PROCESS: Variables :

?destination ?path

Individuals : object ?source

object ?destination heat-path ?path

Pconditions: heat-flow-aligned ?path

Qconditions: greater-than (a (temperature ?source))

(a (temperature ?destination))

Relations : Q+ (heat-flow-rate) (temperature ?source)

Q- (heat-flow-rate) (temperature ?destination)

Influences : 1+ (heat ?destination) (a (heat-flow-rate))

Revision Log: 1 specialize preconditions

PROCESS: Variables : Individuals : Pconditions: Qconditions: Relations : Influences : Revision Log; heat-flow ?source ?destination ?path object ?source object ?destination heat-path ?path flow-aligned ?path

greater-than (a (temperature ?source)) (a (temperature ?destination))

Q+ (heat-flow-rate) (temperature ?source) Q- (heat-flow-rate) (temperature ?destination) 1+ (heat Tdestination) (a (heat-flow-rate)) 1 specialize preconditions

Figure 8.8: The revisions generated by MID using the BKRB above

preferred to the others by dynamic selection. Since b o th revisions are the result of the same single o perato r application, however, MED uses th e static selection m echanism to distinguish betw een them . According to the criteria and weighting described above, MID uses conservatism , simplicity, generality, m odesty and evidential support to perform the selection.

D etails of the scoring in static selection are given in Table 8.5. Rem em ber th a t the difference between th e counts th a t are generated for each revision are com pared against the lim it values for th e set of revisions being com pared, and the difference betw een p ar­ ticular revisions and th e lim iting revision for each criterion is used instead of the original count. Here, the revisions differ on counts of disjunctions of predicates. Simplicity and m odesty b o th prefer a smaller count, bu t generality prefers a larger one. Thus for gener­ ality, the lim iting num ber is th e larger one, which is for th e h e a t - f lo w - a lig n e d revision, while for m odesty an d simplicity, the lim iting num ber is th e sm aller one, which is for the

flow-aligned heat-flow-aligned criterion, c weight, ujc difference in count. Tie _«C Tic X Wc Tic Tic X Wc

K A K A K A

conservatism 0.8 0.4 no. of changes 0 0 0 1 0.8 0.4

simphcity no. of processes 0 0

0.8 0.2 no. of disjunctions 1 0

no. of conditions 0 0.8 0.2 0 0 0

generality no. of conditions 0 0

0.8 0.1 no. of disjunctions 0 1

no. of effects 0 0 0 0 0.8 0.1

m odesty no. of conditions 0 0

0.2 0.9 no. of disjunctions 1 0

no. of effects 0 0.2 0.9 0 0 0

support 0.1 0.1 tim e since last revision 0 0 0 0 0 0

/(A )

Table 8.5: Scores for révisions in static selection. f lo w - a lig n e d revision.

U nder knowledge m otivations, Specialize-Condition: f lo w - a lig n e d gives

(0.8 X 0) -t- (0.8 X (0 + 1 -H 0)) -H (0.8 X (0 -|- 0 -f 0)) -f (0.2 x (0 -H 1 + 0)) -}- (0.1 x 0) = 1.0 in com parison to Specialize-Condition: h e a t - f lo w - a lig n e d

(0.8 X 1)4- (0.8 X (0-1-0-1-0)) 4- (0.8 x (0 -h 1 4- 0)) 4- (0.2 x (0 4-0 4 - 0 ) ) -k (0.1 x 0) = 1.6 Rem ember th a t only the difference betw een the revisions and th e m inim um count values for zmy revision is used, f lo w - a lig n e d scores b e tte r for conservatism since it involves only one lin k by specialization ra th e r th a n tw o, zmd for generality since it has more disjunctions of conditions and hence greater coverage) This meéins th a t it scores worse for simplicity and modesty, however. It thus scores 1.0 while h e a t - f lo w - a lig n e d scores 1.6. The combined effects of conservatism and generality outweigh the effects of simplicity and modesty, and MID prefers the lower value which is associated w ith the specialization to f lo w - a lig n e d . This solution will subsequently need further revision to specialize th e f lo w - a lig n e d predicate to h e a t - f lo w - a lig n e d , bu t it has a ttem p ted to maximize the u tility of the theory in m aintaining a wide coverage.

Under action m otivations, Specialize-Condition: f lo w - a lig n e d gives

(0 .4 X 0 )-k (0.2 X (0 4 -1 + 0 )) 4- (0.1 x (0 + 0 + 0 )) + (0 .9 x (0 + 1 + 0 )) + (0.1 X 0) = 1.1

in com parison to th e TniniTmiTn values of Specialize-Condition: h e a t - f lo w - a lig n e d

Here, f lo w - a lig n e d scores 1.1 while h e a t - f lo w - a lig n e d scores 0.5. In this case where m odesty is valued m ore highly th a n generality, th e h e a t - f lo w - a lig n e d revision is preferred. Note th a t evidential support is the sam e for b o th cases, since they revise the same process. Also, the assum ption has been m ade th a t consistency is not an issue here, and it has not been considered. However, we can see th a t th e f lo w - a lig n e d revision would be inconsistent w ith sm appropriate history, and would require further search (or revision).

