Symmetry-Based Reasoning about Equations ofPhysical Laws

Yoshiteru IshidaNara Institute of Science and Technology,

8916-5 Takayama, Ikoma, Nara 630-01 Japanphone: +81-7437-2-5351 fax: +81-7437-2-5359

E-mail : ishida@is .aist-nara.ac.jpAbstract : Based on the view that symmetryplays an essential role in human reasoning aboutthe laws of physical phenomena, we propose areasoning paradigm in which symmetry assistsin the reasoning about equations physical laws .Within this paradigm, symmetries are used asconstraints which enable us to specify, derive andgeneralize these equations . The symmetry-basedreasoning is extracted and formalized fromEinstein's work on relativity. We claim that thereasoning procedure thus formalized provides ageneral reasoning architecture that is common todimensional analysis in engineering,mathematical proofs, and commonsensereasoning. This symmetry-based reasoningsystem has been implemented as a symbol-processing system with a production system and aformula-processing system . Using the symmetry-based reasoning system, the equation of Black'slaw of specific heat is demonstrated to bespecified .

1 IntroductionOne aspect of symmetry that has not yet been

fully addressed is its function in humanreasoning. Despite the fact environment is fullof symmetry and that humans depend symmetryfor perception, memory and reasoning, relativelylittle work on symmetry has been done inartificial intelligence . Leyton's work on the roleof symmetry in reasoning about shape [Leyton, 88]and Liu and Popplestone's work on spatialreasoning [Liu and Popplestione, 90] areexceptions to this general trend.The ultimate goal of our research is to

introduce a reasoning that uses symmetry toenhance the intelligence of the system . In thispaper, we focus on a specific reasoning based onsymmetry found in physics. We formalize thereasoning that we can trace in Einstein's 1905paper [Einstein, 05] which emphasizes that thephysical law should be expressed in a wayinvariant under reference frame . Based on thisprinciple, we specify and derive the equations ofphysical laws by using symmetry as a guidingconstraint .The explicit investigation of symmetry-based

reasoning in physics is relatively modern. Such

84

reasoning is evident in the special and generalrelativity theory, the theory of quantummechanics, the theory of elementary particles,and superstring theory . Early attempts to relatesymmetries in physical laws to conservation lawscan be seen in the 1918 work of Noether or eventhe older work by Hamel in 1904. Nevertheless,the origin of this investigation may be tracedback to Newtonian physics, which implementedsymmetries such as space translation, timetranslation, time reversal and parity .

Further, this investigation is not restricted tophysics. If we take a broad definition ofsymmetry, such common sense reasoning as "if ittakes one hour to go from A to B, then it wouldtake also one hour from B to A," canbe recognizedas examples of symmetry-based reasoning . Ingeneral, symmetry-based reasoning seems to be aweak method to which we resort when powerfulknowledge such as the dynamics of a particularsystem is not readily available, as occurs withthe theory of elementary particles .Viewed as a reasoning system, the symmetry-

based reasoning is distinct in that it is reasoningabout modeling rather than reasoning about thebehavior of given models. Although reasoningabout behavior has been widely discussed andimplemented -- primarily in artificialintelligence -- little has been done on theimportant topic of modeling .Viewed as a law-discovery system [Simon, 77],

our implementation is in the same line as that ofBlack System [Langley et al ., 87] in that it istheory-driven and uses a conservation concept.Although Bacon 5 [Langley et al., 81] usessymmetry in part to economize search, theimplementation here relied on symmetry to formthe base of the reasoning paradigm .Although we address symmetry-based

specification and derivation of the equations ofphysical laws, we are fully aware not only thatthe final justification of the law should beagainst experimental data but thatexperimental data are required to bridge the gapbetween physical laws and symmetry .Applying within a specific symbol system

[Simon, 69] would depend on the system's abilityto identify symmetry in the representation

adopted. Currently, it is possible for a computersymbol system to identify symmetry to someextent in symbolically expressed formulae, --something which, up to now has been possibleonly for a human symbol system using paper andpencil .

