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A Generalization of Knowledge Space Theory to Problems with More Than Two Answer Alternatives
Journal of Mathematical Psychology MP1169 journal of mathematical psychology 41, 237243 (1997) article no. MP971169
A Generalization of Knowledge Space Theory to Problems with More Than Two Answer Alternatives Martin Schrepp Ladenburg, Germany
In the theory of knowledge spaces (Doignon 6 Falmagne, 1985) a knowledge domain is represented by a set Q of problems. The knowledge of a subject in the knowledge domain is described by the subset of problems from the domain that the subject is able to solve. A central assumption is therefore that the answer of a subject to a problem can be classified as either correct or incorrect. We show that the central concepts of the theory can be generalized to problem domains in which solutions can be evaluated on a linear scale concerning their quality. ] 1997 Academic Press
THEORETICAL BACKGROUND
In the theory of knowledge spaces (Doignon 6 Falmagne; 1985) a knowledge domain is represented as a finite set Q of questions. The subset W of questions from Q that a subject is capable of solving is called the knowledge state of that subject. The set of all possible knowledge states is called the knowledge structure on Q. A knowledge structure W which is closed under union, i.e., W1 , W2 # W implies W1 _ W2 # W, is called a knowledge space on Q. A knowledge space which is additionally closed under intersection, i.e., W1 , W2 # W implies W1 & W2 # W, is called a quasi-ordinal (q.-o.) knowledge space on Q. The closure properties of knowledge spaces and q.-o. knowledge spaces permit an alternative description of such spaces. A reflexive and transitive relation = C on Q is called a surmise relation on Q. For p, q # Q the interpretation of pC =q is ``Every subject who is able to solve q is also able to solve p.'' Let ^(Q) be the power set, i.e., the set of all subsets, of Q. A mapping $ : Q ^(^(Q)) is called a surmise function if it has the following properties: 1. W # $(q) q # W,
Theorem 1 (Birkhoff, 1937). The formula pC =q \W # W (q # W p # W) defines a one-to-one correspondence : between QKS(Q) and SR(Q). Therefore, every q.-o. knowledge space W on Q is completely characterized by a surmise relation = C and vice versa. The following result establishes a similar connection between knowledge spaces and surmise functions. Theorem 2 (Doignon 6 Falmagne, 1985).
The formula
K # $(q) _W # W (q # W 7 K W)
2. W # $(q) 7 p # W _W$ # $( p)(W$ W ),
7 \W # W (q # W W/3 K )
3. W, W$ # $(q) 7 W W$ W=W$. I am grateful to professor J. C. Falmagne and four anonymous referees for their suggestions concerning the mathematical argumentation and to Diana Bursy for improving the English style. Correspondence and reprint requests should be sent to Martin Schrepp, Am Bahndamm 26, 68526 Ladenburg, Germany. 237
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The interpretation of a surmise function $(q)=[W1 , ..., Wn ] is ``Every subject who is able to solve q is able to solve all problems from at least one element Wi of $(q).'' The elements of $(q) are called clauses of q. The clauses of q are interpreted as sets of prerequisites for q. A subject is able to solve a problem q if she or he is able to solve all problems from at least one prerequisite set Wi # $(q) for this problem. Define QKS(Q) to be the set of all q.-o. knowledge spaces on Q, KS(Q) the set of all knowledge spaces on Q, SR(Q) the set of all surmise relations on Q, SF(Q) the set of all surmise functions on Q, REL(Q) the set of all binary relations on Q, and, SM(Q) the set of all mappings from Q to ^(^(Q)). Notice that a surmise function is a proper generalization of a surmise relation, since a surmise function $ for which every q # Q has exactly one clause K q corresponds to a surmise relation = C through pC =q W p # K q . The connection of q.-o. knowledge spaces with surmise relations is established by the following theorem.
establishes a one-to-one correspondence # between KS(Q) and SF(Q). Therefore, every knowledge space W on Q is completely characterized by a surmise function $ and vice versa. The 0022-249697 25.00 Copyright 1997 by Academic Press All rights of reproduction in any form reserved.
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clauses of a problem q are the -minimal subsets of Q which contain q. There are two main applications of knowledge space theory. First, knowledge spaces can be used for an efficient assessment of knowledge. Since a knowledge structure does not in general contain all possible elements of the power set of Q it can be used to infer the answers of subjects to particular problems from the answers they have already given. Therefore, an adaptive procedure for the assessment of knowledge can be established (Falmagne 6 Doignon, 1988a or Falmagne 6 Doignon; 1988b). Second, knowledge spaces can be used to test psychological models of problem solving processes. In this approach a knowledge space is derived from assumptions on the problem solving processes of subjects in a knowledge domain. The derived space is then compared to observed response patterns of subjects. Such tests of psychological models are described for problem solving in chess (Albert, Schrepp 6 Held; 1994), geometry (Korossy; 1993), probability theory (Held; 1993), and letter series completion (Schrepp; 1993, 1995). Psychological models of problem solving processes describe the basic cognitive abilities subjects must possess in order to solve problems from the underlying knowledge domain. Interindividual differences are described by different shapings of these abilities. Every shaping of abilities or skills corresponds to a solution pattern. Such a detailed analysis of the cognitive processes allows one often not only to predict whether or not a subject with specific abilities will solve a problem, but also to predict how far the subject will come in hisher effort to solve the problem. Therefore, the assumption that every problem is solved either correctly or incorrectly by a subject is too restrictive. We must differentiate between several degrees of correctness of a given solution. These degrees of correctness should correspond to the subject's understanding of the domain. The quality of a solution may, for example, be judged by points. The better the solution is the more points are assigned to it. Another example is speed problems in which every subject is able to find the correct solution if enough time is available. The performance of a subject on such speed problems is measured by the amount of time required to find the solution. The faster the problem is solved, the better is the performance of the subject. In the following section we describe a generalization of the central concepts of knowledge space theory to problems of this type. GENERALIZATION OF THE THEORY
Let Q be a finite set of problems. We assume that the quality of a solution to a problem q # Q can be measured by an element l # L, where L is a linear ordered set. We assume
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that L contains at least two elements. Let L be the linear order on L. The performance of a subject on the problems in Q can be described by the values in L the subject obtains, i.e., by a mapping W : Q L. We call such a mapping a knowledge state. A set W{< of knowledge states is calledas abovea knowledge structure on (Q, L). Notice that for L=[0, 1] these generalized concepts are identical with the original concepts, since every subset of Q can be identified with its characteristic function. Let, for example, Q be a set of speed problems. In this case L=[0, [ and L = . An element l # L represents the time required to solve a problem. The knowledge state W : Q L of a subject assigns to every problem q # Q the time the subject needs to solve q. Since surmise relations and surmise functions describe the relative difficulty of problems their formal definitions can be used unchanged. We interpret pC =q as ``Every subject reaches a better (or equal) result in solving p than in solving q,'' i.e., pC =q W(q) L W( p) for every possible knowledge state W. The interpretation of $(q) is ``Every subject reaches in all problems of at least one clause in $(q) a better (or equal) result than in q,'' i.e., for every possible knowledge state W there exists a K in $(q) with W(q) L W( p) for all p # K. These interpretations are the reason for our assumptions of a linear order on L. We have to ensure that the solutions given to problems q # Q are comparable in quality in order to be able to relate two problems by a surmise relation respectively surmise function. It is easy to determine the knowledge structures on (Q, L) corresponding to a given surmise relation = C and surmise function $. Definition 1. Let = C be a binary relation on Q and let $ be a mapping $ : Q ^(^(Q)). We define W(C = )=[W : Q L | \p, q # Q( pC =q W(q) L W( p))], W($)=[W : Q L | \q # Q_K # $(q) \p # K(W(q) L W( p))]. In analogy to the previous section we call W (C = ) the q.-o. knowledge space corresponding to = C and W ($) the knowledge space corresponding to $. In the original case (L=[0, 1]) knowledge spaces and q.-o. knowledge spaces are defined by closure properties (closure against union, closure against union, and intersection). The question arises of how these closure properties can be generalized. A natural assumption would be that a knowledge structure which is closed under maximum, i.e., W1 , W2 # W implies max(W1 , W2 ) # W, is in a one-to-one
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correspondence with a surmise function and that a knowledge structure which is additionally closed under minimum, i.e., W1 , W2 # W implies min(W1 , W2 ) # W, is in a oneto-one correspondence with a surmise relation. The following example shows that this is not true if L contains more than two elements.
