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A logical approach to problem representation
Journal of Economic Dynamics and Control 14 (1990) 451-464. North-Holland
A LOGICAL APPROACH TO PROBLEM REPRESENTATION Paul BOURGINE* CEMAGREF, 92160 Antoy, Frunce GRASCE. 13627 Aix-en-Provence, Chdex I, France
Received February 1989, final version received October 1989 The cognitive sciences distinguish between symbolic and nonsymbolic problems. Here we consider problems that can be represented in terms of symbolic statements. We propose to represent a symbolic problem as a pair comprised of a partial structure and a theory, thereby situating the issue in the domain of trivalent logics and model theory. More specifically, the solution of a problem is a matter of finding the models of the theory which extend the initial partial structure. Such an approach makes it possible to see the solution of a problem as a process of reformulation whereby reasoning progresses in leaps and bounds. It provides a powerful tool for the symbolic description of economic agents’ decision-malting processes.
1. Introduction Economics has been based for quite some time now on the hypothesis of homo oeconomicus. But the postulate of expected maximum utility, which is
generally taken for granted, can hardly account for the large variety of description models and representations that economic agents use in reasoning about the problems they perceive. According to Simon (1988), if we want to solve problems, it is a good idea to have a good representation of them. He thus thinks that a theory of problem representation is essential to research on decision making. The goal of the present article is to set forth such a theory. Recent economic doctrines have tried to correct the image of homo oeconomicus by borrowing certain features from homo cogitans. They try to take the agent’s mental representations and computations over them explicitly into account. Muth (1961) and Lucas (1972) take a first step in this direction by bringing expectations into the picture. Stigler’s search theory (1961) takes another step by noting that optimization presupposes the cost of the search for information and calculation. Finally, with Simon (1976), homo oeconomicus is a full-fledged homo cogitans having limited cognitive capacities and decisions to make in complex environments. Modem economic theories do not only *Thanks to Jean-Louis Le Moigne for his comments on an earlier draft of this paper, and to Paisley Livingston for his translation. 0165-1889/90/$3.5001990,
Elsevier Science Publishers B.V. (North-Holland)
P. Bourgine.
452
A logical approach
to problem
representation
concern instrumental rationality, for they also take into account the cognitive rationality of economic agents [e.g., Walliser (1989)]. The very description of agents’ behavior becomes more and more complex as we try to produce more and more accurate models of the agents’ observations of economic phenomena. As u result, we should expect the theory of problem representation to guarantee its homogeneity by adopting a very large framework. This is why we have chosen to make use of the branch of mathematical logic known as model theory. Model theory [e.g., Chang and Keisler (1973)] studies the relations between structures and theories. Structures are defined in terms of sets, functions, and relations (thought to represent a ‘perceived reality’ or a phenomenology); they belong to a semantic universe. Theories (thought to represent a constrained or intended reality) are defined as statements contained within a formal language and belonging to a syntactic universe. In the next section, we survey the main cognitive approaches to problem solving. A distinction between symbolic and nonsymbolic problems emerges. In section 3, we use some examples to set forth the fundamental hypothesis of this article, which is that a symbolic problem can be represented as a pair of a partial structure and a theory (in an appropriate logical system). In the fourth section, we develop a more rigorous formulation of this intuition and discuss its advantages. 2. Cognitive sciences and problem solving A problem arises in the space between the reality that an economic agent perceives and the reality that he wishes to realize [Le Moigne (1977)]. When the difference between the two is identified, the agent begins to look for ways of eliminating it. The cognitive sciences distinguish in general between three phases of this search procedure: problem finding, problem setting, and problem solving. Different authors emphasize different phases: _ Newell and Simon’s (1972) contribution to cognitive psychology focuses on how subjects solve symbolic problems (fig. 1). _ Gestalt psychologists look at the search process as a series of reformulations of the global mental representation of the problem, ending when a fully structured solution appears (fig. 2).
perception 4 action
-
distance
statement b
finding 4
solution
setting
solving
-
Fig. 1. Processes of problem
formulation
-
and solving.
P. Bourgine, A logical approach to problem repre.sentution
perception A finding 4 action -
distance +solution
Fig. 2. Processes of problem
453
(re’ setting
reformulation.
_ Both approaches postulate the existence of a phase in which the problem
receives an explicit symbolic formulation. Varela (1987) on the other hand considers a class of problems that subjects identifv and solve instantaneously without going through a phase of formulation (fig. 3). Such a process is viable because of the large knowledge that subjects have acquired thorough learning, and that they eventually ‘share’ (common knowledge). In cases where the problem can be solved without a stage of formulation, we will speak of a nonsymbolic problem. It must be noted that each of these theories share certain presuppositions with the others: solutions based on statements necessarily involve a formulation of that statement; all forms of solution presuppose that there is a finding and presenting of the problem and that the problem can either be ‘representable’ or not. The movement from one theory to the next (see our three figures) is, then, a matter of differentiation. The three theories are largely complementary. They hold for observational data pertaining to complementary time scales: less than l/10 second for Varela, from l/10 second to l/2 hour for Newell and Simon, and for even longer periods for the processes of reformulation studied by Gestalt psychologists. Thus a single process of problem solving may very well be described as the succession and overlapping of periods interpretable in one of the complementary theories. Some typical examples may help us to understand the differences between these diverse classes of problems. Varela (1986) presents driving an automobile in a city as a nonsymbolic problem: when we are driving, we deal with most of the difficulties presented by traffic in less than l/lOth of a second. This is even an essential condition of a proper traffic flow! Simon (1988) gives examples of other problems that would be difficult to represent: ‘The challenge presented by Japanese and other Far Eastern competition is on the agenda; the problem is on the agenda, but finding an appropriate problem representation is a more difficult task.’ The same holds for the problem of unemployment.
Em Fig. 3. Process
nonsymbolic
solving.
454
P. Bourgine, A logical approach fo problem representation
At the end of this article we shall return to the question of the distinction between symbolic and nonsymbolic, representable and nonrepresentable problems.
3. Examples of modeling of economic reasoning In this section various examples of situations are presented. We clarify the semantics of problems and propose a representation of problems in an adequate logical system, the goal being to show that many economic reasonings can be modeled symbolically once we have logical systems adapted to them.