The literature of physics includes severalprinciples [Bridgman, 62, Feynman, 65, Wigner,67, Rosen, 83, Brandmuller, 86, van Fraassen, 89,Froggatt, 91], which allow us to use symmetry inreasoning . The principle we used in order tospecify and derive the equations of physicallaws is Symmetry as Guiding Constraints . Atthis point, we need to formally state what wemean by symmetry . The following definition ismerely a restatement of that by Weyl [Weyl,52]: An object Ohas symmetry under a mapping Tthat can operate on O if T(O) = O. We use theword transformation when the mapping isdefined on the symbolic formulae .To illustrate the use of these principles we

will consider the problem of specifying figures byrotational symmetry around a center of rotationon a plane. Let a symmetry defined by a rotationofa/naround the center point be denoted by C� .As an example of using Symmetry as GuidingConstraints, consider the problem of specifyingfigures by the constraint of C6 rotationalsymmetry . Obviously, this symmetry cannotcompletely specify the object as a hexagon,although it can eliminate the possibility ofpentagon or octagon.

Section 2 discusses the extraction of reasoningwith symmetry from Einsteins's work onrelativity . Derivation of the formula ofPythagorean theorem and dimensional analysisare discussed as one of the symmetry-basedreasoning . Section 3 presents primitives ofsymmetry-based reasoning . In section 4, thesymmetry-based reasoning is formalized as asymmetry-based specification and derivation,and the example of specifying the equations ofBlack's law is presented.

2. Overview of Symmetry-BasedReasoning

2.1 . Extraction of Symmetry-BasedReasoning from Einstein's WorkThis work is motivated by Einstein's work on

relativity : Einstein's reasoning can beformalized as symmetry-based reasoning thatfully uses symmetry as a constraint, and thereasoning can be applied not only to motion butalso to other physical reasoning or even commonsense reasoning which is applied to the domainother than physics.

Hence, we first discuss Einstein's work ofrelativity and formalize the reasoningparadigm as symmetry-based reasoning .Then, as an application to artificial

intelligence, the symmetry-based reasoning canbe implemented as deriving the symbolic formswith the symbolic rewriting system .When one reviews Einstein's work on special

and general relativity from the viewpoint ofreasoning paradigm, one would realize theyhave a reasoning style in common: Both specialand general theory of relativity have twocomponents, i .e. , the principle of relativity andthe entity carrying physical meaning. As willbe discussed in section 4 in the more generalcontext of symmetry-based reasoning, weidentify the former as symmetries and the latteras object . In order to derive the symbolic formseither for the mapping defined in symmetries orfor the object, symmetries are used as constraintsthat the object must satisfy.

Specifically, the principle of relativity (asin special theory of relativity) states that thephysical law should be invariant when viewedfrom different coordinate systems; one moveswith a constant velocity relative to the other.The application of the symmetry (specialrelativity) to the object (constancy of the speedof light) leads to the following: The light speedmust be invariant viewed from differentcoordinate systems. This fact is used to derivethe Lorentz transformation, which correspond tothe mapping defining symmetry . Afterwards,the Maxwell-Hertz equations are checkedagainst the Lorentz transformation to seewhether or not the equations are invariantunder the transformation . Further, the samestyle of symmetry-based reasoning is used togeneralize the equations for the composition ofvelocity and those for the Doppler effect . Inthis reasoning, symmetry is the same as theabove (i.e ., special relativity), but it applied tothe object different from the constancy of thespeed of light.

In summary, symmetry-based reasoning, asdemonstrated by Einstein's work on relativity,has two components : symmetry as shown in theprinciple of relativity (in the special theory ofrelativity) and objects carrying physicalmeaning as in the constancy of the speed light(in the special theory of relativity). Thesymmetry-based reasoning is carried out by usingthe symmetry as constraint to specify, derive,andmodify the object. In the next subsection, wewill show an example which use symmetry-based reasoning and which is outside of the

physics domain. This will show the symmetry-based reasoning (as extracted from Einstin'swork) is so general that it can be applied tophysics as well as to other areas such asgeometry and modeling in engineering.

2.2. Geometric reasoning as a symmetry-based reasoning

It is interesting that the same reasoning asthat explored in relativity theory can beapplied to many fields . One of the fields isgeometry .Example 2.2 . (specifying the formula of

Pythagorean theorem)An important symmetry for geometric objects is

that the interrelation among the components ofgeometric objects is invariant underdilation(expansion of the size). In other words,what defines the geometric object is not theabsolute length of the components but therelation of the lengths among them. Forexample: for the right triangle shown in Figure1, the relation between the length of the threeedges must be invariant under dilation . Therelation, thus, should be represented in asymbolic form which is invariant underdilation . This is completely parallel to thereasoning in Einstein's relativity that thesymbolic form of the physical law should beinvariant under the different frame of reference .In our context of symmetry-based reasoning, thefollowing two components are given.O (Object to be specified) : c = f(a,b).T (Transformation of the symmetry): c-> kc,

a-> ka, b-> kb where k is any real number andv i -> g(v j ) denotes a substitution.