Definition 2. Let W be a knowledge structure on (Q, L). W is called strictly closed if for all F : Q L the condition
Example 1. Let L=[1, 2, 3] and Q=[ p, q]. The knowledge structure W=[(1, 1), (3, 3)] is closed under minimum and maximum. But W does not correspond to a surmise relation or surmise function, since (2, 2) W and every knowledge structure on (Q, L) corresponding to a surmise relation or surmise function contains all constant mappings from Q to L.
implies F # W. W is called closed if for all F : Q L the condition
Definition 1 shows how a knowledge structure on (Q, L) can be derived from a surmise relation respectively surmise function on Q. The following result shows how a surmise relation and a surmise function can be derived from a knowledge structure on (Q, L). Lemma 1. Then
Let W be a knowledge structure on (Q, L).
\p, q # Q
\W # W (W(q) L W( p) F(q) L F( p))
\q # Q _W # W \p # Q(W(q) L W( p) F(q) L F( p)) implies F # W. Define WS(Q, L) to be the set of all knowledge structures on (Q, L), KS(Q, L) the set of all closed knowledge structures on (Q, L), and, QKS(Q, L) the set of all strictly closed knowledge structures on (Q, L). Let W be a strictly closed knowledge structure on (Q, L) and F : Q L with \q # Q _W # W \p # Q(W(q) L W( p) F(q) L F( p)).
pC =q \W # W (W(q) L W( p)) defines a surmise relation = C on Q and K # $(q) _W # W (K=[ p | W(q) L W( p)]) 7 \W$ # W ([ p | W$(q) L W$( p)] /3 K )
For all p, q the condition \W$ # W (W$(q) L W$( p)) implies F(q) L F( p) and since W is strictly closed, F # W. Therefore, every strictly closed knowledge structure is closed. Example 2. Let Q=[q 1 , q 2 , q 3 ], L=[0, 1, 2] and L = . Define
defines a surmise function on Q. Proof. First, it will be shown that = C is a surmise relation. The reflexivity of = C follows immediately. Let pC =q r. For each W # W this implies W(q) W( p) and and qC L = W(r) L W(q). Since L is transitive, W(r) L W( p) for C is transitive. each W # W. Hence, pC =r and = Second, it will be shown that $ is a surmise function. The condition K # $(q) q # K follows immediately. Let K # $(q) and p # K. There exists W # W with K= [r | W(q) L W(r)]. The conclusion is that W(q) L W( p). There exists K$ [r | W( p) L W(r)] with K$ # $( p). Since L is transitive, K$ K. Let K, K$ # $(q) and K$ K. There exists W, W$ # W with K=[ p | W(q) L W( p)] and K$= [ p | W$(q) L W$( p)]. Finally, [ p | W$(q) L W$( p)] /3 K for each W$ # W implies K$=K. Therefore, $ is a surmise function on Q. GENERALIZED CLOSURE PROPERTIES
As Example 1 shows, a knowledge structure on (Q, L) must fulfill stronger closure properties to be in a one-to-one correspondence with a surmise function or surmise relation on Q.
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W1 =[(0, 0, 0), (2, 1, 0), (2, 2, 2)], W2 =[(0, 0, 0), (1, 1, 1), (2, 2, 2), (1, 0, 0), (2, 0, 0), (2, 1, 2), (2, 1, 1), (1, 0, 1), (2, 0, 2), (2, 0, 1), (1, 2, 2), (1, 1, 0), (0, 1, 1), (2, 1, 0), (2, 2, 0), (2, 2, 1), (0, 2, 2)], W3 =[(0, 0, 0), (1, 1, 1), (2, 2, 2), (1, 0, 0), (1, 1, 0), (1, 0, 1), (2, 0, 0), (2, 1, 0), (2, 2, 0), (2, 0, 1), (2, 0, 2), (2, 1, 1), (2, 2, 1), (2, 1, 2)]. Let W1 =(0, 0, 0) and W2 =(1, 1, 1). Then for each q # Q we have W1(q) L W1( p) W2(q) L W2( p) for all p # Q, but W2 W1 . Therefore, W1 is not closed. W2 is closed, but not strictly closed. Choose p, q # Q with p{q. From (2, 1, 0), (1, 0, 1), (0, 1, 1) # W2 we can conclude c\W # W2(W(q) L W( p)). For W3 =(0, 1, 0) this implies W(q) L W( p) W3(q) L W3( p) for each W # W2 , but W3 W2 . W3 is strictly closed.
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The following result shows that the (quasi-ordinal) knowledge space corresponding to a surmise function (relation) is (strictly) closed. Lemma 2. Let = C be a binary relation on Q and let $ be a mapping $ : Q ^(^(Q)). W ($) is closed and W (C = ) is strictly closed. Proof.
