Towards a knowledge level formalization of learning Denis PIERRE and Joel QUINQUETON
LIRMM, 161 Rue ADA, 34392 Montpellier Cedex 5, FRANCE fpierre,
[email protected] Abstract
Our work is an attempt to de ne a knowledge level model of a learning process, between a full integration of machine learning and knowledge level modelling and the use of inference structures to bias the learning process. We de ne a representation formalism, called mental scheme , to modelize the increase of the knowledge during the learning process. This is a relation between elements in a language and a set of beliefs. We show how this representation scheme can be used as belief to represent nominal or structured variables, and how we increase knowledge through reasoning mechanisms and argumentation mechanisms.
1 Introduction As stated in the announcement of this workshop, three important requirements are postulated by the so-called model of expertise by the knowledge acquisition community : the separation of the symbol level and the knowledge level, the use of generic problem solving methods, the necessity of di erent modelling primitives for epistemologically di erent kinds of knowledge. A widespread approach to model-based knowledge acquisition is the KADS project [Schreiber et al., 1993]. The KADS model of expertise is intended to provide pre-de ned modelling primitives to allow an implementation-independent description of the knowledge using several layers. Several works have attempted to ll the cultural gap between the machine learning methods and model-based knowledge acquisition, in three directions :
Laboratoire d'Informatique, de
Robotique et de Microelectronique de Montpellier
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the integration of machine learning and knowledge level modelling [van de Velde and Aamodt, 1992; Graner and Sleeman, 1993], the use of inference structures to bias the learning process [Rouveirol and Albert, 1994], the investigation of the diculties arising when applying machine learning techniques to learn knowledge for a diagnostic task to be solved by heuristic classi cation [Fensel et al., 1993].
Our approach is between the rst two items of this classi cation. In the KADS framework, the domain layer is clearly separated from the inference layer. The machine learning techniques are both in the inference layer and the task layer. Our goal is to formalise the interactive use of machine learning system as shared inferences and tasks between a user and a system. Then we propose a decomposition of learning techniques into inference rules and tasks, as well as a description in this formalism of the interactive use of such a system, through argumentative tasks. We try to modelize the "increasing" of the knowledge during the machine learning process, by de ning a representation formalism called mental scheme lying upon a general de nition of a language and a belief management scheme. Our model is illustrated on a toy example of structured objects.
2 The mental scheme The mental scheme is a relation between statements, built to represent knowledge during acquisition. It could be de ned as a graph of which vertices would be statements and arcs would be labeled with beliefs.
2.1 belief
We need to measure knowledge during the learning process. In that way, we de ne a set of beliefs . The idea is to use two order relations de ned on this set to evaluate a belief in both its knowledge meaning and truth meaning. Ginsberg's bilattice exactly matches this property [Ginsberg, 1991].
De nition. A bilattice is a sextuple (B; ; ; ; +; ) where : ^ _
1. (B; ; ) and (B; ; +) are both lattices 2. : B B is de ned as : ^ _
:
!
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:
(a) (b)
=1 is a lattice homomorphism from (B; ; ) to (B; ; ) and from (B; ; +) to itself.
:
2
:
^ _
_ ^
And we de ne a set of beliefs as a bilattice. The minimal set of beliefs is de ned from the lower and upper bounds of each order relation : respectively False and True for the relation, and Silence and Contradictory for the relation. T
K
2.2 Language
For any language, usually de ned from a grammar, we can de ne a partial order on statements in the language. So, we minimally de ne a language as a partially ordered set on statements , and note the order relation , interpreted as \less general than" (or \more speci c than"). L
Our aim is to build acceptable properties from a set of interpretations on a language. An interpretation of a language L is a monotonic application I from L to B, set of beliefs, for one of the order relations on B. When I is such that if a b, then I(a) I(b), we call it a T-interpretation. We can de ne a K-interpretation in the same way. L
T
2.3 Mental scheme
The mental scheme is the knowledge base for our framework. Relations between statements are stored in this table as belief values.
De nition. Let L and L be languages, and B a set of beliefs. A mental scheme is a quadruple (L ; L ; B; ), where is an 1
2
application from L1 L2 to B.
1
2
The expression (a; b) = v is interpreted as if a is true, then b has v for belief value . The silent belief value expresses an undecided knowledge on a relation between two statements, this de nes as an application.
3 Mental schemes as beliefs We will assume that each statement in our base language has a domain that contains all possible values it can take. From this and for each statement e we de ne an associated language, l(e), containing the entire domain of e and the value is-a. Note that, in regard to our de nition of a language, a structured statement has a tree-structured associated language, due to the partial order on it. - III.2.3 -
Let L be a base language and B a set of beliefs, we can build the set of all mental schemes obtained from them : 8 9 [ < l(x) = L1 ; L2 SM(L; B) = :(L1 ; L2; B; ) :L L2 B ; 1
x2L
!