3.1. Reasoning
about and with bounded resources
Economic reasoning deals with situations in which resources are limited. Moreover, the cognitive resources that agents use in these reasonings are themselves limited. In general, economic theories try to describe such situations by using quantitative models of the agent’s preferences. In what follows, we propose a qualitative model based on linear logic [Girard (1987)]. The approach to be adopted may be particularly well suited to reasonings that involve limited cognitive resources. To understand the meaning of the connectors of linear logic, we may refer to an example having to do with limited physical resources [Lafont (1987)]. Menu for $20 Starter:
Ham or grapefruit, according to the season
Main course:
Sirloin steak
French fries:
All you can eat
Cheese or dessert
The translation of this menu in linear logic is as follows: $20
--0
(ham @ grapefruit) 0 steak 8 !french-fries (cheese & dessert)
In linear logic, a formula A does not represent a stable situation in which there is a truth that has been settled once and for all subsequent situations. Instead, the formula represents an action that, when realized, gives rise to a certain number of consequences while depleting a certain amount of resources.
P. Bourgine, A logical approach
toproblemrepresentation
The inference rule of connector --o corresponds classical logic: A
455
to the modus ponens of
A --oB B
*
In the case of the menu, the interpretation specifies that ‘if I have $20, I can have the menu (but will no longer have my $20)‘. The expression !A means that I still have A after I have used it (A then is an unlimited resource). The connectors @ and 0 represent a choice: A 0 B means I can have either A or B (but not both), and the choice is mine; A $ B also means I can have either A or B (but not both), but the choice is not mine. 3.2. Qualitative reasoning with order of magnitude Qualitative reasoning was initially introduced in order to model certain forms of reasoning in physics [e.g., De Kleer and Brown (1984)]. For an application of it to economics and for an introduction to a reasoning with order of magnitude, the reader may consult Bourgine and Raiman (1987). We illustrate this kind of modeling here by means of an example of ‘temporary supply shock’ taken from Barro (1984, pp. 130, 262). The ‘ . . . ’ indicate that other factors are left out, and the signs are those of the partial derivations.
where Ys CD ID W P r
= supply of commodities, = consumption, = net investment, = nominal wage rate, =price, = real interest rate;
(b)
LD( ~~~,...) =Ls(~~;;~: ,...),
where LD = demand of work Ls = supply of work; ’
456
cc>
P. Bourgtne. A logtcal upprouch to problem representution
M/P=H
Y
,
R (-)
i (+I
,
where M = aggregate amount of money, R = nominal interest rate. To describe a change of equilibrium (if there is one), we introduce relations of order of magnitude A Vo B (read: A is close to B) and A Ne B (read: A is negligible to B). These relations are easily interpreted in terms of infinitely smalls (designated by ‘0’ below) of nonstandard analysis [Robinson (1966)]: AVoB=A=(l+o)B, ANeB=A=oB. A change
of equilibrium
satisfies the conditions:
AYSVo(ACD+AID),
ALDVoALS,
AM/PVoAH(Y,
R).
a temporary supply shock amounts to a modification For Barro, A,Ys, A,CD, A,ZD of the curves such that A,CD, A,LD are negligible before A,Ys and the sign of A,Ys is negative ([xl represents the sign + , - , 0, ? of x). But the change in the curves of the equilibrium leads to variations (noted as A,) in all the magnitudes: [A,YD]
= -,
A,CD NeA,YS.
A,lD NeA,Ys.
A,Ys+ A,Ys Vo ( A,CD + A,CD + A,lD + A,ZD), A,LDVoA,LS,
A,M/PVoA,H(
Y, R).
A qualitative reasoning with order of magnitude assigns meaning to the variation of certain variables of the new equilibrium and leaves undetermined the meaning of the variation of the other variables [Bourgine and Raiman (1987)]:
[r] = [A,Ys] [W/P] [A,M/P] The interest
= +.
= [A&‘“]
= [A,ID]
= [A,R]
of qualitative
= [A,LD]
= [A,L’]
= -,
= ?.
reasoning
with order of magnitude
is double.
On the
P, Bourgine, A logical approach
to problemrepresentation
457
one hand, it makes it possible to arrive at certain consequences even if the laws are quantitatively poorly known, provided that there is a consensus about the qualitative properties of these laws and of the orders of magnitude. On the other hand, the reasoning remains wholly intelligible and close to the reasoning done by economic experts. 3.3. Common knowledge and the emergence of conventions and institutions By introducing the expectations of the agents, recent economic theories have given up working with a single equilibrium. The determination of a particular equilibrium is not the result of a purely economic mechanism, but must be explained in terms of such sociological mechanisms as conventions or institutions. Thus it becomes necessary to explain in turn how conventions emerge. Following Lewis (1964), the role of common knowledge (CK) is given an essential role in this process. To illustrate the emergence of conventions, we borrow an example from Dupuy (1989). In table 1 a game is given defined in terms of the payoff matrix for two players. This game has two Nash equilibria, uA and bB. But both of them require the two agents to coordinate their strategies so as to avoid the disastrous cases aB and Ab. This coordination problem may be understood in cognitive terms, for it is easily solved if each agent wants to coordinate his action with the other agent, and if each agent knows that the other wants to realize such a coordination, and knows that the other wants to realize it, . . . Thus we have a figure of infinite specularity that can be represented by introducing a new knowledge operator, KOthcr, defined as
The Other knows proposition P if and only if all agents know that the Other knows P. In that case we say that proposition P is CK (common knowledge). If we now designate as P the proposition ‘every agent knows that the other wants to coordinate his actions with those of the first agent’, the fact
Table 1 Player 1 Action LI Action a
Action 10
10
b
0 0
Player 2 Action
b
10
0
0
10
P. Bourgrne, A logud
458
upproach to problem representation
that P is CK will have a stabilizing influence, making it possible to solve the coordination problem. A possible interpretation of the game’s structure is found in the example of driving automobiles on the right or left side of the road: the two conventions are equivalent, yet the choice between them is necessary. Once one of these conventions has been adopted - such as driving on the left - it is obvious that driving on the left is facilitated if the proposition ‘everyone drives on the left’ is CK. The three examples have the goal of showing that there is a link between a certain kind of economic reasoning and the logical system in which this reasoning may be realized. Many contributions to the modeling of reasoning could also be cited, but we will limit ourselves to a few references useful to the modeling of problems confronted by economic agents; for example, reasoning related to situations and attitudes [Bar-wise and Perry (1983)], to belief change [Shoham (1989)], and belief revision [Gardenfors (1988)]. 4. Problem representation
and the process of reformulation
A process of setting a problem produces the statement of the problem. A process of reformulation is a rewriting of this statement. Every scientific discipline must define its object domain. To make the hypothesis of the preceding section more precise, we must now define what is meant by the statement of a problem. By the same stroke, we create a mode of representing a problem and define a class of representable problems. Next we characterize the set of solutions and take a look at the topologies of this set.