First, it will be shown that
W (C = )=[W | \p, q # Q( pC =q W(q) L W( p))] is strictly closed. Let F be a mapping from Q to L. Assume W(q) L W( p) F(q) L F( p) for all p, q # Q and each W # W (C = ). For all p, q with pC =q and for each W # W (C =) we have W(q) L W( p). Therefore, F(q) L F( p). So pC = q F(q) L F( p) for all p, q # Q, which implies F # W (C = ). Therefore, W (C = ) is strictly closed. Second, it will be shown that W ($)=[W | \q # Q _K # $(q) \p # K(W(q) L W( p))] is closed. Let F be a mapping from Q to L. Assume that for each q # Q there is a W # W ($) so that W(q) L W( p) F(q) L F( p) for each p # Q. Let r # Q. Because of our assumptions on F, there exists W # W ($) with W(r) L W( p) F(r) L F( p) for each p # Q. Therefore, there exists K # $(r) with W(r) L W( p) for each p # K, which implies F(r) L F( p) for each p # K. Thus, F # W ($) and W ($) is closed. The following results show that the (strictly) closed knowledge structures on (Q, L) are a generalization of the (quasi-ordinal) knowledge spaces on Q, since for the special case L=[0, 1] these concepts can be identified. Lemma 3. Let W be a closed knowledge structure on (Q, L). Then W1 , W2 # W implies max(W1 , W2 ) # W. If W is strictly closed, then W1 , W2 # W implies min(W1 , W2 ) # W. Proof. Let W be a closed knowledge structure on (Q, L), q # Q and W1 , W2 # W. Assume without loss of generality max(W1 , W2 )(q)=W1(q). Therefore, W1(q) LW1( p) implies max(W1 , W2 )(q) L max(W1 , W2 )( p) for each p # Q. Since W1 # W and W is closed, max(W1 , W2 ) # W. Let W be strictly closed and W1 , W2 # W. Let p, q # Q with W(q) L W( p) for each W # W. Therefore, W1(q) L W1( p), W2( q) L W2( p), and min(W1 , W2 )(q) L min(W1 , W2 )( p). The conclusion is that for all p, q # Q and for each W # W we have W(q) LW( p) min(W1 , W2 )(q) L min(W1 , W2 )( p). Since W is strictly closed, min(W1 , W2 ) # W. Define for L=[0, 1] the mappings < L , Q L : Q L by < L(q)=0 for all q # Q and Q L(q)=1 for all q # Q.
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Lemma 4. Let L=[0, 1] and W be a knowledge structure on (Q, L). If < L , Q L # W and W1 , W2 # W implies max(W1 , W2 ) # W, then W is closed. If additionally W1 , W2 # W implies min(W1 , W2 ) # W, then W is strictly closed. Proof. Assume < L , Q L # W and that W1 , W2 # W max(W1 , W2 ) # W. Let F : Q L be a mapping with \q # Q _W # W \p # Q(W(q) L W( p) F(q) L F( p)). We have to show F # W. Define M 1 =[q | F(q)=0] and M 2 =[q | F(q)=1]. Let q # M 2 , which implies F( p)< L F(q) for all p # M 1 . By assumption on F it follows that there exists W # W, so that W(q) L W( p) F(q) L F( p) for each p # Q. For p # M 1 this implies W( p)< L W(q) for every such W, since F( p)< L F(q). Therefore, for every q # M 2 there exists Wq # W with Wq( p)=0 for all p # M 1 and Wq(q)=1. Define W$=max[Wq | q # M 2 ]. W contains with two and therefore with any finite number of elements also their maximum. Thus, W$ # W, W$(q)=0 for all q # M 1 and W$(q)=1 for all q # M 2 . In conclusion, W$=F # W and W is closed. Assume additionally that W1 , W2 # W min(W1 , W2 ) # W. Let F: Q L be a mapping with \p, q # Q
\W # W (W(q) L W( p) F(q) L F( p)).
We have to show F # W. Let M 1 , M 2 be defined as above. Let p # M 1 , thus F( p)< L F(q) for all q # M 2 . By assumption on F we have c(\W # W (W(q) L W( p))), and for every q # M 2 there exists Wq # W with Wq( p)< L Wq(q), i.e., Wq( p)=0 and Wq( q)=1. Define Wp =max[Wq | q # M 2 ]. Since W contains with every finite number of elements also their maximum this implies Wp # W. By definition of Wp we can conclude Wp( p)=0 and Wp(q)=1 for all q # M 2 . Define W$=min[Wp | p # M 1 ]. As above this implies W$ # W. In conclusion, W$( p)=0 for p # M 1 and W$( p)=1 for p # M 2 . Therefore, W$=F # W and W is strictly closed. As a consequence we call a closed knowledge structure on (Q, L) a knowledge space on (Q, L) and a strictly closed knowledge structure on (Q, L) a q.-o. knowledge space on (Q, L). GALOIS CONNECTIONS
In the following it will be shown that there is a one-to-one correspondence between the surmise relations on Q and the q.-o. knowledge spaces on (Q, L). This one-to-one correspondence isas in the special case L=[0, 1] (see Doignon 6 Falmagne, 1985)introduced by a Galois connection. The concept of a Galois connection is described in the Appendix.