A further extended example is given in A ppendix A, giving comparisons for selection w ith and w ithout use of the observation grouping heuristic.

8.7

D iscu ssio n

8 .7 .1 R e la te d W o rk

Of th e m any systems concerned w ith discovery, only a lim ited consideration has been given to selection. This section reviews a num ber o f systems in discovery and other areas th a t have been im plem ented and which use selection criteria in some form or other.

T h a g a r d ’s P I a n d e c h o

Thagard [115] identified the im portance of simplicity, consilience and analogy in theory evaluation and selection, and went on to develop com putational models. In PI, a program th a t attem p ts to provide a ‘m odel of problem solving and inductive inference’ [41, 117], he provided computationed measures of simplicity and consilience b u t was unable to incorporate analogy in such a way (though still noting its significance). More recently, he has developed e c h o, a connectionist program th a t com paratively evaluates theories

principally on the basis of a Theory o f E xplanatory Coherence which includes these three criteria [118,121]. T h ag ard ’s model requires th a t a set o f facts F be explained by a theory T and a set of auxiliary hypotheses A. The auxiliary hypotheses are claimed to be the source of complexity in th a t they are not p a rt of th e original theory, b u t are introduced as assum ptions in order to explain some of the facts.

S im p lic ity In P I, one theory is simpler th a n another if it has a lower ratio o f cohy­ potheses to facts explained. This ensures th a t a hypothesis th a t does not explain anything is no t preferred to one th a t does b u t uses auxiliary hypotheses to do so.

e c h o’streatm en t is a little different. The Theory of E xplanatory Coherence (TEC ) states th a t th e coherence of a theory decreases as the num ber of propositions in­ creases. The propositions here are the cohypotheses th a t are used to explain the facts, including ad hoc assum ptions. A lthough this m irrors the use of simplicity in P I, the connectionist n atu re of e c h o avoids the need for explicit values and formulae, using instead inhibitory and excitatory links to adjust coherence levels appropriately.

C o n s ilie n c e Consilience is a form of generality. A theory which explains all th e facts is m axim ally consilient, b u t if this is achieved by m eans of auxiliary hypotheses, then it is unsatisfactory because it adds to complexity. Here we see the need for simplicity and consilience (or generality) to be taken in tandem when evaluating theories, w ith a decrease in simplicity — am ongst o ther criteria — being used to offset the increase in consilience. In P I, th e facts are weighted w ith associated im portance. Thus P I considers consilience to be th e sum of all the weights of the facts th a t it explains. If all facts are equally im p o rtan t, then degree of consilience can be taken to be ju st the num ber of facts explained. TE C states th a t w hat explains coheres w ith what is explained, and hence the m ore th a t is explained, the greater th e coherence. In addition, T E C undermines th e acceptability of hypotheses th a t explain only a small p a rt of the relevant data. 'If m any results of relevant experim ental observations are unexplained, then th e acceptability of a proposition P th a t explains only a few of them is reduced.’

A n a lo g y The m erits of analogy as a selection criterion are particularly contentious. It is clear th a t analogy is significant in the form ation of theories, b u t T hagard contends th a t analogy is also im p o rtan t in the support of hypotheses already discovered. Analogy is not used in P I, b u t in e c h o an analogy betw een two propositions will increase their coherence.

C o n s e r v a tis m Almost as a side-effect of th e im plem entation of e c h o as a connectionist system, conservatism is tagged on to the three criteria explicitly specified above. In justification of this, T hagard suggests th a t conservatism is a consequence of explanatory coherence, not a separate factor, e c h o does not tre a t new evidence th a t does not cohere w ith existing accepted evidence as equally. For example, if a hypothesis H i explains E i and E2, and subsequently a new hypothesis H2 th a t

contradicts H i b u t explains E i and E2 is advanced, th en the system will not resettle

in to a state in which H i and H2 receive equal activation, rath e r one in which H i

has a higher activation th a n H2.

From th e descriptions above, it should be clear th a t T hagard has considered th e use of selection criteria in some depth. His system is certainly very attractiv e, and has been dem onstrated in a num ber of domains including legal reasoning [118], adversarial problem solving [120] [122], smd a variety of scientific fields [125] [126], b u t is also very lim ited. It is only used in the evaluation of com peting theories (revisions) which are presented to it by some unknown means, and ignores other issues of theory form ation and revision. The connectionist im plem entation is im m ediately attractiv e, bu t leads to problem s over the justification of th e weights given to evidence and links in th e network. They m ust represent som ething, bu t what and why? P a rt of the problem would seem to lie in the natu re of the system , th a t it is designed only to evaluate theories in isolation from the rest of a wider process involving theory form ation, revision and so on, which m ight yield some justification for setting weights in any particu lar way. A nother im p o rtan t issue is th a t th e evaluation th a t takes place in e c h o is not variable, b u t is designed w ith some vague notion of trying to acquire knowledge. As O ’Rorke [83] points o ut, this evaluation should n ot be fixed, since in any real world problem , an ag en t’s goals and priorities play im p o rtan t roles in evaluation.