A Constructivist Approach to Rule Bases

Giovanni Sileno, Alexander Boer and Tom Van Engers

Leibniz Center for Law, University of Amsterdam, Amsterdam, The Netherlands

Keywords:

Rule Bases, Constructivism, Rules Revision, Rules Optimization, Adaptation, Regulated Systems.

Abstract:

The paper presents a set of algorithms for the conversion of rule bases between priority-based and constraint-

based representations. Inspired by research in precedential reasoning in law, such algorithms can be used for

the analysis of a rule base, and for the study of the impact of the introduction of new rules. In addition, the

paper explores an optimization mechanism, built upon assumptions about the world in which the rule-based

system operates, providing a model of environmental adaptation. The investigation is relevant to practical

reasoning, agent modeling and agent programming.

1 INTRODUCTION

Practical reasoning in everyday life is often not based

on repeated deliberation, but on behavioural scripts

constructed and reﬁned by some adaptation process.

Let us consider this simple story: Raphael, who lives

in a rainy country, always checks before he goes to

work whether it will rain, in which case he will bring

his umbrella. Samuel, who lives in a sunny country,

usually goes out without checking the weather. Both

of them believe in taking an umbrella if it rains, how-

ever. Intuitively, if Raphael or Samuel moved from

one country to the other, we would expect that they

would change their habits eventually.

According to traditional optimization theory,

adaptation comes from the agent’s efforts to obtain

a better overall pay-off. In contrast, Heiner’s theory

of predictable behaviour (Heiner, 1983) explains how

behavioural regularities arise in the presence of unre-

solvable uncertainty about the “right” course of action

to follow: being predictable seems to pay off by itself.

Regardless of its explanation, we acknowledge the

existence of a sort of structuring process in the agent,

which results in behavioural rules or scripts that are

followed without consciously reﬂecting on them. The

foundations of agent modeling, agent programming,

and before those, of expert systems, are for the most

based on such behavioural rules or scripts.

A similar analysis can be performed on collective

agencies, whose behaviour is described/prescribed

via formalized artifacts and procedures. Focusing on

administrative organizations, Boer and Van Engers

recognize in (Boer and Engers, 2013) three spheres of

activity: operations, planning/design, policy making.

The three spheres are concerned by different activi-

ties and often utilize different resources. For instance,

people at the operations provide services depending

on the procedures and the resources assigned by the

design department.

For simplicity reasons, this paper divides the

agency/system in only two levels: the regulated (non-

reﬂective) component and the regulatory (reﬂective)

component. In contrast to approaches like machine

learning, theory induction, etc. our agent is not con-

structing the rules that govern his behaviour from the

facts he observes, but he follows the prescription a

given set of rules (cf. agent programming), possibly

making its operationalization more efﬁcient.

Two research questions are mainly addressed.

First, how to revise (the operational knowledge of) a

rule base if a new rule is introduced? This problem is

common in practical reasoning, statutory law and case

law. Horty recently explored in (Horty, 2011) the dy-

namics of normative systems based on common law,

examining phenomena like distinguishing on binding

precedent rules. Inspired by his work, we generalize

the analysis of revision in law to a generic intelligent

system whose operational knowledge is described by

a rule base. Focusing on two ways to represent a

system of rules, priority-based and constraint-based

(§2), we identify and organize the algorithms to pass

from one representation to the other (§ 3).

Then, analyzing the interactions of the evaluation

of rules with default assumptions, we extend the anal-

ysis to consider the rewriting of the rule base due to

new informational commitments, in order to tackle

540

Sileno G., Boer A. and van Engers T..

A Constructivist Approach to Rule Bases.

DOI: 10.5220/0005280205400547

In Proceedings of the International Conference on Agents and Artiﬁcial Intelligence (ICAART-2015), pages 540-547

ISBN: 978-989-758-074-1

 2015 SCITEPRESS (Science and Technology Publications, Lda.)

down the second question: how to reﬁne rules when

certain expectations are ascribed to the environment?

(§4) As the change of informational commitment and

the beginning of the rule revision process are only

asynchronously linked, the coupling is worth further

analysis (§5).

Both the rules that govern the practical behaviour

of the agent and the default assumptions about the

world may change. For these reasons, our work is

afﬁne to the spirit of defeasible logic. Rather than

focusing on a formal framework following this tra-

dition, however, we prefer at this stage to propose a

direct computational implementation.