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Define :(R)=[W: Q L | \p, q # Q( pRq W(q) L W( p))] ;(W)=[( p, q) | \W # W (W(q) L W( p))] for all binary relations R on Q and all knowledge structures W on (Q, L). Theorem 3. (:, ;) establishes a Galois connection between (REL(Q), ) and (WS(Q, L), ). The closed subsets for the closure operators : b ; and ; b : are the surmise relations on Q respectively the strictly closed knowledge structures on (Q, L). The mappings :, ; define a one-to-one correspondence between the surmise relations on Q and the q.-o. knowledge spaces on (Q, L). Proof. It will be shown that (:, ;) has properties (a)(f) of a Galois connection. Since REL(Q) and WS(Q, L) are partially ordered by properties (a) and (b) follow immediately. C2 # REL(Q) with = C1 C Assume = C1 , = =2 and W # ). Therefore, pC q implies W(q) W( p) for all p, :(C 2 2 L = = we can conclude that pC q # Q. From = C1 C 2 = =1 q implies W(q) L W( p) for all p, q # Q and therefore W # :(C =1 ). ) :(C ) and (:, ;) has property (c). Thus, :(C 2 1 = = Let W1 , W2 # WS(Q, L) with W1 W2 and let ( p, q) # ;(W2 ). Therefore, W(q) L W( p) for each W # W2 . From W1 W2 we can conclude W(q) L W( p) for each W # W1 and therefore ( p, q) # ;(W1 ). Hence, ;(W2 ) ;(W1 ) and (:, ;) has property (d). C . Therefore, W(q) L Assume = C # REL(Q) and (p, q) # = W( p) for each W # :(C = ). This implies ( p, q) # ;(:(C = )). Thus, = C ;(:(C = )) and (:, ;) has property (e). Let W # WS(Q, L) and W # W. Therefore, ( p, q) # ;(W) W(q) L W( p) for all p, q # Q. This implies W # :(;(W)). Hence, W :(;(W)) and (:, ;) has property (f). By Theorem 1 (see Appendix), : b ; is a closure operator on WS(Q, L) and ; b : is a closure operator on REL(Q). What are the closed elements concerning these closure operators? Let R be a binary relation on Q with R SR(Q). By Lemma 1, ;(W) # SR(Q) for each W # WS(Q, L). Therefore, ; b :(R) # SR(Q) and ; b :(R){R. Assume now R # SR(Q) and ( p, q) # R. Thus, W(q) L W( p) for each W # :(R) and therefore ( p, q) # ;(:(R)). Let ( p, q) R. There exists at least one W # :(R) with W(q)>L W( p) and thus ( p, q) ;(:(R)). Therefore, R # SR(Q) implies R=;(:(R)). Hence, the closed elements concerning ; b : are the elements of SR(Q). Let W QKS(Q, L). By Lemma 2, :(R) # QKS(Q, L) for every binary relation R. Therefore, : b ;(W) # QKS(Q, L) and W{: b ;(W). Let W # QKS(Q, L) and W # W. For every ( p, q) # ;(W) we have W(q) L W( p), which implies W # : b ;(W).
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Assume W W. There exists p, q # Q with W$(q) L W$( p) for each W$ # W and W(q)>L W( p). Thus, ( p, q) # ;(W) and W : b ;(W). Therefore, W : b ;(W) and W= : b ;(W). Hence, the closed elements concerning : b ; are the elements of QKS(Q, L). By Theorem A (see Appendix), (:, ;) defines a one-toone correspondence between the elements of SR(Q) and QKS(Q, L). Example 3. Let Q, L, L and W3 be defined Example 2. Define a surmise relation = C on Q by [(q 1 , q 1 ), (q 2 , q 2 ), (q 3 , q 3 ), (q 1 , q 2 ), (q 1 , q 3 )]. :(C = )=W3 .
as in C == Then
It will be shown in the following theorem that there is a similar one-to-one correspondence between the surmise functions on Q and the knowledge spaces on (Q, L). Define #($)=[W : Q L | \q # Q _K # $(q) \p # K(W(q) L W( p))] {(W)=[(q, [K 1 , ..., K n ]) | _W # W (K i =[ p | W(q) L W( p)]) 7 \W$ # W ([ p | W$(q) L W$( p)] / 3 K)] for all mappings $ : Q ^(^(Q)) and knowledge structures W on (Q, L). SM(Q) is quasi-ordered by the relation R (see Doignon 6 Falmagne, 1985) defined by $ 1 R$ 2 \q # Q \K # $ 2(q)
_K$ # $ 1(q)(K$ K).
Theorem 4. (#, {) establishes a Galois connection between (SM(Q), R ) and (WS(Q, L), ). The closed elements for the closure operators # b { and { b # are the surmise functions on Q respectively the closed knowledge structures on (Q, L). The mappings #, { define a one-to-one correspondence between the surmise functions on Q and the knowledge spaces on (Q, L). Proof. It will be shown that (#, {) has properties (a)(f) of a Galois connection. Since WS(Q, L) is partially ordered by properties (a) and (b) follows immediately. Let $ 1 , $ 2 # SM(Q) with $ 1 R$ 2 and W # #($ 2 ). Therefore, for each q # Q there exists a K # $ 2(q) with W(q) L W( p) for each p # K. The condition $ 1 R$ 2 implies that for each q # Q and each K # $ 2(q) there exists a K$ # $ 1(q) with K$ K. This implies that for each q # Q there is a K$ # $ 1(q) so that W(q) L W( p) for each p # K$. Thus, W # #($ 1 ) and therefore #($ 2 ) #($ 1 ). Hence, (#, {) has property (c).
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Let W1 , W2 # WS(Q, L) with W1 W2 . Let q # Q, {(W1 )=$ 1 and {(W2 )=$ 2 . Let K # $ 1(q). Then there exists W # W1 with K=[ p | W(q) L W( p)] and K is minimal with this property. Since W1 W2 this implies the existence of W # W2 with K=[ p | W(q) L W( p)] and therefore there is a K$ K with K$ # $ 2(q). This implies $ 2 R$ 1 and (#, {) has property (d). Let $ # SM(Q) and q # Q. Let K # {(#($))(q). Therefore, there exists W # #($) with K=[ p | W(q) L W( p)] and K minimal with this property. This implies the existence of K$ # $(q) with W(q) L W( p) for all p # K$, i.e., K$ K. Therefore, $R{(#($)) and (#, {) has property (e). Let W # WS(Q, L) and W # W. For every q # Q there exists C q [ p | W(q) L W( p)] with C q # {(W). For each p # C q we have W(q) L W( p), i.e., W # #({(W)). This implies W #({(W)) and (#, {) has property (f). Therefore, (#, {) is a Galois connection. By Theorem A (see Appendix), # b { is a closure operator on WS(Q, L) and { b # is a closure operator on SM(Q). Again we have to determine the closed elements concerning these closure operators. Assume ${SF(Q). By Lemma 1, {(W) # SF(Q) for all W # WS(Q, L). Therefore, {(#($)) # SF(Q), and ${{ b #($). Assume now $ # SF(Q). Let K # $(q) for q # Q. For every W # #($) this implies that for each q # Q there is a K # $(q) with W(q) L W( p) for all p # K. Since K is a clause of q there exists no W # #($) with [ p | W(q) L W( p)]/K. Let l 1 , l 2 # L with l 1 L l 2 . Then WK :QL defined by WK (p)= l 2 for p # K and W K ( p)=l 1 for p K is an element of #($). This implies K # {(#($))(q). Therefore, K # $(q) implies K # {(#($))(q). Let K $(q) for q # Q. Assume that there exists a W # #($) with K=[ p | W(q) L W( p)]. This implies by definition of #($) that there exists K$ # $(q) with W(q) L W( p) for each p # K$ and therefore K$/K. Thus, K {(#($))(q). In conclusion $ # SF(Q) implies $={(#($)). Hence, the closed elements concerning { b # are the elements of SF(Q). Assume W KS(Q, L). By Lemma 2, #($) # KS(Q, L) for every $ # SM(Q). Therefore, # b {(W) # KS(Q, L) and # b {(W){W. Assume now W # KS(Q, L). Let W # W and q # Q. There exists K # {(W)(q) with K [ p | W(q) L W( p)]. Therefore, for each q # Q there is a K # {(W)(q) with W(q) L W( p) for each p # K. This implies W # #({(W)). Let W W. Since W # KS(Q, L) there exists q # Q so that for each W$ # W there exists a p # Q with W$(q) L W$( p) 7 W(q)>L W( p). Therefore, there is no K # {(W)(q) with K=[ p | W(q) L W( p)] which implies W # b {(W). Thus, W # KS(Q, L) implies W=#({(W)). Hence, the closed elements concerning # b { are the elements of KS(Q, L). By Theorem A (see Appendix), (#, {) define a one-to-one correspondence between SF(Q) and KS(Q, L).