Thus, O =T(O) gives the constraint : k f(a,b) =f(ka, kb) that is used to specify the symbolicform of f(a,b) . For three right triangles, the bigone and two internal ones, equivalent forms musthold : c = f(a,b), a = f(a2 /c, ab/c), b= flab/c,b2 /c). The length c can be obtained by addingthe two edges from the two internal righttriangles; c = a2/C + b2 /c.

Thus, separating cinto the left hand side yields : c =

a2 +b2 .

aFigure 1 Geometric object for Pythagorean

theorem2 .3 . Symmetry-based reasoning as a

generalization of dimensional analysis

One of the important difference betweengeometry and physics can be found in the scalesymmetry . In physics the scale symmetry doesnot hold (as pointed out, for example, by[Feynman 65]) : i.e ., if you make a miniaturesystem whose size is, say one tenth of the realsystem, then you cannot expect everything issame for the miniature as the real one eventhough everything is made same as the originalone except size (known as scale effect). However,the scale symmetry for unit of measure ofindependent dimension must hold for physicalsystem as known from Buckingham's II theorem[Buckingham 1411 .Again, it should be noted that this

Buckingham's Il theorem is understood as theresult of applying same reasoning as Einstein'stheory of relativity : the representation ofphysical law (the equation describing a physicalsystem in this case) must be represented as a forminvariant under change of frame of reference (thesystem of dimension in this case). In fact,0(7ri,7C2, . . .7r�_,) is such an invariant form since7r1,762, . . .7r,_, are invariant under scale changeof unit of measure for each independentdimension.The next example illustrates not only that the

dimensional analysis can be done within theframework of symmetry-based reasoning but thatthe symmetry of scale change of unit of measurefor independent dimension in dimensionalanalysis can be treated as constraints insymmetry-based reasoning, similarly to theother symmetries .Example 2.3 . (Dimensional analysis) [Sedov 43]

Consider the motion of a simple pendulumwhere the particle with mass m is suspended bya string as shownin Figure 2.

Figure 2 A simple pendulum where theparticle with mass m is suspended by a string

1 If a physical system is described by f(zl, x2 9 . . ., X")where z,,x2, . . . I x,, are n variables that involve r basicdimensions, then the physical system will be describedonly by n-r independent dimensionless products

The equation describing the physicalsystem is, thus, reduced to be 0(n1,7r2,. . .7r._,) .

The object to be specified is the formula of thesystem f(t, 1, g, m, o) where t: time(T), 1: length ofstring(L), g: gravity constant (L / (T 2 )), m: massof the pendulum(M), and o : angle of the string(dimension is indicated inside the parentis) .The given symmetries are:" (1) scale symmetries due to the basic

dimensions T, L, M.," (2) phase translatory symmetry in terms of

time t, and" (3) mirror symmetry in terms of the angle 0.The symmetry (1) specify the form f(t, 1, g, m,

On one hand, insight of what symmetries areinvolved in physical phenomena ormathematical statement is important element.Using these symmetries in deriving andspecifying the formulae by symbolic processingis a core of our symmetry-based reasoning.On the other hand, it should be noted that

finding the object with domain specific contentis an important core in scientific discovery .However, this process of finding the object withcontent is far from automating or even fromformalizing at this stage.As will be seen in section 4.2 ., if the given

object is specific enough, the symmetry-basedreasoner carry out deductive reasoning .However, if the given object is not specific, thereasoner carry out inductive reasoning byproposing candidate forms satisfying the givensymmetry and modifying the current forms tomeet the other given symmetries .

3 . Primitives of Symmetry-BasedReasoning

3.1 . Fixed point and symmetryOur method of specifying the equations of

physical laws is a generalization of that indimensional analysis . Buckingham's r1 theoremcan be generalized along the following lines(where the symmetry used is scale symmetry forunit of measure of an independent dimension andthe invariant form is a dimensionless product inthe Buckingham theorem):

gives the constraint :f(t 9,0)=f((t+Tc)0) . The symmetry (3) further gives the constraint :

f(t (t19, 0)=f(t N1 , -0) .Our method of specifying the equations of

physical law can be viewed as a generalizationof that in dimensional analysis . Table 1summarizes the parallelism of the symmetry-based reasoning among Einstein's theory ofrelativity, derivation of the form ofPythagorean theorem and dimensional analysis .