2 REPRESENTATION TYPES

We summarize some practical deﬁnitions that will be

used throughout the paper.

• A fact f is a true proposition.

• A rule r is a conditional statement relating a

premise with a conclusion. Premise and conclu-

sion are propositions, or expressions of proposi-

tions. They are called also body and head.

• Given a rule r, Premise(r) is a function returning

the set of atomic propositions that should be true

when its premise is true. These components are

also called factors. Conclusion(r) is a function

returning the atomic propositions asserted by the

rule when the premise is true.

• A situation Σ is a set of facts: Σ = { f

, f

, . . . , f

• The applicability condition is expressed by Σ |=

Premise(r).

• A rule base is a set of rules: ∆ = {r

, r

, . . . , r

possibly partially ordered.

Evidently, different rules may share part of the

premises, or stated equivalently, share part of their

domain of applicability. In general, a set of facts can

apply on several rules. Furthermore, when attacking

each other conclusions, rules are potentially in con-

ﬂict. When conﬂicting rules apply, we have the prob-

lem of determiningwhich is the right conclusion. Two

solutions are possible:

• at horizontal level, modifying adequately the con-

tent of rules, i.e. redeﬁning the constraints of ap-

plicability of mutually conﬂicting rules;

• at vertical level, deﬁning relations of hierarchy or

priority between rules.

Two representational attitudes are directly related

to those approaches: constraint-based and priority-

based.

2.1 Constraint-based

A rule base written following a strict constraint-based

approach is a non-ordered set. The relative position

of rules (e.g. the position in the artifact describing the

regulatory system, the chronological time of creation,

etc.) is not relevant. A rule is required to transport all

what is needed to check its applicability, i.e. all the

relevant factors have to be made explicit.

On the down side, however, the complete set of

rules has to be (maintained) consistent as a whole.

The problem can be explained referring to the pro-

gramming construct switch...case, which separates

execution paths depending on a certain condition:

switch (condition) {

case value : [...]; break;

[...]

}

In order to be functional, the condition variable has to

index precisely one value position in the list.

As the application of a rule is triggered by the rel-

evant factors in the premise being true, the most basic

encoding is built upon the permutations of all factors

described in the rule base. If the rule base is sensitive

to N factors, there are 2

situations to be encoded.

Such full-tabular encoding implies that the requests

for a rule are inﬂected also in terms of factors which

are irrelevant for the rule in itself, but are relevant to

other rules. Other more efﬁcient encoding are possi-

ble.

2.2 Priority-based

In all acts of communication there is an intrinsic

strictly ordered dimension, usually referred to as the

discourse. It can be temporal, as in the case of inter-

actions; or positional, as with texts.

Priority-based representations take advantage of

the meta-information intrinsic to the composition of

content to reduce its redundancy. In programming,

something similar occurs with the construct if...else:

if (condition1) [...];

else if (condition2) [...];

else [...];

In order to avoid to repeat the same calculations, a

good programmer implements the conditions to be

checked in a convenientway, conscious of the sequen-

tial processing of the code. For instance, he knows

that if condition2 is being evaluated, condition1 is

necessarily false.

Considering such structured enchaining in the

evaluation of rules, each new evaluation could remove

the conjunction of factors which have been already

AConstructivistApproachtoRuleBases

541

computed. The resulting rewriting of the rules is more

efﬁcient(in size terms) than the original version. Con-

versely, the application of a rule may require more

computational effort than a constraint-basedrepresen-

tation, as the applicability must be strictly checked

following the given ordering.

So far, we have considered priority as directly re-

lated to ordering (positional or chronological). In

such conditions, if we introduce a new rule with a cer-

tain domain of application, we overwrite all existing

rules whose domains subsume that of the new rule.

There are situations in which it is useful to avoid this

mechanism. This occurs for instance in law with the

principle called lex specialis derogat legi generali: a

more speciﬁc law overrules the more general. Princi-

ples like this create meta-rules: their conclusion is not

a fact, but a priority constraint. In the following we

suppose that we are always able to apply meta-rules

to obtain the relative priority between rules.

3 INCREMENTAL DYNAMICS

Depending on the underlying representation, inter-

ventions on a rule base require a different attention:

• in a constraint-based rule-base, we need some

kind of rule revision process (Horty calls it reﬁne-

ment) on the rules whose applicability is affected

by the addition, so that after the update each rule

is again consistent with the whole, and can be also

read separately;

• in a priority-based rule-base, the addition of rules

or modiﬁcation of priorities is straightforward;

however, we cannot infer the resulting practi-

cal consequences without taking into account the

whole rule-base.