We de ne the set SM(L; B) as a set of beliefs from the two following orders : for two mental schemes SM1 = (L11 ; L12; B; 1) and SM2 = (L21 ; L22; B; 2 ), SM1 SM2 i L11 L21 , L12 L22 and x L11 ; y L12; 1 (x; y) 2 (x; y) SM1 SM2 i L11 L21 , L12 L22 and x L11 ; y L12; 1 (x; y) 2 (x; y) The bounds of these two order relations on SM(L; B) are respectively the following mental schemes false = ( is-a ; is-a ; B; = false), true = (L; L; B; = true) and silence = ( is-a ; is-a ; B; = silence), contradictory = (L; L; B; = contradictory), and we de ne the operators , , +, and from the same operators on B : SM1 SM2 = (L11 L21; L12 L22 ; B; ) with (x; y) = 1(x; y) 2 (x; y) SM1 SM2 = (L11 L21; L12 L22 ; B; ) with (x; y) = 1(x; y) 2 (x; y) SM1 + SM2 = (L11 L21 ; L12 L22; B; ) with (x; y) = 1(x; y) + 2 (x; y) SM1 SM2 = (L11 L21; L12 L22; B; ) with (x; y) = 1(x; y) 2(x; y) SM1 = (L11 ; L12; B; ) with (x; y) = (x; y) We have de ned a bilattice of mental schemes which is usable as a set of beliefs. We express a relation between two statements from the base language with a mental scheme representing the relations between all possible values of these statements.
T
8
2
8
2
T
K
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2
8
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K
f
f
g f
g f
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g
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0
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4 Reasoning in a mental scheme Our idea of a learning process on a mental scheme is to build a kernel of knowledge among it. That kernel is composed of rules and facts with monotonic belief values, that is, their evolutions are monotonic in the meaning of truth (the relation). The main of the learning process is to maintain that kernel, monotonically modifying the belief values in it, as much as possible. T
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Initially de ned from interpretations on a base language building from structured variables, the mental scheme, through reasoning and argumentation mechanisms , computes such a kernel of knowledge, abstracting these datas, and try to maintain it, including new interpretations. When too many facts or rules can't be kept in the kernel because non monotonic truth belief values are necessary, the kernel is destroyed and built again. We de ne those mechanisms. This supposes that we can verify properties on interpretations of the base language. Thus, to compute the result of the deduction color(A) = y, color(x) = y form(x) = z form(A) = z, the property \A is an example of the concept form" needs to be true. We de ne an initial value of the base language as L0 = isExampleOf, isCounterExampleOf and complete the mental scheme on these statements when a new interpretation occurs. !
`
f
g
The results of the reasoning mechanisms are obtained by majority computation to allow exceptions in the mental scheme. The belief value of a rule express both its quality (by its knowledge level) and its truth/falsi cation ratio. At last, we introduce some noise in the results. Each of the mechanisms has its own noisy function. The idea is that we want express that a deduction is a secure operation as vector of information (and the noisy function attached to deduction is identity), that we consider an abduction as a hypothesis and must reduce the computed value, and that the belief in a rule is linked to the number of interpretations used to compute it.
4.1 Reasoning mechanisms
We use three kinds of operations : induction, abduction and deduction.
Deduction spreads knowledge. This mechanism closes the mental scheme, that is, all was possible to learn from datas was. More formally, we talk about consistency principle : each fact is at least as well known as the deduction result on it. We use two deduction methods : modus ponens (a; a b b) and modus tollens ( b; a b a). !
:
!
`
` :
Abduction (b; a b a) is seen as a hypothesis, insecure operation. We apply a strong noisy function on its result, evolving with datas and results correctness. !
`
Induction computes new rules from interpretations. We select only examples of a concept to compute a rule on it (i.e. where this concept is the premise of the computed rule). From the noisy function we have presented above, we obtain
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belief values expressing interpretation heterogeneousness (ratio of examples and counter-example of the rule conclusion).