4.1. What is a problem statement? The existence of a problem presupposes, on the one hand, that there is something being sought (an x) and, on the other, that this x is not just anything (in which case, there would be no problem). Thus the x is subject to certain constraints. For this reason, it is possible to write a form of statement, which remains very general since we set no restriction on the set X and function K(x): ‘Find
x E X satisfying
constraints
K ( x) ‘.
This formulation may be an excellent internal representation, but it is far too ‘flat’ to be a very good instrument for representing knowledge, even for small-scale statements. Even if it is only two lines long, yet we have to state that the object sought is not an undifirentiated vector of variables, but for example a function of one set on to another, in other words a partially or totally unknown mathematical structure.
P. Bourgine, A logicul approach io problem representdon
459
Thus we arrive at another very general form of problem statement, borrowed from model theory: ‘The statement of a symbolic problem may be expressed in terms of the pair (S, T) of a partial structure and a theory ‘.
The concept of a partial structure is logical value is noted u. Given a theory and functional and relational symbols), result of a domain D of individuals and
taken from trivalent logic. The third T (utilizing constants for individuals a partial structure S = (D, F) is the a function, F, where this function
-
makes an individual d s = F(d) belonging to D correspond to each symbol of an individual d,
-
makes a partial function, noted fs = F(f) functional symbol f,
-
makes a partial relation, noted rs = F(r) of D” + { t, f, u }, correspond to each relational symbol r.
of D” + D, correspond to each
One may arrive at total functions by means of the following procedure: the domain D is completed with element u, to obtain a domain D*. A partial function f: D + D can now be considered a total function of D in D*. This total function is extended to D*“, to obtain a function f*: -ViE[l,n],
di=u~f*(d,,...,d,)=f(d,,...,d,),
_q~[l,n],
di=u*f*(dl,...>d,)
=u.
In the same manner, we can extend a particular partial predicate r as a total function r* of D*” * (t, f, u}. The functions and the predicates thus have the propriety of being strict (their value is u as soon as one of the arguments has the value of u). In what follows, we assume that these extensions have been made, but we omit the asterisks so as to simplify the notation. To make our definition more readily intelligible. we have assumed (without, however, any loss of generality) that the individual domain D is unique. But we are generally working within typed logics where the variables may take on diverse types Di and where the functions and relations have domains and codomains constructed on the basis of these 0,. Now we must set forth the semantics of a trivalent logic. In the above equations, the logical connectors (the right member) are interpreted by operators (denoted partially in the left member) of Kleene, Lukasiewicz, or Bochvar, defined by their truth tables [e.g., Turner (1984)]. The calculation of the truth value of a closed formula (or of a theory) is obtained with the help of the
460
P. Bourgtne,
seven following [r(t,
A logicd
approach
to problent
representutiort
rules: ,...,
f,)]S=~S(rlS
if
t;=d,
if
ti=f(r;
,...,
,...,
t;),
f:)
with
then
fis = dS,
then
rs=fS(t;S
[?4]S=‘[A]S.
(1) ,...,
t;‘), (2)
[A ABIS=
[A]SA
[B]?
(3)
[A v BIS=
[AISV
[BIS.
(4)
[A *BIS=
[AIS=,
[BIS,
(5)
[VXA(X)]~=
r\[A(d)lS,
dED.
(6)
[3xA(~)]~=
v[A(d)lS,
dED.
(7)
In keeping with the different modelizations proposed by various economic theories, it will be necessary to introduce other logical or arithmetical operators and to complete the foregoing list of semantic rules. For example, if the model includes calculations performed on the knowledge or beliefs of the agents, it is necessary to use knowledge or belief operators for each agent and to add rules corresponding to possible words semantics [Kripke (1983)]. Similarly, if one wants to build models using a qualitative economics [Bourgine and Raiman (1987)], it is necessary to introduce symbols for the signs (+, - ,O) and a calculus for these signs. If one also wants to reason in terms of an order of magnitude, it is necessary to introduce infinitely small (nonstandard) terms and the corresponding rules. There is no point here in continuing to list the different kinds of semantics that may be useful in economic modeling; instead, we will propose a rule for the choice of a logical system:
‘The choice of a logical system is determined by the choice of a semantics, which is itself a function of the way in which the problem is modeled.’ 4.2. What is the set of solutions? Two preliminary definitions are necessary before we can offer a definition of the set of solutions of a symbolic problem. First we must define the fundamental relation of satisfaction between structures and theories; then we must define the relation of extension between partial structures.
P. Bourgine. A logicui upprouch to problem representation
461
A structure S (partial or total) is said to satisfy a theory T iff [T]’ = t (‘the result of the interpretation of this theory in S is true’). In such a case it is said that S is partial or total model of T. One may give D a partial order, in which u is dominated by all d E D. Similarly, u is dominated by the truth values t and f: u < t,
u
and
VdED
(u
Given two structures S and S’ associated with a single theory T and with a single domain D. It is said that the structure S’ is an extension of a structure S if and only if the result of the interpretation of the functional and relational symbols in S’ is dominated by that of the interpretation in S:
V(d,,..., A), fS(4,...,d,)
‘fS’(d,
,...,
d,),
V(d, ,...,
,...,
d,).
d,),
rS(d, ,...,
d,)
Naturally, we quantify over the appropriate domains. We may now define the resolution of a symbolic problem having statement (S, T) as the search for partial models M extending the initial structure S and satisfying theory T. Now our attention is drawn to the set of these models: E={MIM>S
and
[~]~=t}.