0) to : f(t 9F11, 0) . The symmetry (2) furtherTable 1 Parallelism of the symmetry-based reasoning among Einstein's theory of relativity, derivation

ofPythagorean theorem and dimensional analysis .of the form

If a formula is to be described with variables :X1,X2, . . .,xn and if the formula should have asymmetry that can be attained only by a formF(xl,x2, . . ., x� ), then the target formula can bereduced to the form f(F(x1,z2, . . .,xn) ).Example 3.1.1 .If the given symmetry is the translatory

symmetry such that the form should beinvariant under the translation: x1,x2, .. .,Xn ->

X1 + C,x2 + C, . . .,x� +C where C is an arbitraryreal number, then the unique 2 form is :F(x1,X2, . . .,Xn) = alxl +a2x2+, . . .,+anxn wherea,,a2, . . .,a. are constants satisfyingal +a2+, . . .,+an = 0. Therefore, the formulawith this translatory symmetry can be reducedto the form

f(F(x1 , x2,. . ., xn ) ) by the aboveargument. (This form will be used in theexample 4 .2.1 . to specify Black's law.)However, if the given symmetry is thepermutation symmetry : x i <-> x i (vi <-> v idenotes a permutation), then there are manyforms satisfying the symmetry . Examples are:

2We do not derive the uniqueness here because thederivation process itself is irrelevant with thesymmetry-based reasoning; we use the heuristicsassociating the given symmetry with possible formssatisfying the symmetry, as seen in the followingsections .

Einstein's theory ofrelativi

Geometric reasoning inPythagorean theorem

Dimensional analysis

Symmetries Symmetry with frame Scale symmetry Scale symmetry forofreference unit of measure of

independent dimensionObject with the Constancy of the speed Geometric Involved variables andContent of the Domain of light Configuration their physical

u ( I dimensions

F(xi,xj)= x i xi,F(x i ,x 1 )= x i+x j and F(x,,x j )= x ix i +x i +x i .The process of specifying the equations of

physical laws is a process of finding an object(formula) whose symmetry is as close as possibleto the strength of the symmetries given asconstraints. Due to the absence of symbolicversion of the fixed point theorem, specificationand derivation process is driven by observinghow T(O) differs from O and by using theequation O = T(O) . There is no continuousmeasure that indicates how close the currentform is to the target solution in the search forthe symbolic form in our procedure of symmetry-based reasoning . The discrete measure indicatinghow close the current form is to the targetsolution is the number of how many symmetries(out of the symmetries given as the constraints)the current form satisfies . One heuristic toincrease the efficiency of the search is that ifthe current solution satisfies the symmetry, thenall the symmetries weaker than the symmetrycan be disregarded .Example 3.1.2Since a regular triangle already has the

symmetry C3, this symmetry does not provideany information to the regular triangle.However, both the symmetries C4 and C6 willprovide the information, for they require thefigure to be a regular 12-gon and a hexagon,respectively . C6 is stronger symmetry than C3because C3 cannot provide information for allthe objects to which C6 can provideinformation . C4 is orthogonal to C3 because C3can provide information for some objects towhich C4cannotprovide information . In thissense, the continuous symmetry C� , whichspecifies a circle, is the strongest possiblesymmetry .

In section 4.2., we will look at Black's law ofspecific heat as an example of symmetry in aphysical law .

3.2 . Formula-processing for symmetry-based reasoning

The formula-level operations required for thesymmetry-based reasoning are : symmetryidentification, equation building by symmetry,and equation solving . These operations arepossible in the commercial formula-processingsystems such as Mathematica, Maple, Macsyma,Reduce, and so on. In our study, we usedMathematica [Wolfram, 88] for formula-processing . The intrinsic problem in symmetry

identification comes from the lack of an ultimatecanonical form of the equations, which makes itdifficult for the system to identify two formulaeas equivalent. Thus, even if the two forms are notidentified as equivalent by the system, it may besimply because the system cannot identify theequivalence .We used a production system to implement the

symmetry-based reasoning . The reasons weadopted the production system architecture aretwofold : First, we formalized symmetry-basedreasoning as based on a human reasoningparadigm rather than a computationalalgorithm. Second, symmetry-based reasoningrequires much heuristic knowledge which is notexplicitly included in the formula .We implemented a production system on

Mathematica so that the formula-leveloperations mentioned above could be done withinthe production system .Like many other pattern-directed

applications of Ma th em a t i ca, formula-processing for symmetry-based reasoning uses thechain of (conditional) rewriting of the formulae .Other than built-in functions such as solve andEliminate, the following components areimplemented for symmetry-based reasoning :transformation of formula, symmetryidentification, proposing formula by symmetry,and symmetry derivation from constraint . Wepresent the syntax and examples of each of thesecomponents in the following subsections.