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Example 4. Let Q, L, L and W2 be defined as in Example 2. Define a surmise function $ on Q by $(q 1 )= [[q 1 ]], $(q 2 )=[[q 1 , q 2 ], [q 2 , q 3 ]], and $(q 3 )=[[q 1 , q 3 ], [q 2 , q 3 ]]. Then #($)=W2 . CONCLUSIONS
It was shown that is possible to generalize the central concepts of knowledge space theory to problem domains in which the answer of a subject to a problem can be classified by an element of a linear scale concerning its quality. For the special case of a linear scale with only two elements the generalized concepts can be identified with the original ones. The connections between the central concepts (surmise function, surmise relation, knowledge space, quasiordinal knowledge space) are valid in our generalization. If a knowledge structure on (Q, L), i.e., the set of all possible response patterns of subjects, fulfills certain closure properties it can be represented by a surmise function or surmise relation. The generalization of the theory of knowledge spaces was motivated by difficulties in testing psychological models of problem solving processes. One such approach is described in Schrepp (1993). In this paper a strictly closed knowledge structure, i.e., a q.-o. knowledge space, was derived from a model on problem solving processes in the field of inductive reasoning. The q.-o. knowledge space describes the set of all patterns of response times to a set of 24 letter series completion problems which are consistent with the model. The q.-o. knowledge space was compared to response patterns of subjects obtained in an experiment. The results show that the model is able to describe the general answer behavior but fails for a special subclass of problems. Thus the model is unable to explain some major influences of the problem structure on the difficulty to solve a letter series completion problem. APPENDIX
Below is a summary of some basic concepts of Galois connections (see Doignon 6 Falmagne, 1985) which are necessary for the understanding of the proofs for Theorems 3 and 4. Definition A. Let ( X, OX ), (Y, OY ) be quasi orders. Let f : X Y and g: Y X be mappings with: (a)
xOX x$ 7 x$OX x f (x)= f (x$)
(b)
yOY y$ 7 y$OY y g( y)= g( y$)
(c)
x OX x$ f (x$) OY f (x)
(d)
yOY y$ g( y$) OX g( y)
(e)
x OX g b f (x)
(f )
y OY f b g( y).
KNOWLEDGE SPACE THEORY
Then ( f, g) is called Galois connection between (X, OX ) and Y( OY ). Notice that conditions (a) and (b) are satisfied if OX or OY is a partial order. Theorem A. Let (X, OX ) and (Y, OY ) be quasi-orders and ( f, g) a Galois connection between (X, OX ) and (Y, OY ). Then: (a)
g b f is a closure operator on ( X, OX ),
(b)
f b g is a closure operator on ( Y, OY ).
Define X 0 =[x | g b f ( g)=x] and Y 0 =[ y | f b g( y)= y]. Then (c) For f 0 = f |X0 : X 0 Y 0 and g 0 = g |Y0 : Y 0 X 0 we have f &1 0 = g0 . REFERENCES Albert, D., Schrepp, M., 6 Held, T. (1994). Construction of knowledge spaces for problem solving in chess. In G. H. Fischer 6 D. Laming
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(Eds.), Contributions to mathematical psychology, psychometrics, and methodology. Springer: BerlinHeidelbergNew York. Birkhoff, G. (1937). Rings of sets. Duke Mathematical Journal, 16, 443454. Doignon, J.-P., 6 Falmagne, J.-C. (1985). Spaces for the assessment of knowledge. International Journal of Man-Machine Studies, 23, 175196. Falmagne, J.-C., 6 Doignon, J.-P. (1988a). A Markovian procedure for assessing the state of a system. Journal of Mathematical Psychology, 32, 232258. Falmagne, J.-C., 6 Doignon, J.-P. (1988b). A class of stochastic procedures for assessing the state of a system. British Journal of Mathematical and Statistical Psychology, 41, 123. Held, T. (1993). Establishment and empirical validation of problem structures based on domain specific skills and textual properties. Unpublished dissertation, University of Heidelberg. Korossy, K. (1993). Modellierung von Wissen als Kompetenz und Performanz [Modeling of knowledge as competence and performance]. Unpublished dissertation, University of Heidelberg. Schrepp, M. (1993). Uber die Beziehung zwischen kognitiven Prozessen und Wissensraumen beim Problemlosen [On the connection between cognitive processes and knowledge spaces in problem solving]. Unpublished dissertation, University of Heidelberg. Schrepp, M. (1995). Modeling interindividual differences in solving letter series completion problems. Zeitschrift fur Psychologie, 203, 173188. Received: March 21, 1997
A Generalization of Knowledge Space Theory to Problems with More Than Two Answer Alternatives Martin Schrepp Ladenburg, Germany
In the theory of knowledge spaces (Doignon 6 Falmagne, 1985) a knowledge domain is represented by a set Q of problems. The knowledge of a subject in the knowledge domain is described by the subset of problems from the domain that the subject is able to solve. A central assumption is therefore that the answer of a subject to a problem can be classified as either correct or incorrect. We show that the central concepts of the theory can be generalized to problem domains in which solutions can be evaluated on a linear scale concerning their quality. ] 1997 Academic Press
THEORETICAL BACKGROUND
In the theory of knowledge spaces (Doignon 6 Falmagne; 1985) a knowledge domain is represented as a finite set Q of questions. The subset W of questions from Q that a subject is capable of solving is called the knowledge state of that subject. The set of all possible knowledge states is called the knowledge structure on Q. A knowledge structure W which is closed under union, i.e., W1 , W2 # W implies W1 _ W2 # W, is called a knowledge space on Q. A knowledge space which is additionally closed under intersection, i.e., W1 , W2 # W implies W1 & W2 # W, is called a quasi-ordinal (q.-o.) knowledge space on Q. The closure properties of knowledge spaces and q.-o. knowledge spaces permit an alternative description of such spaces. A reflexive and transitive relation = C on Q is called a surmise relation on Q. For p, q # Q the interpretation of pC =q is ``Every subject who is able to solve q is also able to solve p.'' Let ^(Q) be the power set, i.e., the set of all subsets, of Q. A mapping $ : Q ^(^(Q)) is called a surmise function if it has the following properties: 1. W # $(q) q # W,
Theorem 1 (Birkhoff, 1937). The formula pC =q \W # W (q # W p # W) defines a one-to-one correspondence : between QKS(Q) and SR(Q). Therefore, every q.-o. knowledge space W on Q is completely characterized by a surmise relation = C and vice versa. The following result establishes a similar connection between knowledge spaces and surmise functions. Theorem 2 (Doignon 6 Falmagne, 1985).