A solution is not necessarily a total model; it may be a partial model and not completed defined. The idea of a solution that is not completely defined is not very familiar in operational research and mathematical programming. It is, however, much more common in temporal decision processes where the degrees of freedom that may be used have to be managed in terms of new constraints. 4.3. Topology of the set of solutions
It is useful to observe right away that it is rarely necessary to generate a complete extension of the set E of partial models as set forth in the previous section. Often it even suffices to verify whether there is at least one model and to stick to this result (Simon’s satisficing): this is a matter of giving the set of solutions a very rough topology since any two solutions may be considered as neighboring (and thus as non-distinct). However, much less rough topologies can be constructed, and these attempt to bring forth only solutions that are significantly different from each other (for example, radically contrasting scenarios). Let us suppose that we know
P. Bourgine, A logad
462
upprouch 10 problem represenrurron
how to give every point in a set E of solutions a neighborhood of non-significantly different solutions. To construct a subset of significantly different partial solutions, which would cover E, we may use the following iterative principle (where each step corresponds to Simon’s satisficing): (a) Find partial
Mk > S and [r,] M*= t.
(b) Construct
W, = union
(c) Construct
theory
of neighborhoods
of points
Tk+l = T k A ‘M E W, and return
in Mk. to instruction
(a).
Here we shall not deal with the conditions under which such a procedure terminates in a finite number of steps. When it does not terminate, it is still possible to fix a maximum number of steps or of significantly different solutions. 4.4. Generating
solutions by reformulation
The reformulation of a statement (S, T) provides a statement (S’. T’). A problem-solving process by means of reformulation is thus a tree or a series of statements (S,, 7;) by means of which the reasoning is conducted, with or without hypotheses. In the framework of this article, we need not argue for any single mode of reformulation: it may be a matter of a set of rewrite rules or, yet again, of a set of inference rules (such as in natural deduction systems). But it is possible to note that the reformulations can either rely on the theorems of the logical system in which T is written or on the properties of the mathematical structures available for stating S. In both cases, we may expect to deal with a large number of rules if we want to be in a position to deal with both global and local reformulations. The reader who would like a more exact depiction of a resolver functioning by means of reformulations based on a large number of rewrite and heuristic rules will find an excellent example in the resolver ALICE [Lauriere (1978), Bourgine (1986)]. 5. Conclusion What problems tions (for the other tion (for proposed structure reasoning,
we have been looking for is a mode for representing symbolic which would, on the one hand, make possible very broad formulaeconomic theories that work with homo oeconomicus) and which, on hand, could serve as a framework for efficient systems of reformulaeconomic theories that work more with homo cogitans). We have a way of representing problems as a pair (S, T) comprised of a and a theory. This approach provides a framework that allows which relies on certain particularities of mathematical structures
P. Bourgine, A logicul approuch
to problem
representutiou
463
and constraints, to progress by leaps and bounds. We thereby have a mode of solving problems by reformulation. It is familiar, intelligible, and communicable, for the bounded cognitive capacities of homo cogitans indeed involve many feats of reasoning that move in this way. If we set no limit on the type of mathematical structures contained in S, not on the kind of logic and number theory used in expressing T, the representation of a problem by the pair (S, T) effectively provides a descriptive theory that covers the diverse symbolic modelings of the behavior of homooeconomicus.
Yet it is not a general descriptive theory to the extent that economic agents can resolve many of their problems without necessarily having recourse to a formulation and representation of them. But the boundary between symbolic and nonsymbolic problems is not water-tight in human affairs. On the one hand, our ability to arrive at solutions without a stage of problem setting is sometimes only the result of a long period of learning how to deal with symbolic problems, the fruits of which are then compiled. On the other hand, it may be economically necessary to make a poorly represented problem the object of an explicit representation (an example is the problem of unemployment), and this may be necessary even if it is a task that is cognitively very difficult and costly. Thus we should look at problems as objects that admit of varying degrees of representation. At a given moment, the boundary between what is and is not explicitly symbolized is the result of a decision that is in turn subject to economic constraints and bounded rationality.
References Barro, R.J.. 1984, Macroeconomics (Wiley, London). Barwise, Jon and John Perry, 1983 Situations and attitudes (M.I.T./Bradford Books, Cambridge, MA). Bourgine, Paul, 1986, Propagation et extraction en resolution de problemes. in: Colloque IA. Strasbourg (CF. Picard, Paris) 105-130. Bourgine, Paul and Oliver Raiman, 19R7. Economics as reasoning on a qualitative model, in: J.L. Ross, ed., Economics and artificial intelligence (Pergamon, Oxford) 121-126. Chang, CC. and H.J. Keisler, 1973, Model theory (North-Holland, Amsterdam). De Kleer. J. and J.S. Brown, 1984. Qualitative physics based on confluences, Artificial Intelligence 24. 7-83. Dupuy, Jean-Pierre, 1989. Common knowledge, common sense, Theory and Decision 27, 37-62. Gardenfors, Peter, 1988, Knowledge in flux (M.I.T./Bradford Books, Cambridge, MA). Girard, J.Y., 1987, Linear logic, TCS 50. Kripke. Saul, 1963, Semantical considerations on modal logic, Acta Philosophica Fennica 16. 83-94. Lafont. Y., 1987, Logique. categories et machines, Doctoral thesis (University of Paris 7. Paris). Lauritre. Jean-Louis, 1978, A language and a program for stating and solving combinatorial problems, Artificial Intelligence 10, 29-127. Le Moigne, Jean-Louis, 1977, La theorie du systeme general (P.U.F.. Paris). Lewis, David. 1969, Conventions: A philosophical study (Harvard University Press, Cambridge, MA).