3.2 .1 . Transformation of formulaThe syntax of transforming an expression is :

Trans_ [exp_,{parameter-list)] . Thereare five types of transformations commonly used :Tranal, Perm, Dilat, Com, ASCom .The symmetry defined by the transformationwill be expressed as follows :Trans-{parameter-list) ." Transl [exp_, (x1,x2, . . . ),c] willtranslate the listed variables xl,x2, . . . toxl+c,x2+c , . . . in the expression specified in exp ." Perm[exp_,(x1,x2,y1,y2 . . .)] willpermute x1 with x2, yl with y2,and so on" Dilat[exp_,(x1,x2, . . .),c] willdilate the listed variables xl,x2, . . . intocxl,c x2, . . . ." Com [exp_, (xl, x2) ]

will permute xlwith x2." ASCom[f_[xl,x21,(x1,x2)] willmake the function f [xl,x2] into1- f_[1 /x1,1 /x2] .

Other transformations which are specific tothe problems will be described at each example .

In the examples of the rest of paper, the input tothe system is in boldface . The output from thesystem is not.Example 3.2 .1Transl[-(cl mi t1) - (c2 m2 t2)+ cf (ml + m2) tf,(tf,t1,t2),c]-(cl ml (c + tl) )

- c2 m2

(c + t2) +cf (m1 + m2) (c + tf)

ASCom[f[m2/mi,c2/cl],(m2/ml,c2/c1)]

Ml Cl1 - f [--,

-m2 c2

3.2.2 . Symmetry identificationThe symmetry of the formula can be

mathematically identified by investigating theequivalence between the original formula andthe formula after transformed as describedabove . The syntax of the symmetryidentification is:

SymQ[{},exp-,Trans_�{parameter-listl]which will identify the expression exp_ is equalto the expression after the transformation :Trans_[exp,{parameter-list)] . It will returnTrue if the equality is identified, otherwise itreturns False. As we have mentioned, it mayreturn False even if the expression and that aftercomplicated reduction are equal.

Example 3.2.2SYMQ[(),-(1 - (1 + (c2ml))"(-1) + t2/((1 + (c2

t 1 ) )

m2)/(clm2)/(cl

m 1 ) )tf/t1,Perm,(m1,c1,m2,c2)]TrueSYMQI(),f[cl, c2, ml, m2, tf,t1, t2],Transl, (tf, ti, t2)]False

3.2 .3 . Proposing formula by symmetrySymmetries can be used to propose possible

forms that the target object may have . Sincesuch proposals are made based on heuristics, theresulting object must be evaluated (see section4.1 . for the process) to see whether if theysatisfy all the given symmetries . The syntax ofproposing possible forms is :Trans_[prop,exp_,{parameter-list)] .

Example 3.2.3Transl [prop, f[c1,

c2,

m1,tf, t1, t2],(tf, t1, t2)]tf - (ti (1 - f[cl, c2, ml, m2]) +

t2 f[c1, c2, m1, m2l)

m2,

Dilat[prop,tf - (tl (1 - f[cl,c2, mi, m2]) + t2 f[cl, c2, ml,m2]),(cl, c2)]

tf -

(t1

(1 - f [l,c2

f [l,

--, ml, m2l )Cl

c2

clml, m2l ) + t2

3.2.4. Symmetry derivation from constraintSymmetries on the formula such as

permutation symmetry can provide a constraintwhich can be used to derive internal symmetries(symmetry on the part of the original formula),as seen in the example of symmetry-basedspecification of the equation of Black's law. Tocarry out a symmetry derivation,

*First, Eliminate irrelevant variables fromthe constraint (given by the symmetry).

*Then, use pattern-matching to search thesymmetry for the extracted part of the originalformula.

*After the internal symmetries are detected,they are recast to the symmetry-based reasoningsystem by creating a subgoal of specifying thepart of the original formula by the internalsymmetries detected .

4 Symmetry-based reasoning4.1 . Procedure of symmetry-based reasoningIn his 1905 paper [Einstein, 05], Einstein used

the symmetry-based reasoning. It should benoted that he used this reasoning to derivesymmetry of the Lorentz transformation [Lorentz,04] and to derive objects3.