4.2 Argumentation mechanisms
We have presented in the previous section a learning process on a mental scheme. It de nes a computed apprentice-agent. We now put it into a learning environment containing a human expert-agent able to criticize some argumentation from the apprentice. This apprentice-expert interaction through argumentations and criticisms is made possible by argumentation mechanisms produced by the apprentice in order to control the abduction and induction results. This ensures the relevance of the results. Deduction computes an unveri ed result, due to its vector of information property. To accept an induced rule in the mental scheme, we verify the knowledge growth from it : a rule doesn't increase knowledge if the deductive closure of the mental scheme with this rule is near to the deductive closure of the mental scheme without it. We call an interpretation I, closed , when it veri es the following properties : 1. I( p ) I(p ) 2. p; I( p) = I(p) 3. if (p; q) = V; then I(q) I(p) So, we consider that a mental scheme is closed when all of its interpretations are closed. ^i i
8
K ^i
:
i
:
T
To control an abducted result, we had to introduce the notions of lemma, general objection, and nally, proof. The result, and the proof of the result are submitted to the expert. A lemma is a necessary condition to prove that a conjecture is true (we call conjecture the statement to prove). So, it's an element of the proof. We build a lemma for a given conjecture from the statements veri ed by its examples : let k and c be two statements, we de ne a lemma for k relatively to c as the disjunction of the elements from the domain of c veri ed by the examples of k. A general objection is a sucient condition to prove that a conjecture is false. As a lemma, we build a general objection from the statements falsi ed by the conjecture examples : let k and c be two statements, we de ne a general objection for k relatively to c as the conjunction of the negation of the elements from the domain of c falsi ed by the examples of k. As the examples and counter-examples, we complete the mental scheme on the statements isaLemmaFor isanObjectionFor when a new one is computed. At last, we de ne a proof for k as the conjunction of lemmas and general objections for k. - III.2.6 -
lk c is a lemma for k : c L; c = k, 8
;
2
6
8 < 9 : v2l(c)
9
I is an example of k = v I; and l = I(c) is an example of v ; ok c is a general objection for k : c L; c = k, 8 I is an example of k ^ < I; v and o = : I(c) is a counter-example of v ( ) pk is a proof for k. _
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8
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2
6
9
v2l c
p =
^
k
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k;c ^
9 = ;
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We de ne a proof, in a semi-empirical theory, as a mental scheme : let S be the mental scheme de ned on two base language L1 and L2. S contains our datas, interpretations on the base language L2, and has mental schemes as belief values. We build a proof-mental scheme on the same languages but the second language of the mental schemes-beliefs is de ned as the set of lemmas and general objections of each of the statements of the base language L1. The submission of a proof of the result of an abducted fact (value of the interpretation I(a) for example) to an expert consists in submitting a mental scheme containing the proof of the conjecture and the interpretation I. The answer from the expert is a mental scheme at least as well known as the mental scheme-proof. The proof and the abducted result are accepted if no contradictory occur during the deductive closure of the mental scheme-answer.
5 An example We illustrate these notions on a simple example : consider many geometric forms and a position relation between them. We build two sets of geometric forms : the examples of a conjecture and the counter-examples. We want to discover a generalisation of the examples of the conjecture. Consider the rst item E1.
E1
A B
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We have to describe such an example using values on the two statements
form and position . Each of these has a non empty domain, a lattice, partially
drawn on the two following gures.
position
form ellipsoid
polygon
right
convex polygon
circle
below above
non convex polygon
triangle square The mental scheme is represented as a table which each square stores a relation between two statements as a mental scheme-belief value. So (E1; Form) represents the belief value for the relation between the two statements E1 and Form (form of the example E1) in a mental scheme ( A,B , square,circle,triangle , V,F , ) storing the possible values for A and B, components of E1, to be square or circle. With our formalism, we represent the item \E1 is an example of the conjecture K" on the language containing the statements form and position by the following mental scheme :
f
f
g f
g
g
Form Position square circle triangle above below left right E1 A V A V B B V V This table contains the following informations : E1 is an example of K ((E1; K) =V) E1 is compounded of two elements A and B The form of A is square and its position is above And the form of B is triangle and its position is below Here is our sample :
E1
E2
CE1
A
C
B
D
CE2
CE3
E F
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I G
H
J
K V
It contains two examples of the conjecture K (E1 and E2) and three counterexamples (CE1, CE2 and CE3). Here is the mental scheme representing it :
Form Position square circle triangle above below left right E1 A V A V B V B V square circle triangle above below left right E2 C V C V D V D V square circle triangle above below left right CE1 E V E V F V F V square circle triangle above below left right CE2 G V G V H H V V square circle triangle above below left right CE3 I V I V J V J V From this sample, we can compute the induced rules (K; Form) and (K; Position) de ning the properties of an example of the conjecture K (i.e. what are its form and position). Form Position polygon circle above below left right K X V X V Y Y V That sub-mental scheme is a generalization of the examples of K which does not match any counter example of K. Its belief values are obtained from the sample and the associated language lattices of the statements in the base language.
6 Conclusion As stated in the introduction, our approach is an attempt to de ne a knowledge level model of a learning process, between the works on integration of machine learning and knowledge level modelling [van de Velde and Aamodt, 1992; Graner and Sleeman, 1993] and on the use of inference structures to bias the learning process [Rouveirol and Albert, 1994]. We have focussed on the modelization of the "increasing" of the knowledge during the machine learning process, by de ning a representation formalism - III.2.9 -
K V V F F F
called mental scheme lying upon a general de nition of a language and a belief management scheme. This formalism appears to be rather general. The dynamic aspect of the knowledge acquisition process, involving the learning steps, is carried out through reasoning mechanisms and argumentation mechanisms. The argumentation mechanisms are classical (deduction, abduction and induction), but the argumentation mechanisms are basically cooperative. We hope to have convinced the reader that the usual techniques used to validate a reasoning (closure, proof) become argumentation techniques in the case when knowledge has been produced by a learning technique. Then we have de ned additional items like lemma and objections. We have illustrated our model on a toy example of structured objects. But we think that our reasoning mechanisms can be used as speci cations of how an existing machine learning methods or softwares can be divided into functionnal parts to allow our kind of control to be used. In that sense, our work can be a bridge between learning methods and knowledge acquisition models.
References
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