The formula
K # $(q) _W # W (q # W 7 K W)
2. W # $(q) 7 p # W _W$ # $( p)(W$ W ),
7 \W # W (q # W W/3 K )
3. W, W$ # $(q) 7 W W$ W=W$. I am grateful to professor J. C. Falmagne and four anonymous referees for their suggestions concerning the mathematical argumentation and to Diana Bursy for improving the English style. Correspondence and reprint requests should be sent to Martin Schrepp, Am Bahndamm 26, 68526 Ladenburg, Germany. 237
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The interpretation of a surmise function $(q)=[W1 , ..., Wn ] is ``Every subject who is able to solve q is able to solve all problems from at least one element Wi of $(q).'' The elements of $(q) are called clauses of q. The clauses of q are interpreted as sets of prerequisites for q. A subject is able to solve a problem q if she or he is able to solve all problems from at least one prerequisite set Wi # $(q) for this problem. Define QKS(Q) to be the set of all q.-o. knowledge spaces on Q, KS(Q) the set of all knowledge spaces on Q, SR(Q) the set of all surmise relations on Q, SF(Q) the set of all surmise functions on Q, REL(Q) the set of all binary relations on Q, and, SM(Q) the set of all mappings from Q to ^(^(Q)). Notice that a surmise function is a proper generalization of a surmise relation, since a surmise function $ for which every q # Q has exactly one clause K q corresponds to a surmise relation = C through pC =q W p # K q . The connection of q.-o. knowledge spaces with surmise relations is established by the following theorem.
establishes a one-to-one correspondence # between KS(Q) and SF(Q). Therefore, every knowledge space W on Q is completely characterized by a surmise function $ and vice versa. The 0022-249697 25.00 Copyright 1997 by Academic Press All rights of reproduction in any form reserved.
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clauses of a problem q are the -minimal subsets of Q which contain q. There are two main applications of knowledge space theory. First, knowledge spaces can be used for an efficient assessment of knowledge. Since a knowledge structure does not in general contain all possible elements of the power set of Q it can be used to infer the answers of subjects to particular problems from the answers they have already given. Therefore, an adaptive procedure for the assessment of knowledge can be established (Falmagne 6 Doignon, 1988a or Falmagne 6 Doignon; 1988b). Second, knowledge spaces can be used to test psychological models of problem solving processes. In this approach a knowledge space is derived from assumptions on the problem solving processes of subjects in a knowledge domain. The derived space is then compared to observed response patterns of subjects. Such tests of psychological models are described for problem solving in chess (Albert, Schrepp 6 Held; 1994), geometry (Korossy; 1993), probability theory (Held; 1993), and letter series completion (Schrepp; 1993, 1995). Psychological models of problem solving processes describe the basic cognitive abilities subjects must possess in order to solve problems from the underlying knowledge domain. Interindividual differences are described by different shapings of these abilities. Every shaping of abilities or skills corresponds to a solution pattern. Such a detailed analysis of the cognitive processes allows one often not only to predict whether or not a subject with specific abilities will solve a problem, but also to predict how far the subject will come in hisher effort to solve the problem. Therefore, the assumption that every problem is solved either correctly or incorrectly by a subject is too restrictive. We must differentiate between several degrees of correctness of a given solution. These degrees of correctness should correspond to the subject's understanding of the domain. The quality of a solution may, for example, be judged by points. The better the solution is the more points are assigned to it. Another example is speed problems in which every subject is able to find the correct solution if enough time is available. The performance of a subject on such speed problems is measured by the amount of time required to find the solution. The faster the problem is solved, the better is the performance of the subject. In the following section we describe a generalization of the central concepts of knowledge space theory to problems of this type. GENERALIZATION OF THE THEORY
Let Q be a finite set of problems. We assume that the quality of a solution to a problem q # Q can be measured by an element l # L, where L is a linear ordered set. We assume
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that L contains at least two elements. Let L be the linear order on L. The performance of a subject on the problems in Q can be described by the values in L the subject obtains, i.e., by a mapping W : Q L. We call such a mapping a knowledge state. A set W{< of knowledge states is calledas abovea knowledge structure on (Q, L). Notice that for L=[0, 1] these generalized concepts are identical with the original concepts, since every subset of Q can be identified with its characteristic function. Let, for example, Q be a set of speed problems. In this case L=[0, [ and L = . An element l # L represents the time required to solve a problem. The knowledge state W : Q L of a subject assigns to every problem q # Q the time the subject needs to solve q. Since surmise relations and surmise functions describe the relative difficulty of problems their formal definitions can be used unchanged. We interpret pC =q as ``Every subject reaches a better (or equal) result in solving p than in solving q,'' i.e., pC =q W(q) L W( p) for every possible knowledge state W. The interpretation of $(q) is ``Every subject reaches in all problems of at least one clause in $(q) a better (or equal) result than in q,'' i.e., for every possible knowledge state W there exists a K in $(q) with W(q) L W( p) for all p # K. These interpretations are the reason for our assumptions of a linear order on L. We have to ensure that the solutions given to problems q # Q are comparable in quality in order to be able to relate two problems by a surmise relation respectively surmise function. It is easy to determine the knowledge structures on (Q, L) corresponding to a given surmise relation = C and surmise function $. Definition 1. Let = C be a binary relation on Q and let $ be a mapping $ : Q ^(^(Q)). We define W(C = )=[W : Q L | \p, q # Q( pC =q W(q) L W( p))], W($)=[W : Q L | \q # Q_K # $(q) \p # K(W(q) L W( p))]. In analogy to the previous section we call W (C = ) the q.-o. knowledge space corresponding to = C and W ($) the knowledge space corresponding to $. In the original case (L=[0, 1]) knowledge spaces and q.-o. knowledge spaces are defined by closure properties (closure against union, closure against union, and intersection). The question arises of how these closure properties can be generalized. A natural assumption would be that a knowledge structure which is closed under maximum, i.e., W1 , W2 # W implies max(W1 , W2 ) # W, is in a one-to-one
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correspondence with a surmise function and that a knowledge structure which is additionally closed under minimum, i.e., W1 , W2 # W implies min(W1 , W2 ) # W, is in a oneto-one correspondence with a surmise relation. The following example shows that this is not true if L contains more than two elements.