Our objective is therefore to provide a conversion

method from one representation to the other, in order

to cope with the respective limitations. While the nat-

ural way of handling the incremental construction of

a rule-base is via priority-based representations, the

more direct way to access the actual applicability of a

rule comes from a constraint-based representation.

From Compound to Simple Rules. In the follow-

ing, we will often refer to simple rules, i.e. rules

with no disjunction in the body, nor conjunction in

the head. They can be seen as the atomic components

For instance, the implicit ordering of rules with fac-

tors expressed at different levels of generalization can be

obtained via Pearl’s system Z (Pearl, 1990).

of the conditional. Compound rules can be reduced to

simple rules applying the following properties:

(a → c∧ d) ⇔ (a → c) ∧ (a → d) (1)

(a∨ b → c) ⇔ (a → c) ∧ (b → c) (2)

Relevance. Given two simple rules r

and r

, we

say that r

is relevant in respect to r

, if the propo-

sitional content of their conclusions are positively or

negatively aligned: Conclusion(r

) = Conclusion(r

)

or Conclusion(r

) = ¬Conclusion(r

Illustrative Example. Let us assume the existence

of a principle like lex posterior derogat legi prior,

i.e. the last rule has higher priority than the previ-

ous ones. Consider a rule base consisting of just one

rule r

: A → P. With one factor (A), we can have two

situations, but our rule refers only to one of them. The

associated representations are:

Constraint-Based

Priority-Based

A → P A → P

[¬A → ?]

At this point, we add a second rule: r

: B → ¬P.

Considering two factors (A, B), we have four possi-

ble situations. In a constraint-based representation,

we have to reﬁne the previous rule in order not to lose

consistency, as r

and r

have opposite conclusions,

and therefore may be in conﬂict. Removing the do-

main of applicability of r

from the domain of r

obtain the following representations:

CB (full-tabular)

CB (intermediate) PB

A∧ ¬B → P A∧ ¬B → P A → P

A∧ B → ¬P B → ¬P B → ¬P

¬A∧ B → ¬P

[¬A∧ ¬B →?]

The table distinguishes two types of constraint-

based representations: full-tabular, where the con-

ﬁguration of all factors is written explicitly for each

rule, and intermediate, where redundancy is reduced

via boolean simpliﬁcation.

We follow this illustrative example to propose

some algorithms to transform one representation into

another. This is sufﬁcient to handle the problem of in-

cremental construction. Given a constraint-based rule

base ∆, the reﬁnement required for the introduction

This is a “narrow” deﬁnition of relevance. In general,

given a rule, we care also of the rules whose conclusions are

premise for the rule, etc.

For completeness, we use the expression → ? to iden-

tify the lack of consequence, for situations which do not

imply any conclusion.

ICAART2015-InternationalConferenceonAgentsandArtificialIntelligence

542

of a new rule r

is equivalent to the conversion of the

priority-based rule-base ∆+{r

}, where r

has higher

priority than all other rules.

3.1 From Priority-based to

Intermediate Constraint-based

Representation

Suppose we have a strictly ordered set of simple rules

∆ = {r

, r

, .., r

}, where r

is the rule with lowest pri-

ority, r

with the highest. The following algorithm

overwrites the elements of ∆ with their translation in

a constraint-based intermediate representation:

for each r

∈ ∆, starting from r

down to r

for each r

∈ ∆, starting from r

i−1

down to r

if Conclusion(r

) = ¬Conclusion(r

) then

Premise(r

) ∧ ¬

Premise(r

)

→ Conclusion(r

)

Stated in words, each cycle adds the complement

of the domain of applicability of the rule with higher

priority to the rule with lower priority.

The negation in the core formula usually applies to

sequence of multiple and expressions, so it is equiv-

alent a sequence of or expressions. Using the body

disjunction formula (2) we can divide such compound

rule in its simple components, and some of those may

be redundant with other rules in the rule base. This is

not the only potential cause of redundancy: for histor-

ical or clarity reasons, redundant rules may have been

introduced already in the priority-based account (ex.

: A ∧ ¬B → P). As a consequence, the previous

algorithm may produce subsumed or cloned rules

which can be removed without harm.

