

**DH=αlog Cp + (β’)**

**DH=αCp****β****+(δ’)**

**TS=αH + β**

**TDS=αDH + (β’)**

**G-G****0****=αlog Ka + β- αlog Ka****0**** - β**

## Background Knowledge used

**Chemical features of amino-acids****:**
**21 natural amino-acids**

**Volume**

**Length**

**Solvable**

**Unsolvable**

**Aromatic**

## Result of Analysis

**Change of H and G between before and after reaction (DH,DG)**

- DH

- DG

- *:298K
- +:303K
- x:308K

**DH, DG:interval scale**

**Correlation coefficient: 0.690  ⇒**** ****Relation is unclear.**

**DG**

**DH**

## Result of Analysis: regression of Eq.

**Change of H and G between before and after reaction (DH,DG)**

- To a(solvable,small)

- To d(solvable,acid,middle)

- To l(unsolvable,middle)

- To e(solvable,acid,middle)

- DH

- DH

- DH

- DH

- DG

- DG

- DG

- DG

## Summary of Result

- For 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.