Definition 2. Let W be a knowledge structure on (Q, L). W is called strictly closed if for all F : Q L the condition
Example 1. Let L=[1, 2, 3] and Q=[ p, q]. The knowledge structure W=[(1, 1), (3, 3)] is closed under minimum and maximum. But W does not correspond to a surmise relation or surmise function, since (2, 2) W and every knowledge structure on (Q, L) corresponding to a surmise relation or surmise function contains all constant mappings from Q to L.
implies F # W. W is called closed if for all F : Q L the condition
Definition 1 shows how a knowledge structure on (Q, L) can be derived from a surmise relation respectively surmise function on Q. The following result shows how a surmise relation and a surmise function can be derived from a knowledge structure on (Q, L). Lemma 1. Then
Let W be a knowledge structure on (Q, L).
\p, q # Q
\W # W (W(q) L W( p) F(q) L F( p))
\q # Q _W # W \p # Q(W(q) L W( p) F(q) L F( p)) implies F # W. Define WS(Q, L) to be the set of all knowledge structures on (Q, L), KS(Q, L) the set of all closed knowledge structures on (Q, L), and, QKS(Q, L) the set of all strictly closed knowledge structures on (Q, L). Let W be a strictly closed knowledge structure on (Q, L) and F : Q L with \q # Q _W # W \p # Q(W(q) L W( p) F(q) L F( p)).
pC =q \W # W (W(q) L W( p)) defines a surmise relation = C on Q and K # $(q) _W # W (K=[ p | W(q) L W( p)]) 7 \W$ # W ([ p | W$(q) L W$( p)] /3 K )
For all p, q the condition \W$ # W (W$(q) L W$( p)) implies F(q) L F( p) and since W is strictly closed, F # W. Therefore, every strictly closed knowledge structure is closed. Example 2. Let Q=[q 1 , q 2 , q 3 ], L=[0, 1, 2] and L = . Define
defines a surmise function on Q. Proof. First, it will be shown that = C is a surmise relation. The reflexivity of = C follows immediately. Let pC =q r. For each W # W this implies W(q) W( p) and and qC L = W(r) L W(q). Since L is transitive, W(r) L W( p) for C is transitive. each W # W. Hence, pC =r and = Second, it will be shown that $ is a surmise function. The condition K # $(q) q # K follows immediately. Let K # $(q) and p # K. There exists W # W with K= [r | W(q) L W(r)]. The conclusion is that W(q) L W( p). There exists K$ [r | W( p) L W(r)] with K$ # $( p). Since L is transitive, K$ K. Let K, K$ # $(q) and K$ K. There exists W, W$ # W with K=[ p | W(q) L W( p)] and K$= [ p | W$(q) L W$( p)]. Finally, [ p | W$(q) L W$( p)] /3 K for each W$ # W implies K$=K. Therefore, $ is a surmise function on Q. GENERALIZED CLOSURE PROPERTIES
As Example 1 shows, a knowledge structure on (Q, L) must fulfill stronger closure properties to be in a one-to-one correspondence with a surmise function or surmise relation on Q.
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W1 =[(0, 0, 0), (2, 1, 0), (2, 2, 2)], W2 =[(0, 0, 0), (1, 1, 1), (2, 2, 2), (1, 0, 0), (2, 0, 0), (2, 1, 2), (2, 1, 1), (1, 0, 1), (2, 0, 2), (2, 0, 1), (1, 2, 2), (1, 1, 0), (0, 1, 1), (2, 1, 0), (2, 2, 0), (2, 2, 1), (0, 2, 2)], W3 =[(0, 0, 0), (1, 1, 1), (2, 2, 2), (1, 0, 0), (1, 1, 0), (1, 0, 1), (2, 0, 0), (2, 1, 0), (2, 2, 0), (2, 0, 1), (2, 0, 2), (2, 1, 1), (2, 2, 1), (2, 1, 2)]. Let W1 =(0, 0, 0) and W2 =(1, 1, 1). Then for each q # Q we have W1(q) L W1( p) W2(q) L W2( p) for all p # Q, but W2 W1 . Therefore, W1 is not closed. W2 is closed, but not strictly closed. Choose p, q # Q with p{q. From (2, 1, 0), (1, 0, 1), (0, 1, 1) # W2 we can conclude c\W # W2(W(q) L W( p)). For W3 =(0, 1, 0) this implies W(q) L W( p) W3(q) L W3( p) for each W # W2 , but W3 W2 . W3 is strictly closed.
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The following result shows that the (quasi-ordinal) knowledge space corresponding to a surmise function (relation) is (strictly) closed. Lemma 2. Let = C be a binary relation on Q and let $ be a mapping $ : Q ^(^(Q)). W ($) is closed and W (C = ) is strictly closed. Proof.