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A logicul upprouch
IO problem
repre~rentution
Lucas. R.E., 1972, Expectations and the neutrality of money, Journal of Economic Theory 4. 103-124. Newell, Alan and Herbert A. Simon, 1972. Human problem solving (Prentice-Hall, Englewood Cliffs, NJ). Muth, J.F.. 1961. Rational expectations and the theory of price movements, Econometrica 29, 315-335. Robinson. Abraham, 1966. Non-standard analysis (North-Holland, Amsterdam). Shoham. Y.. 1989, Reasoning about change (M.I.T.. Cambridge, MA). Simon, Herbert A., 1976. From substantive to procedural rationality, in: S.J. Latsis. ed., Method and appraisal in economics (Cambridge University Press, Cambridge) 129-148. Simon, Herbert A.. 1988, Problem formulation and alternative generation in the decision making process, Technical report AIP 43 (Carnegie Mellon University, Pittsburgh, PA). Stigler, G.J.. 1961, The economics of information, Journal of Political Economy TOME, 213-225. Turner, Raymond, 1984, Logics for artificial intelligence (Ellis Horwood) 24-27. Varela. Francisco. 1987, Trends in cognitive science and technology. in: J.L. Roos, ed., Economics and artificial intelligence (Pergamon, Oxford) l-8. Walliser. Bernard, 1989, Instrumental rationality and cognitive rationality, Theory and Decision 27. 7-36.
A LOGICAL APPROACH TO PROBLEM REPRESENTATION Paul BOURGINE* CEMAGREF, 92160 Antoy, Frunce GRASCE. 13627 Aix-en-Provence, Chdex I, France
Received February 1989, final version received October 1989 The cognitive sciences distinguish between symbolic and nonsymbolic problems. Here we consider problems that can be represented in terms of symbolic statements. We propose to represent a symbolic problem as a pair comprised of a partial structure and a theory, thereby situating the issue in the domain of trivalent logics and model theory. More specifically, the solution of a problem is a matter of finding the models of the theory which extend the initial partial structure. Such an approach makes it possible to see the solution of a problem as a process of reformulation whereby reasoning progresses in leaps and bounds. It provides a powerful tool for the symbolic description of economic agents’ decision-malting processes.
1. Introduction Economics has been based for quite some time now on the hypothesis of homo oeconomicus. But the postulate of expected maximum utility, which is
generally taken for granted, can hardly account for the large variety of description models and representations that economic agents use in reasoning about the problems they perceive. According to Simon (1988), if we want to solve problems, it is a good idea to have a good representation of them. He thus thinks that a theory of problem representation is essential to research on decision making. The goal of the present article is to set forth such a theory. Recent economic doctrines have tried to correct the image of homo oeconomicus by borrowing certain features from homo cogitans. They try to take the agent’s mental representations and computations over them explicitly into account. Muth (1961) and Lucas (1972) take a first step in this direction by bringing expectations into the picture. Stigler’s search theory (1961) takes another step by noting that optimization presupposes the cost of the search for information and calculation. Finally, with Simon (1976), homo oeconomicus is a full-fledged homo cogitans having limited cognitive capacities and decisions to make in complex environments. Modem economic theories do not only *Thanks to Jean-Louis Le Moigne for his comments on an earlier draft of this paper, and to Paisley Livingston for his translation. 0165-1889/90/$3.5001990,
Elsevier Science Publishers B.V. (North-Holland)
P. Bourgine.
452
A logical approach
to problem
representation
concern instrumental rationality, for they also take into account the cognitive rationality of economic agents [e.g., Walliser (1989)]. The very description of agents’ behavior becomes more and more complex as we try to produce more and more accurate models of the agents’ observations of economic phenomena. As u result, we should expect the theory of problem representation to guarantee its homogeneity by adopting a very large framework. This is why we have chosen to make use of the branch of mathematical logic known as model theory. Model theory [e.g., Chang and Keisler (1973)] studies the relations between structures and theories. Structures are defined in terms of sets, functions, and relations (thought to represent a ‘perceived reality’ or a phenomenology); they belong to a semantic universe. Theories (thought to represent a constrained or intended reality) are defined as statements contained within a formal language and belonging to a syntactic universe. In the next section, we survey the main cognitive approaches to problem solving. A distinction between symbolic and nonsymbolic problems emerges. In section 3, we use some examples to set forth the fundamental hypothesis of this article, which is that a symbolic problem can be represented as a pair of a partial structure and a theory (in an appropriate logical system). In the fourth section, we develop a more rigorous formulation of this intuition and discuss its advantages. 2. Cognitive sciences and problem solving A problem arises in the space between the reality that an economic agent perceives and the reality that he wishes to realize [Le Moigne (1977)]. When the difference between the two is identified, the agent begins to look for ways of eliminating it. The cognitive sciences distinguish in general between three phases of this search procedure: problem finding, problem setting, and problem solving. Different authors emphasize different phases: _ Newell and Simon’s (1972) contribution to cognitive psychology focuses on how subjects solve symbolic problems (fig. 1). _ Gestalt psychologists look at the search process as a series of reformulations of the global mental representation of the problem, ending when a fully structured solution appears (fig. 2).
perception 4 action
-
distance
statement b
finding 4
solution
setting
solving
-
Fig. 1. Processes of problem
formulation
-
and solving.
P. Bourgine, A logical approach to problem repre.sentution
perception A finding 4 action -
distance +solution
Fig. 2. Processes of problem
453
(re’ setting
reformulation.
_ Both approaches postulate the existence of a phase in which the problem
receives an explicit symbolic formulation. Varela (1987) on the other hand considers a class of problems that subjects identifv and solve instantaneously without going through a phase of formulation (fig. 3). Such a process is viable because of the large knowledge that subjects have acquired thorough learning, and that they eventually ‘share’ (common knowledge). In cases where the problem can be solved without a stage of formulation, we will speak of a nonsymbolic problem. It must be noted that each of these theories share certain presuppositions with the others: solutions based on statements necessarily involve a formulation of that statement; all forms of solution presuppose that there is a finding and presenting of the problem and that the problem can either be ‘representable’ or not. The movement from one theory to the next (see our three figures) is, then, a matter of differentiation. The three theories are largely complementary. They hold for observational data pertaining to complementary time scales: less than l/10 second for Varela, from l/10 second to l/2 hour for Newell and Simon, and for even longer periods for the processes of reformulation studied by Gestalt psychologists. Thus a single process of problem solving may very well be described as the succession and overlapping of periods interpretable in one of the complementary theories. Some typical examples may help us to understand the differences between these diverse classes of problems. Varela (1986) presents driving an automobile in a city as a nonsymbolic problem: when we are driving, we deal with most of the difficulties presented by traffic in less than l/lOth of a second. This is even an essential condition of a proper traffic flow! Simon (1988) gives examples of other problems that would be difficult to represent: ‘The challenge presented by Japanese and other Far Eastern competition is on the agenda; the problem is on the agenda, but finding an appropriate problem representation is a more difficult task.’ The same holds for the problem of unemployment.
