# Presentation: 000990

## Conditions of Law Equations as Communicable Knowledge

_**Takashi Washio**_
**Hiroshi Motoda**
**I.S.I.R., Osaka University.**

**Symposium on Computational Discovery of Communicable Knowledge**
**March, 24****th ****-25****th****, 2001**

## What are the conditions of communicable law equations?

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

**Question to clarify criteria and knowledge which can **
**be implemented in computational discovery systems**

## 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 an 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 syntactic 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.)**
**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 (consistency).
-  	   Law of gravity force
-     F=G M1M2/R2
-   **→** It is general (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 )**

- Conditions on law equation formulae

- MDL, AIC, .....

- 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 heuristics**

## Domain dependent conditions for communicable law equations

- (1) 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,fb,t,ir) is relevant to the****	****	****	****	****	****	****  interest of stock ****	****	****	****	****	****	****  fund managers.**

## Domain dependent conditions for communicable law equations

- (2) 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**
**    **

**veiw**

## Domain dependent conditions for communicable law equations

- (3) Clarity 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

- (4) 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**

## Domain dependent conditions for communicable law equations

- (5) 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**

## 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 techniques **
**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 (Empirical Knowledge)**

**-**

**Anomaly?**

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

**no**

**yes**

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

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

**abduction**

**consistency**
**checking**

**model **
**diagnosis**

## Trial of Communicable Knowledge Discovery 

**Mathematical Admissibility**** (e follows S and A)**

- Conditions to be 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 )**

- Conditions on law equation formulae

- scale-types

*Trial of Communicable Knowledge Discovery *
*using mathematical constraints and BK*

**Application of SDS**

##  Example: Antigen=Antibody Reaction Data

**Antibody consists of 20 types of natural amino-acid.**
**H-chain****:****a chain of 110 amino-acid (VH 1-110)**
**L-chain****:****a chain of 120 amino-acid (VL 1-120)**
**An amino-acid is replaced by another type of amino-acid in a anti-body. Its thermo-dynamical features are measured.**
**Total data: 35X3=105**

**Change of quantity values before **
**and after the reaction with antigen**
**Reaction constant:Ka****,**
**Change of free energy:DG****,**
**Change of enthalpy:DH****,**
**Change of entropy:TDS**
**Change of specific heat:DCp**

**Reaction with Antigen**

**L-chain**

**H-chain**

**Antibody**

* **Example: Antigen=Antibody Reaction Data*
*Japanese domestic KDD challenge (Sep.,2000)*

**Data are provided by a biologist.**

## Objective of Analysis

- Discovery of generic physical relations in data  and its physical interpretation by domain experts
- Discovery of (semi-)quantitative physical relations in data under the consideration of chemical features of amino-acid and its interpretation by domain experts

*Trial of Communicable Knowledge Discovery using scale-type constraints (SDS) and BK*

## Mathematical scale-type constraints

**Ratio scale**

- Absolute origin and invariance of ratio(length)

- Arbitrary origin and invariance of ratio of difference (temperature in Celsius, Fahrenheit)

**Absolute scale**

- Invariance of value (radian angle)

- x,y :ratio scale

**Shift of origin****,****contradictory**

- unit conversion

- x’,y’ :ratio scale

- x’ = kx
- y’ = Ky

- y = log x

- y’ = log x’

- y’ = Ky = log x + log k

**Mathematical scale-type constraints**

## Mathematical scale-type constraints 

- [R.D.Luce 1959][T.Washio 1997]

**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

## 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**

**G-G****0**** =αKa****β****+δ- αKa****0****β****-δ**

**The biologist is interested in bi-variate relation.**

## Background Knowledge used

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

**Volume**

**Length**

**Solvable**

**Unsolvable**

**Aromatic**

## Result and Evaluation

**Result and Evaluation**

- A generic relation independent of replacement conditions

- Ka:ratio scale,DG:interval sacle

**DG=αlog Ka + β**

**DG=αKa****β****+δ**

- F=547200>4.196

- F=49240>4.96

- log Ka

- log Ka

- DG

- DG

- (Biologist:definition of Ka)

## Result and Evaluation

- A generic relation independent of replacement conditions

- DH, TDS:interval sacle

**TDS=αDH + β**

- F=770.5>4.196

- ( Biologist:physically
-                 deducible relation)

- DH

- TDS

## 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.

**Biologist : Comprehensible discovery for experts.**
**The relation for unsolvable amino-acid may show clear tendency,**
**since they do not change the molecule shape in solvent very much. **

## -

**Proposing framework:**

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

**Hypothesis**
**Model**

**Background Knowledge (Empirical Knowledge)**

**-**

**Anomaly?**

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

**no**

**yes**

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

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

**abduction**

**consistency**
**checking**

**model **
**diagnosis**

## 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 computer-human collaboration.