By extracting thereasoning from Einstein's work and making it tofit as a procedure, the symmetry-based reasoningmay be formalized as:*(step 0) Given: transformation T and somepiece of information for an object O.*(step 1) Checking symmetry : if the objectis given in symbolic form, then checkwhether the object satisfies the givensymmetry . If O = T(O) , stop; otherwisego to (step 3)*(step 2) Proposing object: If object Odoes symmetry, then propose a candidateformula of the object O by heuristics onthe basis of the given list of symmetries T.*(step 3)

Modifying object:*(step 3.1) if O = T(O) can derive newtransformation T' for the part of objectO' then go back to (step 0) with theseT' and O'."(step 3.2) if

O = T(O) includeunknown parameters, solve the equationO = T(O) .

3The generalized equation of Doppler's principle, theequation of pressure of light, and the equation for arelativistic mass[Einstein, 05] .

*(step 3.3) modify O by heuristics sothat O = T(O) is satisfied. If there is notenough knowledge to do this, go back to(step 2) for new proposal.The rules of the symmetry-based reasoner can

be divided into the procedual steps above. Therules of the step 1 see if given symmetries aresatisfied by the given object using SymQ describedabove. If all the given symmetries are satisfiedby the object, then the rules of this step simplyterminate the task. A simple symmetry checkcan be done within this step . We call this modesymmetry identification .

If any of the symmetries are not satisfied bythe object and if there is any knowledgeavailable to propose modifications to the objectso that the unsatisfied symmetry may besatisfied after the modification, then the rules ofthe step 2 will make those modifications . Amongseveral symmetries which can proposemodifications, permutation symmetries work in aunique manner. Permutation symmetries do notpropose the modification, but rather deriveconstraints at the step 3.1 and then derive theinternal symmetries found in this step . Next, thepermutation symmetries create a subgoal ofspecifying the part of the object using the newsymmetries found at this step . After thesemodifications and after working memoryelements have been preprocessed, the object isrecast to the symmetry check process returningthe control to the step 1 . We call this modesymmetry-based specification .

If the modification is just a specification ofunknown parameters in the object, then the rulesof the step 3.2 are evoked after the step 1 . In thisstep, the given symmetries are used as buildingthe equation whose solution will specify theunknown parameter . The modification on theobject is made by substituting the solution to theunknown parameter at the next step 3.3 . Afterthis modification, the same process as that ofsymmetry-based specification follows. We callthis mode symmetry-based derivation . In thefollowing, we present sample sessions for thesethree modes.

4.2 . Examples of symmetry-basedreasoningIn this section, we will give only example of

symmetry-based specification, since symmetry-based derivation and symmetry identificationuse only a part of steps of symmetry-basedspecification . The examples of symmetryidentification . mode demonstrated in ourreasoner include that the angular momentum isinvariant under spatial rotation, that the

quadratic form of -(c 2 t 2 ) + x2

+ y'-

+ z 2 i sinvariant under the Lorentz transformation, andthat the Maxwell equation is invariant underthe Lorentz transformation . The examples ofsymmetry derivation of parameters includethat the specific heat from the symmetries usedin the next example 4.2.1 when given object hasonly specific heat left as unspecified parameter,the parameters in Lorentz transformation,

theangle between spin and velocity for mass-lessparticle is zero4, which is more complex thanthose shown here .Example 4.2.1 (Specification of the equation

of Black's Law)Black's law of specific heat can be stated as

follows: If two entities, whose initial statedescribed by temperature

t i and t2 .

specificheat c i and c,, and mass mi and m2 ,

arethermally coupled, then the final equilibriumtemperature will be

tf = (c emit 1

+ c, m2 t ,)/(c i in i +c 2 m2 ).

In the following example, the equation ofBlack's law: t f = (cimiti + c 2m2 t 2 )/(cirri +c2m2 ) will be specified starting from scalesymmetry (dimensional analysis), translatorysymmetry of temperature, permutation symmetryof interacting entities, and permutationsymmetry of specific heat and mass. Whenpermutation symmetry exists in the list ofsymmetries in the input, three more steps areevoked after the step of P r o p o s e For mu 1 aDeriveConstraint, DeriveSymmetry andFocusintobj . As seen in the trace of the rulefiring, these additional steps first deriveconstraint from permutation symmetry and findthe internal symmetry that this part of the objectexhibits .