Em Fig. 3. Process
nonsymbolic
solving.
454
P. Bourgine, A logical approach fo problem representation
At the end of this article we shall return to the question of the distinction between symbolic and nonsymbolic, representable and nonrepresentable problems.
3. Examples of modeling of economic reasoning In this section various examples of situations are presented. We clarify the semantics of problems and propose a representation of problems in an adequate logical system, the goal being to show that many economic reasonings can be modeled symbolically once we have logical systems adapted to them.
3.1. Reasoning
about and with bounded resources
Economic reasoning deals with situations in which resources are limited. Moreover, the cognitive resources that agents use in these reasonings are themselves limited. In general, economic theories try to describe such situations by using quantitative models of the agent’s preferences. In what follows, we propose a qualitative model based on linear logic [Girard (1987)]. The approach to be adopted may be particularly well suited to reasonings that involve limited cognitive resources. To understand the meaning of the connectors of linear logic, we may refer to an example having to do with limited physical resources [Lafont (1987)]. Menu for $20 Starter:
Ham or grapefruit, according to the season
Main course:
Sirloin steak
French fries:
All you can eat
Cheese or dessert
The translation of this menu in linear logic is as follows: $20
--0
(ham @ grapefruit) 0 steak 8 !french-fries (cheese & dessert)
In linear logic, a formula A does not represent a stable situation in which there is a truth that has been settled once and for all subsequent situations. Instead, the formula represents an action that, when realized, gives rise to a certain number of consequences while depleting a certain amount of resources.
P. Bourgine, A logical approach
toproblemrepresentation
The inference rule of connector --o corresponds classical logic: A
455
to the modus ponens of
A --oB B
*
In the case of the menu, the interpretation specifies that ‘if I have $20, I can have the menu (but will no longer have my $20)‘. The expression !A means that I still have A after I have used it (A then is an unlimited resource). The connectors @ and 0 represent a choice: A 0 B means I can have either A or B (but not both), and the choice is mine; A $ B also means I can have either A or B (but not both), but the choice is not mine. 3.2. Qualitative reasoning with order of magnitude Qualitative reasoning was initially introduced in order to model certain forms of reasoning in physics [e.g., De Kleer and Brown (1984)]. For an application of it to economics and for an introduction to a reasoning with order of magnitude, the reader may consult Bourgine and Raiman (1987). We illustrate this kind of modeling here by means of an example of ‘temporary supply shock’ taken from Barro (1984, pp. 130, 262). The ‘ . . . ’ indicate that other factors are left out, and the signs are those of the partial derivations.
where Ys CD ID W P r
= supply of commodities, = consumption, = net investment, = nominal wage rate, =price, = real interest rate;
(b)
LD( ~~~,...) =Ls(~~;;~: ,...),
where LD = demand of work Ls = supply of work; ’
456
cc>
P. Bourgtne. A logtcal upprouch to problem representution
M/P=H
Y
,
R (-)
i (+I
,
where M = aggregate amount of money, R = nominal interest rate. To describe a change of equilibrium (if there is one), we introduce relations of order of magnitude A Vo B (read: A is close to B) and A Ne B (read: A is negligible to B). These relations are easily interpreted in terms of infinitely smalls (designated by ‘0’ below) of nonstandard analysis [Robinson (1966)]: AVoB=A=(l+o)B, ANeB=A=oB. A change
of equilibrium
satisfies the conditions:
AYSVo(ACD+AID),
ALDVoALS,
AM/PVoAH(Y,
R).
a temporary supply shock amounts to a modification For Barro, A,Ys, A,CD, A,ZD of the curves such that A,CD, A,LD are negligible before A,Ys and the sign of A,Ys is negative ([xl represents the sign + , - , 0, ? of x). But the change in the curves of the equilibrium leads to variations (noted as A,) in all the magnitudes: [A,YD]
= -,
A,CD NeA,YS.
A,lD NeA,Ys.
A,Ys+ A,Ys Vo ( A,CD + A,CD + A,lD + A,ZD), A,LDVoA,LS,
A,M/PVoA,H(
Y, R).
A qualitative reasoning with order of magnitude assigns meaning to the variation of certain variables of the new equilibrium and leaves undetermined the meaning of the variation of the other variables [Bourgine and Raiman (1987)]:
[r] = [A,Ys] [W/P] [A,M/P] The interest
= +.
= [A&‘“]
= [A,ID]
= [A,R]
of qualitative
= [A,LD]
= [A,L’]
= -,
= ?.
reasoning
with order of magnitude
is double.