3.2 From Intermediate to Full-tabular

Constraint-based Representation

The same regulation can be described with several in-

termediate representations. A way to compare them

is to fully compute their extensional meaning, i.e.

the underlying full-tabular representation. In order to

do that, we can expand each rule with the unspeci-

ﬁed factors which are relevant (i.e. are part of the

premises) for relevant rules (i.e. having the same con-

clusion or its complement) in the rule base.

For instance, in the illustrative example, r

and r

are relevant to each other, and refer to the two factors

{ f

, f

, ..., f

} ⇔ f

∧ f

∧ ... ∧ f

Given two rules r

and r

such that Conclusion(r

) =

Conclusion(r

), if Premise(r

) ⊇ Premise(r

) then r

sub-

sumes r

; if Premise(r

) = Premise(r

) then r

clones r

A and B. However, r

: B → ¬P does not explicitly

refer to the factor A, but it can be rewritten as two

full-speciﬁed rules using some boolean properties:

B → ¬P ⇒ (B ∧ A → ¬P) ∧ (B∧ ¬A → ¬P)

If a rule has n unspeciﬁed factors in the constraint-

based intermediate representation, then it is written as

rules in the full-tabular representation.

3.3 From Full-tabular to Minimal

Constraint-based Representation

Our objective at this point is to ﬁnd a way to produce

a minimal intermediate form, i.e. removing all redun-

dancy. The solution we propose is to pass by the full-

tabular representation and then simplify using known

algorithms for boolean simpliﬁcation.

We have a set of rules ∆ = {r

, r

, .., r

}, repre-

sented in a full-tabular constraint-based way. Its char-

acterization can be visualized using a table similar to

a truth table. The only difference is that we are not

interested in showing the speciﬁc truth value of the

output variable for all combinations of inputs, but in

mapping where speciﬁc outputs are deﬁned. Consid-

ering the illustrative example, we have:

B P ¬P ?

T T X

T F X

F T X

F F X

To reduce the implicit redundancy of the full-

tabular representation, we can apply then the same

principle of the minterm expansion used for logic cir-

cuits, in order to compute the characteristic function.

First, we aggregate all rules which share the same

conclusion φ into seed rules r

for each r

∈ ∆, with r

6= r

φ := Conclusion(r

)

if r

is not set then r

:= r

else r

Premise(r

) ∨

Premise(r

) → φ

Second, we apply an algorithm for boolean reduc-

tion — e.g. Quine-McCluskey (Mccluskey, 1956)

—

to reduce each r

to its minimal form.

These algorithms take as input the logical addition

(disjuction, or) of the products (conjunction, and) of

factors producing a target φ. The output is a rule r

′

which may be a compound rule. In this case, it can be

The Quine-McCluskey algorithm provides an optimal

solution, but is costly in computational terms. There are

heuristics which provide suboptimal solutions consuming

less computational power exist, e.g. Espresso.

AConstructivistApproachtoRuleBases

543

divided in its components r

′

φ1

, r

′

φ2

, ... using the body

disjunction formula.

Finally, considering both positive and negative

characterizations of a certain conclusion φ, we obtain

∆

= {r

′

φ1

, r

′

φ2

, . . . , r

′

¬φ1

, r

′

¬φ2

, . . .}, which is by con-

struction the minimal constraint-based representation

of the φ-relevant component of the initial rule-base.

3.4 From Constraint-based to

Priority-based Representation

Suppose we have a set of rules ∆, given in a minimal

constraint-based representation. In order to obtain

an associated priority-based representation we need

some partial ordering between the rules. Such order-

ing can be: (a) hard-coded, i.e. provided as input,

(b) obtained by some evaluative function which takes

the rules as inputs (§ 4). In the following we suppose

the ordering is already available; for simplicity, we

assume that the input priorities constructs a strict or-

der on the rule set (we label r

the rule intended to be

with the highest priority, r

with the lowest one). The

following algorithm performs the transformation of ∆

according to such priority labeling:

for each r

in ∆, from r

to r

φ = Conclusion(r

)

factors := RelevantFactors(∆, φ)

if notYetEvalSituations

is not set

notYetEvalSituations

:= Allocate(factors)

if establishedFactors

is not set

establishedFactors

= ∅

create two empty rule bases ∆

and ∆

′

add r

to ∆

convert ∆

to full tabular considering factors

for each r

∈ ∆

if Premise(r

) ∈ notYetEvalSituations

newPremise := Premise(r

); apply := true

for each f ∈ establishedFactors

if ¬ f ∈ Premise(r

)