This example specifies the equations ofBlack's Law when sufficient symmetries aregiven. Both the input and the output have beentranslated to higher-level expression forreadability. Also, some inessential parts areomitted. The step numbers in parentheses refer tothe procedural step described in the previoussubsection . The number before parenthesesindicates the depth as well as the cycle of theprocedure . ==, != and && denote equality,inequality and logical and, respectively .

4"The intrinsic angular momentum of a particle withzero rest-mass is parallel to its direction of motion, thatis parallel to its velocity [Wigner 67] ."

O : f[cl,c2,m1,m2,tf,t1,t2)T :Perm(m1,m2,cl,c2,t1,t2),Perm(mi,c1,m2,c2),Dilat(ml,m2),Dilat{cl,c2),Dilat(t1,t2,tf),Transl(tf, t1, t2)

1(step 1 .1) The system identifies that the symmetryTransl{tf, tl, t2} is not satisfied by the object .. . . . symmetry checking given in the list T, similar to the above . . . . .1(step 2) By the symmmetry Transl{tf, tl, t2}, the

modification from the object :f [c1,

c2,

ml,

m2,

tf,

tl,

t2]to .tf -

(tl

(1 -

f [cl,

c2,

ml,

m2])

+ t2

f [cl,

c2,

m1,

m2))1(step 2) By the symmmetry Dilat{c1, c2}, the

modification from the object :tf - (t1 (1 - f[cl, c2, ml, m2]) + t2 f[cl, c2, m1, m2])to :

c2

c2tf -

(t1

(1

-

f [1,

--,

ml,

m2])

+ t2

f [1,

--,

ml,

m2] )c1

cl. ..modification of the objects by Dilat symmetries, similar to the above . . .1(step 2) By the symmmetry Dilat{ml, m2}, the

modification from the object :c2

t2 f[1, --, m1, m2]tf

c2

c1-_ _ (1 _ f[1, _-, ml, m2] + ________--____-----)ti

clto .

tic2 m2

t2 f[1, --, 1, --]c2 m2 cl m1-_, 1, _-] + --_------_______-_)

t1

c1 m1

t1

tf

system proposed the

system proposed the

system proposed the

1(step 3 .1 .1) By the symmmetry Perm{m1, cl, m2, c2}, the system derived theconstraint :

c2 m2 m2f[1, --, 1, --] != 0 && f[1, --,

cl m1 m11(step 3 .1 .1) By the symmmetry

c21, --]

clPerm{m1,

==c2

f [I, --cl

m2, cl, c2,

m2

m1t1, t2}, the system

derived the constraint :c2 m2 c1 ml c2 m2

f[1, --, 1, --]!=0 && f[1, --, 1, --]== 1 - f[1, --, 1, --]cl ml c2 m2 cl ml

1(step 3 .1 .2) By the constraint :c2 m2 m2 c2 c2 m2

1 , --] _= f[1, -_ 1 --]cl ml ml c1 cl ml

the system derives the symmetry :m2 c2

Com{--, -_}ml c1

for the object :c2 m2

f[1, -_ 1 --]cl ml

1(step 3 .1 .2) By the constraint :c2 m2 cl ml c2 m2

f[1, --, 1, --]!=0 &&cl m1 c2 m2 cl m1

the system derives the symmetry :m2 c2

ASCom{--, --}ml cl

for the object :c2 m2

f[1, --, 1, --]cl m1

1(step 3 .1 .3) The system creates subgoal of checking thederived symmetries :

m2 c2

m2 c2Com{--, --} and ASCom{--, --}

ml c1

ml clfor the internal object :

c2 m2f[1, --, 1, --]

cl ml

1 .1(step 2) By the symmmetrym2 c2

Com{--, --},ml cl

the system proposed the modification from the object :c2 m2

f[1, --, 1, --]cl m1

to :

m2 c2f[1, 1, com[--, --}]

m1 cl1 .1(step 2) By the symmmetry

m2 c2ASCom{--, --},

ml clthe system proposed the modification from the object :

m2 c2f[1, 1, com[--, --}}

ml clto .

that the symmetry

is satisfied by the internal object .1 .2(step 1 .2) The system terminated the subgoal because there is no symmetry

that is not satisfied by the internal object .2 (step 1 .1) The system identifies that all given symmetries are satisfied by

the object with new internal object .2(step 1 .2) The system terminated because there areno other symmetries that are not satisfied by the object with new internal

object .