On the
P, Bourgine, A logical approach
to problemrepresentation
457
one hand, it makes it possible to arrive at certain consequences even if the laws are quantitatively poorly known, provided that there is a consensus about the qualitative properties of these laws and of the orders of magnitude. On the other hand, the reasoning remains wholly intelligible and close to the reasoning done by economic experts. 3.3. Common knowledge and the emergence of conventions and institutions By introducing the expectations of the agents, recent economic theories have given up working with a single equilibrium. The determination of a particular equilibrium is not the result of a purely economic mechanism, but must be explained in terms of such sociological mechanisms as conventions or institutions. Thus it becomes necessary to explain in turn how conventions emerge. Following Lewis (1964), the role of common knowledge (CK) is given an essential role in this process. To illustrate the emergence of conventions, we borrow an example from Dupuy (1989). In table 1 a game is given defined in terms of the payoff matrix for two players. This game has two Nash equilibria, uA and bB. But both of them require the two agents to coordinate their strategies so as to avoid the disastrous cases aB and Ab. This coordination problem may be understood in cognitive terms, for it is easily solved if each agent wants to coordinate his action with the other agent, and if each agent knows that the other wants to realize such a coordination, and knows that the other wants to realize it, . . . Thus we have a figure of infinite specularity that can be represented by introducing a new knowledge operator, KOthcr, defined as
The Other knows proposition P if and only if all agents know that the Other knows P. In that case we say that proposition P is CK (common knowledge). If we now designate as P the proposition ‘every agent knows that the other wants to coordinate his actions with those of the first agent’, the fact
Table 1 Player 1 Action LI Action a
Action 10
10
b
0 0
Player 2 Action
b
10
0
0
10
P. Bourgrne, A logud
458
upproach to problem representation
that P is CK will have a stabilizing influence, making it possible to solve the coordination problem. A possible interpretation of the game’s structure is found in the example of driving automobiles on the right or left side of the road: the two conventions are equivalent, yet the choice between them is necessary. Once one of these conventions has been adopted - such as driving on the left - it is obvious that driving on the left is facilitated if the proposition ‘everyone drives on the left’ is CK. The three examples have the goal of showing that there is a link between a certain kind of economic reasoning and the logical system in which this reasoning may be realized. Many contributions to the modeling of reasoning could also be cited, but we will limit ourselves to a few references useful to the modeling of problems confronted by economic agents; for example, reasoning related to situations and attitudes [Bar-wise and Perry (1983)], to belief change [Shoham (1989)], and belief revision [Gardenfors (1988)]. 4. Problem representation
and the process of reformulation
A process of setting a problem produces the statement of the problem. A process of reformulation is a rewriting of this statement. Every scientific discipline must define its object domain. To make the hypothesis of the preceding section more precise, we must now define what is meant by the statement of a problem. By the same stroke, we create a mode of representing a problem and define a class of representable problems. Next we characterize the set of solutions and take a look at the topologies of this set.
4.1. What is a problem statement? The existence of a problem presupposes, on the one hand, that there is something being sought (an x) and, on the other, that this x is not just anything (in which case, there would be no problem). Thus the x is subject to certain constraints. For this reason, it is possible to write a form of statement, which remains very general since we set no restriction on the set X and function K(x): ‘Find
x E X satisfying
constraints
K ( x) ‘.
This formulation may be an excellent internal representation, but it is far too ‘flat’ to be a very good instrument for representing knowledge, even for small-scale statements. Even if it is only two lines long, yet we have to state that the object sought is not an undifirentiated vector of variables, but for example a function of one set on to another, in other words a partially or totally unknown mathematical structure.
P. Bourgine, A logicul approach io problem representdon
459
Thus we arrive at another very general form of problem statement, borrowed from model theory: ‘The statement of a symbolic problem may be expressed in terms of the pair (S, T) of a partial structure and a theory ‘.
The concept of a partial structure is logical value is noted u. Given a theory and functional and relational symbols), result of a domain D of individuals and
taken from trivalent logic. The third T (utilizing constants for individuals a partial structure S = (D, F) is the a function, F, where this function
-
makes an individual d s = F(d) belonging to D correspond to each symbol of an individual d,
-
makes a partial function, noted fs = F(f) functional symbol f,
-
makes a partial relation, noted rs = F(r) of D” + { t, f, u }, correspond to each relational symbol r.
of D” + D, correspond to each
One may arrive at total functions by means of the following procedure: the domain D is completed with element u, to obtain a domain D*. A partial function f: D + D can now be considered a total function of D in D*. This total function is extended to D*“, to obtain a function f*: -ViE[l,n],
di=u~f*(d,,...,d,)=f(d,,...,d,),
_q~[l,n],
di=u*f*(dl,...>d,)
=u.
In the same manner, we can extend a particular partial predicate r as a total function r* of D*” * (t, f, u}. The functions and the predicates thus have the propriety of being strict (their value is u as soon as one of the arguments has the value of u). In what follows, we assume that these extensions have been made, but we omit the asterisks so as to simplify the notation. To make our definition more readily intelligible. we have assumed (without, however, any loss of generality) that the individual domain D is unique. But we are generally working within typed logics where the variables may take on diverse types Di and where the functions and relations have domains and codomains constructed on the basis of these 0,. Now we must set forth the semantics of a trivalent logic. In the above equations, the logical connectors (the right member) are interpreted by operators (denoted partially in the left member) of Kleene, Lukasiewicz, or Bochvar, defined by their truth tables [e.g., Turner (1984)]. The calculation of the truth value of a closed formula (or of a theory) is obtained with the help of the
460
P. Bourgtne,
seven following [r(t,
A logicd
approach
to problent
representutiort
rules: ,...,
f,)]S=~S(rlS
if
t;=d,
if
ti=f(r;
,...,
,...,
t;),
f:)
with
then
fis = dS,
then
rs=fS(t;S
[?4]S=‘[A]S.
(1) ,...,
t;‘), (2)
[A ABIS=
[A]SA
[B]?
(3)
[A v BIS=
[AISV
[BIS.
(4)
[A *BIS=
[AIS=,
[BIS,
(5)
[VXA(X)]~=
r\[A(d)lS,
dED.
(6)
[3xA(~)]~=
v[A(d)lS,
dED.
(7)
In keeping with the different modelizations proposed by various economic theories, it will be necessary to introduce other logical or arithmetical operators and to complete the foregoing list of semantic rules. For example, if the model includes calculations performed on the knowledge or beliefs of the agents, it is necessary to use knowledge or belief operators for each agent and to add rules corresponding to possible words semantics [Kripke (1983)]. Similarly, if one wants to build models using a qualitative economics [Bourgine and Raiman (1987)], it is necessary to introduce symbols for the signs (+, - ,O) and a calculus for these signs. If one also wants to reason in terms of an order of magnitude, it is necessary to introduce infinitely small (nonstandard) terms and the corresponding rules. There is no point here in continuing to list the different kinds of semantics that may be useful in economic modeling; instead, we will propose a rule for the choice of a logical system:
‘The choice of a logical system is determined by the choice of a semantics, which is itself a function of the way in which the problem is modeled.’ 4.2. What is the set of solutions? Two preliminary definitions are necessary before we can offer a definition of the set of solutions of a symbolic problem. First we must define the fundamental relation of satisfaction between structures and theories; then we must define the relation of extension between partial structures.