apply := false; break

else if f ∈ Premise(r

)

newPremise := newPremise\ { f}

if apply∧ newPremise 6= ∅

′

newPremise → φ; add r

′

to ∆

′

notYetEvalSituations

\ Premise(r

)

establishedFactors

ExtractFacts(notYetEvalSituations

)

apply Quine-McCluskey on ∆

′

obtaining r

′

The general principle of the algorithm is to cre-

ate clusters of rules depending on the conclusion, and

to synthesize the reciprocal dependency only within

these clusters. We leverage the following information:

which factors are relevant, considering the whole rule

base, in respect to the conclusion [Relevant(∆, φ)]; the

set of all possible situations, via the allocation of a

truth value to such relevant factors [Allocate(·)]. Ex-

panding the rule to its full-tabular components, we

prune the situations which are evaluated, and we ex-

tract the set of factors which have been established on

the not yet evaluated situations [ExtractFacts(·)].

4 INTEGRATING ASSUMPTIONS

In a constraint-based representations all relevant and

discriminatory factors are made explicit. In an opera-

tional setting, however, it is difﬁcult and not efﬁcient

to evaluate all of them: the check of conditions may

require external investigations.

Following optimization principles, it is important

to put in the beginning checks on conditions which

cost less (more in general, have better pay-off) and

provide more discrimination in the set of possible ex-

ecution paths (i.e. are more informative, in Shannon’s

terms). This means that knowledge about the world

and of the operational mechanisms of the system do

provide elements to create a priority in the evaluation.

Consider for instance this example:

if (f()) [...];

else if (g()) [...];

else [...];

This implementation would be really poor if

f()

very costly and often false, while

g()

is not computa-

tionally expensive and often true.

4.1 Payoff Analysis

In general, the expected payoff in performing an ac-

tion can be decomposed as:

E[payoff] = p(success) · E[payoff of success]

+ p(failure) · E[payoff of failure]

Actions are expected to have some side effects, which

may occur at symbolic level (e.g. acquiring new in-

formation) and at practical level (e.g. modifying the

physical environment). The second aspect may be

critical in certain settings. For instance, trying to get a

certain evidence often brings the risk to destroy other

evidence. Such side effects modify what the agent

could do in the following step. For simplicity, we ne-

glect this dynamic component in this work. Consider-

ing a (non invasive) action, the most important voice

An implementation in Java/Groovy of the algorithms

is available on our site. http://justinian.leibnizcenter.org/

rulebaseconverter

ICAART2015-InternationalConferenceonAgentsandArtificialIntelligence

544

related to side effects remains a quantiﬁcation of the

expected cost.

Suppose that the action is an investigation. The

success corresponds to reach a certain conclusion

about φ. Assuming the cost is known and the same

for both success and failure, the previous equation be-

comes:

E[payoff] = p(success) · E[payoff of concluding φ]

+ p(failure) · E[payoff of non concluding φ]

− cost

The investigation is worth if E[payoff] > 0, i.e. if

E[payoff of concluding φ] >

cost − (1− p(success)) · E[payoff of non concl. φ]

p(success)

In the case in which no positive or negative out-

come is entailed from not concluding φ, the previous

constraint is simpliﬁed to:

E[payoff of concluding φ] >

cost

p(success)

(3)

4.2 Rule Guided Investigation

Suppose our knowledge is described in the form of a

rule base. Let us consider a target matter φ. We know

that the factors concerning φ, i.e. the factors rele-

vant for rules implying φ or ¬φ, are the reasons which

possibly allow us to conclude φ or ¬φ. In Shannon’s

terms, they provide the dictionary of symbols which

should be possibly extracted from the informational

source for our investigative purposes.

4.2.1 Individual Evaluation

Suppose we have a rule whose conclusion is φ, be-

longing to a rule base ∆ described in a constraint-

based representation. The probability that the state

of the world satisﬁes the domain of applicability of r

is given by:

p(applicable(r)) = p(

Premise(r))

An action on investigation performed following a cer-

tain rule is successful if it is able to bring to the in-

tended conclusion φ. Therefore, accounting the set of

known facts K:

p(success) = p(

Premise(r)|K)

Going further, the function quantifying the informa-

tional cost can be written in the form c(D, K), where

D is a set of factors we need to investigate, and K is

the set of known facts.

cost = c(D, K)

Thus, the rule r should be evaluated only if:

E[payoff of concluding φ] >

c(D, K)

Premise(r)|K)

4.2.2 Optimization

In general, the rule base may contain several rules

which entail φ. The choice about which rule should

be evaluated ﬁrst is optimal if the rule maximizes the

investigative payoff.