1

c2 m2

cl ml1 .2(step 1 .1) The system identifies

m2 c2 m2 c2Com{--, --} and ASCom{--, --}

ml c1 m1 c1

1 .1(step 1 .1) The system identifies that the symmetrym2 c2 m2 c2

ASCom{--, --} and Com{--, --}is not satisfied by the internal object .MI. cl ml cl

Specification by symmetry can apply notonly to physics but to mathematics as well . Aswe have seen in section 3.2 ., the Pythagoreantheorem, for example, can be specified by thegeometrical symmetries of a right triangle . Theapplication of this specification method is notlimited to the equations of physical laws. It canalso apply to the equation of a particular systemif the system exhibits symmetry . As we haveseen in section 3.3 ., it can apply to derive theequation of a pendulum that exhibits someobvious symmetry specific to the systemstructure.

In this section, we have discussed thespecification and derivation of an object(formula) . However, the symmetry-basedreasoning can be applied to specify or derive thesymmetry (transformations) as well . Einsteinspecified or derived the Lorentz transformationfrom the fact that it leaves an object (the speedof light) unchanged . He also noted that thetransformation leaves unchanged the otherobject such as Maxwell-Herz equation [Einstein,05] . Our symmetry-based reasoning also cancheck whether some other objects may have thesymmetry .

5 ConclusionsIn this study, we demonstrated how the

Symmetry as Guiding Constraints principle canbe used to automate reasoning in order tospecify, derive, and generalize equations inphysical laws . Nevertheless, symmetry itselfstill needs to be found .

There is not always one-directional mappingfrom symmetries (constraints) to objects(equations). Rather, in reality, the elaborationprocess may go back and forth with symmetriesand objects constraining each other.

The symbolic equation serves as a powerfulrepresentation of knowledge that can be :checked if it has a certain symmetry, solved ifvariables and constants are specified,transformed into canonical form, and so forth . Inorder for these operations to give physicalimplications, the syntax as well as semantics ofthe equation must be included in the knowledgeof the system . In this paper , we implementedthe syntax on the formula-rewriting systemMathematica and the semantics on both thelong-term and short-term memory of aproduction system .

ACKNOWLEDGMENTSMost of this work has been done during the

visit at CMU in 1993 and 1994 . 1 am mostly

grateful to Dr. Herbert A . Simon for theimportant discussions .

ReferencesBrandmuller, J . 1986. An Extension of theNeumann-Minnigerode-Curie Principle, Comp& Math. with Appls. 12B, pages 97-100 .

Bridgman, P.W. 1962 . A Sophisticate's Primer ofRelativity , Wesleyan University Press .

Buchingham, E . 1914 . On Physically SimilarSystems : Illustration of the Use ofDimensional Equations . Physical Review IV,4, pages 345-376 .Einstein, A. 1905 . On the Electrodynamics ofMoving Bodies, in The Principle of RelativityDover Publications, INC.

Feynman, R.P . 1965.The Character of PhysicalLaw � .

Feynman, R.P., and Weingberg, S. 1987 .Elementary Particles and The Laws of PhysicsCambridge Univ . Press .

Froggatt, C.D., and Nielsen, H.B . 1991.Origin ofSymmetries, World Scientific .

Langley, P . ; Bradshaw, G.L . ; and Simon, H.A .1981 . Bacon 5:The Discovery of ConservationLaw, Proc. ofIJCA181, (1981) pages 121-126.

Langley, P . et al . 1987 .Scientific Discovery,The MIT Press .

Leyton, M. 1988 . A Process-Grammar for Shape,Artificial Intelligence, 34 pages 213-247 .

Leyton, M. 1992.Symmetry, Causality, Mind,The MIT Press.

Liu, Y., and Popplestione R.J . 1990 . SymmetryConstraint Inference in Assembly Planning,Proceeding of AAAI 90, pages 1038-1044 .

Lorentz, H.A. 1904 . Electromagnetic Phenomenain A System Moving with Any Velocity LessThan That of Light, in The Principle ofRelativity , Dover Publications, INC.

Rosen, J . 1983 . A Symmetry Primer for ScientistsJohn Wiley & Sons.

Simon, H.A. 1969.The Sciences of the Artificial,The MIT Press .

Simon, H.A . 1977.Models of Discovery, Reidel,Dordrecht.

van Fraassen, B.C. 1989. Laws and Symmetry,Clarendon Press .

Weyl, H. 1952 . Symmetry , Princeton Univ.Press .

Wigner, P. 1967.Symmetries and Reflections,Indiana University Press .

Wolfram, S. 1988 . Mathematica : A System forDoing Mathematics by Computer, WolframResearch Inc .