P. Bourgine. A logicui upprouch to problem representation
461
A structure S (partial or total) is said to satisfy a theory T iff [T]’ = t (‘the result of the interpretation of this theory in S is true’). In such a case it is said that S is partial or total model of T. One may give D a partial order, in which u is dominated by all d E D. Similarly, u is dominated by the truth values t and f: u < t,
u
and
VdED
(u
Given two structures S and S’ associated with a single theory T and with a single domain D. It is said that the structure S’ is an extension of a structure S if and only if the result of the interpretation of the functional and relational symbols in S’ is dominated by that of the interpretation in S:
V(d,,..., A), fS(4,...,d,)
‘fS’(d,
,...,
d,),
V(d, ,...,
,...,
d,).
d,),
rS(d, ,...,
d,)
Naturally, we quantify over the appropriate domains. We may now define the resolution of a symbolic problem having statement (S, T) as the search for partial models M extending the initial structure S and satisfying theory T. Now our attention is drawn to the set of these models: E={MIM>S
and
[~]~=t}.
A solution is not necessarily a total model; it may be a partial model and not completed defined. The idea of a solution that is not completely defined is not very familiar in operational research and mathematical programming. It is, however, much more common in temporal decision processes where the degrees of freedom that may be used have to be managed in terms of new constraints. 4.3. Topology of the set of solutions
It is useful to observe right away that it is rarely necessary to generate a complete extension of the set E of partial models as set forth in the previous section. Often it even suffices to verify whether there is at least one model and to stick to this result (Simon’s satisficing): this is a matter of giving the set of solutions a very rough topology since any two solutions may be considered as neighboring (and thus as non-distinct). However, much less rough topologies can be constructed, and these attempt to bring forth only solutions that are significantly different from each other (for example, radically contrasting scenarios). Let us suppose that we know
P. Bourgine, A logad
462
upprouch 10 problem represenrurron
how to give every point in a set E of solutions a neighborhood of non-significantly different solutions. To construct a subset of significantly different partial solutions, which would cover E, we may use the following iterative principle (where each step corresponds to Simon’s satisficing): (a) Find partial
Mk > S and [r,] M*= t.
(b) Construct
W, = union
(c) Construct
theory
of neighborhoods
of points
Tk+l = T k A ‘M E W, and return
in Mk. to instruction
(a).
Here we shall not deal with the conditions under which such a procedure terminates in a finite number of steps. When it does not terminate, it is still possible to fix a maximum number of steps or of significantly different solutions. 4.4. Generating
solutions by reformulation
The reformulation of a statement (S, T) provides a statement (S’. T’). A problem-solving process by means of reformulation is thus a tree or a series of statements (S,, 7;) by means of which the reasoning is conducted, with or without hypotheses. In the framework of this article, we need not argue for any single mode of reformulation: it may be a matter of a set of rewrite rules or, yet again, of a set of inference rules (such as in natural deduction systems). But it is possible to note that the reformulations can either rely on the theorems of the logical system in which T is written or on the properties of the mathematical structures available for stating S. In both cases, we may expect to deal with a large number of rules if we want to be in a position to deal with both global and local reformulations. The reader who would like a more exact depiction of a resolver functioning by means of reformulations based on a large number of rewrite and heuristic rules will find an excellent example in the resolver ALICE [Lauriere (1978), Bourgine (1986)]. 5. Conclusion What problems tions (for the other tion (for proposed structure reasoning,
we have been looking for is a mode for representing symbolic which would, on the one hand, make possible very broad formulaeconomic theories that work with homo oeconomicus) and which, on hand, could serve as a framework for efficient systems of reformulaeconomic theories that work more with homo cogitans). We have a way of representing problems as a pair (S, T) comprised of a and a theory. This approach provides a framework that allows which relies on certain particularities of mathematical structures
P. Bourgine, A logicul approuch
to problem
representutiou
463
and constraints, to progress by leaps and bounds. We thereby have a mode of solving problems by reformulation. It is familiar, intelligible, and communicable, for the bounded cognitive capacities of homo cogitans indeed involve many feats of reasoning that move in this way. If we set no limit on the type of mathematical structures contained in S, not on the kind of logic and number theory used in expressing T, the representation of a problem by the pair (S, T) effectively provides a descriptive theory that covers the diverse symbolic modelings of the behavior of homooeconomicus.
Yet it is not a general descriptive theory to the extent that economic agents can resolve many of their problems without necessarily having recourse to a formulation and representation of them. But the boundary between symbolic and nonsymbolic problems is not water-tight in human affairs. On the one hand, our ability to arrive at solutions without a stage of problem setting is sometimes only the result of a long period of learning how to deal with symbolic problems, the fruits of which are then compiled. On the other hand, it may be economically necessary to make a poorly represented problem the object of an explicit representation (an example is the problem of unemployment), and this may be necessary even if it is a task that is cognitively very difficult and costly. Thus we should look at problems as objects that admit of varying degrees of representation. At a given moment, the boundary between what is and is not explicitly symbolized is the result of a decision that is in turn subject to economic constraints and bounded rationality.
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Lucas. R.E., 1972, Expectations and the neutrality of money, Journal of Economic Theory 4. 103-124. Newell, Alan and Herbert A. Simon, 1972. Human problem solving (Prentice-Hall, Englewood Cliffs, NJ). Muth, J.F.. 1961. Rational expectations and the theory of price movements, Econometrica 29, 315-335. Robinson. Abraham, 1966. Non-standard analysis (North-Holland, Amsterdam). Shoham. Y.. 1989, Reasoning about change (M.I.T.. Cambridge, MA). Simon, Herbert A., 1976. From substantive to procedural rationality, in: S.J. Latsis. ed., Method and appraisal in economics (Cambridge University Press, Cambridge) 129-148. Simon, Herbert A.. 1988, Problem formulation and alternative generation in the decision making process, Technical report AIP 43 (Carnegie Mellon University, Pittsburgh, PA). Stigler, G.J.. 1961, The economics of information, Journal of Political Economy TOME, 213-225. Turner, Raymond, 1984, Logics for artificial intelligence (Ellis Horwood) 24-27. Varela. Francisco. 1987, Trends in cognitive science and technology. in: J.L. Roos, ed., Economics and artificial intelligence (Pergamon, Oxford) l-8. Walliser. Bernard, 1989, Instrumental rationality and cognitive rationality, Theory and Decision 27. 7-36.