The payoff of reaching a conclusion about φ is in-

dependent of the rule used, therefore can be labeled

as a constant:

G = E[payoff of concluding φ]

Furthermore, in respect to a rule r, the set of desired

factors D consists of the factors in the premise which

are not known:

D = Premise(r) \ K

Neglecting the payoff of not concluding φ, we have:

E[payoff] = G· p(

Premise(r)|K) − c(D, K) (4)

The next paragraphs consider a few simple conﬁgura-

tions.

Equal Probabilities. Suppose that there are n fac-

tors relevant for φ, statistically independent and with

equal probability. Supposing c(D, K) negligible, we

have:

E[payoff] ∝

Suppose also we are ignorant about the world, there-

fore #D = #Premise(r). In a full-tabular constraint-

based representation, all rules have the same num-

ber of premises, therefore the expected payoff is

the same for all rules. Conversely, in a minimal

constraint-based representation, rules have been con-

structed by functional aggregation reducing the num-

ber of premises. Therefore the optimal choice is to

choose the rule with less conditions in the premise.

Unequal Probabilities. In condition of ignorance,

and with statistically independent factors, we have

p(Premise(r)) = p( f

) · p( f

) · . . . · p( f

)

At this point, it is interesting to refer to the theory

of communication introduced by Shannon (Shannon,

1948). Deﬁning the information related to a certain

symbol s as:

I(s) = −log

(p(s))

the information of a message m composed of different

and statistically independent symbols (s

, s

, ..., s

)

is given by the sum of information of each symbol:

I(m) = I(s

) + I(s

) + . . . + I(s

)

AConstructivistApproachtoRuleBases

545

Thus, assigning each factor to a symbol, the premise

of a rule r can be interpreted as a message. As infor-

mation maintains monotonicity, we can consider the

best rule to be chosen not in terms of maximizing the

product of probabilities, but as minimizing the sum of

information required to reach the target conclusion,

possibly integrating the cost component.

4.3 Default Assumptions

Let us consider again the story reported in the intro-

duction. Both Samuel and Raphael think it is normal

to take the umbrella if it is raining. Simplifying

, we

can write this practical rule as:

rain → umbrella

With no prior assumptions, the rule base of both

agents consists of a simple rule. Now, suppose the

agents have a probabilistic model of the world. Ac-

cording to (4), the payoff of the evaluation of that rule

is equal to:

E[payoff] = G· p(rain)− c({rain}, K)

If the agent already knows the fact that it will rain,

supposing retrieval costs negligible, the overall cost c

is null. In such case, the payoff of the investigation is

always positive. It tends to 0 when the agent thinks

that the probability of rain is very low, as in the case

of Samuel, living in a sunny country: p(rain) ∼ 0.

However, if Samuel’s knowledge base does not

contain the “rain” fact, the expected payoff of the in-

vestigation may be negative: if G · p(rain) is small

enough, the payoff constraint is violated, so the rule

shouldn’t be evaluated.

Raphael is in the opposite

condition: as p(rain) is not null, the positive addend

in the expected payoff of investigation makes the cost

of checking the weather negligible.

4.3.1 Prioritization with Default Rules

The analysis of evaluation payoffs provides an op-

timal order of investigation and such ordering can

feed the conversion from constraint-based to priority-

based representations (§ 3.4). However, if the ex-

pected payoff is null or negative, the related investi-

gation is not worth and the agent should rather con-

sider to introduce a default assumption rule. A simple

We put aside all issues related to causality, intentional-

ity, action, etc.

The payoff G of concluding the fact that it rains can be

interpreted in terms of the practical consequences that such

conclusion entails. For instance, the agent will avoid to be-

come wet, by taking the umbrella. How much this is valu-

able to the agent depends on subjective components, and

plausibly changes in time.

way to represent that is to introduce the negation as

failure operator, as deﬁned for instance in Answer Set

Programming (Baral and Gelfond, 1994; Lifschitz,

2008). In practice, when we commit to a default as-

sumption concerning the fact f , we add to the rule

base the following rules:

(not¬ f → f) ∧ (not¬ f ∧ not f → ¬ f)

Note that there is an implicit priority between the

two rules, cf. system Z (Pearl, 1990). Such interven-

tion does not modify the previous rules, but overrides

the investigations concerning f. In other words, they

force the agent not to perform the investigation but

just to refer to his current knowledge, so that the pay-

off constraint is not violated.

5 CONSTRUCTION AND

RECONSTRUCTION

The paper investigated two types of events changing a

rule base. The ﬁrst type of events consisted of incre-

mental modiﬁcations: new rules are added by external

intervention, determining a partial reconﬁguration of

the operationalknowledgeused by the agent. Because

of distinguishing actions, the new rules brings to the

foreground factors left implicit in the previous rules.

The second family of events concerned ad-hoc re-

organizations, aiming for better adaptation. However,

when a rule base is “compiled” to a more efﬁcient

priority-based form, the agent loses the reasons moti-

vating that structure (e.g. probabilistic assumptions),

and therefore he cannot check if those reasons are still

valid.

To rewrite the rule base again, the agent has to re-

ﬂect over the rule base. He has to unveil the underly-

ing constraint-based representation, removing all de-

fault assumptions and recompute the optimal priority-

based indexing.

What may motivate the agent to do that? An in-

tuitive answer is because of (repeated) failures of his

current practical reasoning. For instance, if Samuel

moves in a rainy place, his default assumption about

the rain will lead him to get wet often. If the num-

ber of practical failures exceeds a certain (subjective)

threshold, we expect the agent will look for a better

adaptation to the context, asynchronously starting the

reﬂection cycle.

There may be more radical transformations. For in-

stance, destructive simpliﬁcations can be imagined remov-

ing f from the premises of the rules, removing from the rule

base rules which have ¬ f in the premises, etc.

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6 CONCLUSION

Part of a wider research concerning agility in pub-

lic administration and policy making (Boer and En-

gers, 2013), the present paper starts the development

of a computational framework operationalizing a con-

structivist approach to rule bases. By dividing agency

in a regulated and a regulatory sub-systems, we ex-

plicitly disjoin the processing of facts, depending on

the rule base, from the modiﬁcation of the rule base.

The analysis we presented is not targeting

beliefs—in the traditional sense of the belief revision

literature, e.g. (Ferm´e, 2011)—norbuilt upon a model

of dynamic theory revision of knowledge accounting

for both facts and rules, as in machine learning, see

e.g. (Omlin and Giles, 1996), (Goldsmith and Sloan,

2005). Similarly to Horty (Horty, 2011), our speciﬁc

scope is on rules, as components of a rule base, al-

ready deﬁned at symbolic level.

In psychology, the theory of constructivism is tra-

ditionally related to Jean Piaget (Piaget et al., 2001),

who investigatedthe mechanisms under which knowl-

edge is internalized by learners. He argued that in-

dividuals construct their knowledge through the two

processes of:

• assimilation: the process of framing new experi-

ences through the existent knowledge framework,

without changing it; the structure exists, it is ﬁlled

by data;

• accomodation: the process of re-framing the

agent’s knowledge framework, usually respond-

ing to contradictions or operational failures of

their knowledge framework; the structure is reor-

ganized.

This theory is aligned with our contribution, as:

• rule bases are interpreted as compiled programs

for reactive symbolic processing modules, which

respond to facts (e.g. data fed by sensors), pos-

sibly performing actions (e.g. sending stimuli to

actuators, communicating, etc.);

• construction and reconstruction of rule bases pro-

vide the reﬂective, adaptive processing dimen-

sion, which occurs concurrently to the ﬁrst one.

The paper is a starting operationalization, there-

fore several research directions remain to be inves-

tigated. First, an evaluation of the application of

our proposal on more complex rule bases, possibly

in comparison with other approaches from the expert

systems literature. Related to that, an in-depth analy-

sis of the proposed algorithms is required, measuring

their complexity, and suggesting possible optimiza-

tions. Second, an investigation of possible theoretical

interactions with the insights coming from belief re-

vision, e.g. (Dubois, 2008), but applied on our deﬁni-

tion of rule revision.

Finally, the paper explores adaptation only as a

problem of maximization of payoffs, studied in deci-

sion theory, game theory and similar top-down per-

spectives. In the future, we plan to analyze it through

the lens of Heiner’s theory of predictable behaviour

(Heiner, 1983); in this way, we expect to be able to

model adaptation as a bottom-up mechanism as well.

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