First, it will be shown that
W (C = )=[W | \p, q # Q( pC =q W(q) L W( p))] is strictly closed. Let F be a mapping from Q to L. Assume W(q) L W( p) F(q) L F( p) for all p, q # Q and each W # W (C = ). For all p, q with pC =q and for each W # W (C =) we have W(q) L W( p). Therefore, F(q) L F( p). So pC = q F(q) L F( p) for all p, q # Q, which implies F # W (C = ). Therefore, W (C = ) is strictly closed. Second, it will be shown that W ($)=[W | \q # Q _K # $(q) \p # K(W(q) L W( p))] is closed. Let F be a mapping from Q to L. Assume that for each q # Q there is a W # W ($) so that W(q) L W( p) F(q) L F( p) for each p # Q. Let r # Q. Because of our assumptions on F, there exists W # W ($) with W(r) L W( p) F(r) L F( p) for each p # Q. Therefore, there exists K # $(r) with W(r) L W( p) for each p # K, which implies F(r) L F( p) for each p # K. Thus, F # W ($) and W ($) is closed. The following results show that the (strictly) closed knowledge structures on (Q, L) are a generalization of the (quasi-ordinal) knowledge spaces on Q, since for the special case L=[0, 1] these concepts can be identified. Lemma 3. Let W be a closed knowledge structure on (Q, L). Then W1 , W2 # W implies max(W1 , W2 ) # W. If W is strictly closed, then W1 , W2 # W implies min(W1 , W2 ) # W. Proof. Let W be a closed knowledge structure on (Q, L), q # Q and W1 , W2 # W. Assume without loss of generality max(W1 , W2 )(q)=W1(q). Therefore, W1(q) LW1( p) implies max(W1 , W2 )(q) L max(W1 , W2 )( p) for each p # Q. Since W1 # W and W is closed, max(W1 , W2 ) # W. Let W be strictly closed and W1 , W2 # W. Let p, q # Q with W(q) L W( p) for each W # W. Therefore, W1(q) L W1( p), W2( q) L W2( p), and min(W1 , W2 )(q) L min(W1 , W2 )( p). The conclusion is that for all p, q # Q and for each W # W we have W(q) LW( p) min(W1 , W2 )(q) L min(W1 , W2 )( p). Since W is strictly closed, min(W1 , W2 ) # W. Define for L=[0, 1] the mappings < L , Q L : Q L by < L(q)=0 for all q # Q and Q L(q)=1 for all q # Q.
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Lemma 4. Let L=[0, 1] and W be a knowledge structure on (Q, L). If < L , Q L # W and W1 , W2 # W implies max(W1 , W2 ) # W, then W is closed. If additionally W1 , W2 # W implies min(W1 , W2 ) # W, then W is strictly closed. Proof. Assume < L , Q L # W and that W1 , W2 # W max(W1 , W2 ) # W. Let F : Q L be a mapping with \q # Q _W # W \p # Q(W(q) L W( p) F(q) L F( p)). We have to show F # W. Define M 1 =[q | F(q)=0] and M 2 =[q | F(q)=1]. Let q # M 2 , which implies F( p)< L F(q) for all p # M 1 . By assumption on F it follows that there exists W # W, so that W(q) L W( p) F(q) L F( p) for each p # Q. For p # M 1 this implies W( p)< L W(q) for every such W, since F( p)< L F(q). Therefore, for every q # M 2 there exists Wq # W with Wq( p)=0 for all p # M 1 and Wq(q)=1. Define W$=max[Wq | q # M 2 ]. W contains with two and therefore with any finite number of elements also their maximum. Thus, W$ # W, W$(q)=0 for all q # M 1 and W$(q)=1 for all q # M 2 . In conclusion, W$=F # W and W is closed. Assume additionally that W1 , W2 # W min(W1 , W2 ) # W. Let F: Q L be a mapping with \p, q # Q
\W # W (W(q) L W( p) F(q) L F( p)).
We have to show F # W. Let M 1 , M 2 be defined as above. Let p # M 1 , thus F( p)< L F(q) for all q # M 2 . By assumption on F we have c(\W # W (W(q) L W( p))), and for every q # M 2 there exists Wq # W with Wq( p)< L Wq(q), i.e., Wq( p)=0 and Wq( q)=1. Define Wp =max[Wq | q # M 2 ]. Since W contains with every finite number of elements also their maximum this implies Wp # W. By definition of Wp we can conclude Wp( p)=0 and Wp(q)=1 for all q # M 2 . Define W$=min[Wp | p # M 1 ]. As above this implies W$ # W. In conclusion, W$( p)=0 for p # M 1 and W$( p)=1 for p # M 2 . Therefore, W$=F # W and W is strictly closed. As a consequence we call a closed knowledge structure on (Q, L) a knowledge space on (Q, L) and a strictly closed knowledge structure on (Q, L) a q.-o. knowledge space on (Q, L). GALOIS CONNECTIONS
In the following it will be shown that there is a one-to-one correspondence between the surmise relations on Q and the q.-o. knowledge spaces on (Q, L). This one-to-one correspondence isas in the special case L=[0, 1] (see Doignon 6 Falmagne, 1985)introduced by a Galois connection. The concept of a Galois connection is described in the Appendix.
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Define :(R)=[W: Q L | \p, q # Q( pRq W(q) L W( p))] ;(W)=[( p, q) | \W # W (W(q) L W( p))] for all binary relations R on Q and all knowledge structures W on (Q, L). Theorem 3. (:, ;) establishes a Galois connection between (REL(Q), ) and (WS(Q, L), ). The closed subsets for the closure operators : b ; and ; b : are the surmise relations on Q respectively the strictly closed knowledge structures on (Q, L). The mappings :, ; define a one-to-one correspondence between the surmise relations on Q and the q.-o. knowledge spaces on (Q, L). Proof. It will be shown that (:, ;) has properties (a)(f) of a Galois connection. Since REL(Q) and WS(Q, L) are partially ordered by properties (a) and (b) follow immediately. C2 # REL(Q) with = C1 C Assume = C1 , = =2 and W # ). Therefore, pC q implies W(q) W( p) for all p, :(C 2 2 L = = we can conclude that pC q # Q. From = C1 C 2 = =1 q implies W(q) L W( p) for all p, q # Q and therefore W # :(C =1 ). ) :(C ) and (:, ;) has property (c). Thus, :(C 2 1 = = Let W1 , W2 # WS(Q, L) with W1 W2 and let ( p, q) # ;(W2 ). Therefore, W(q) L W( p) for each W # W2 . From W1 W2 we can conclude W(q) L W( p) for each W # W1 and therefore ( p, q) # ;(W1 ). Hence, ;(W2 ) ;(W1 ) and (:, ;) has property (d). C . Therefore, W(q) L Assume = C # REL(Q) and (p, q) # = W( p) for each W # :(C = ). This implies ( p, q) # ;(:(C = )). Thus, = C ;(:(C = )) and (:, ;) has property (e). Let W # WS(Q, L) and W # W. Therefore, ( p, q) # ;(W) W(q) L W( p) for all p, q # Q. This implies W # :(;(W)). Hence, W :(;(W)) and (:, ;) has property (f). By Theorem 1 (see Appendix), : b ; is a closure operator on WS(Q, L) and ; b : is a closure operator on REL(Q). What are the closed elements concerning these closure operators? Let R be a binary relation on Q with R SR(Q). By Lemma 1, ;(W) # SR(Q) for each W # WS(Q, L). Therefore, ; b :(R) # SR(Q) and ; b :(R){R. Assume now R # SR(Q) and ( p, q) # R. Thus, W(q) L W( p) for each W # :(R) and therefore ( p, q) # ;(:(R)). Let ( p, q) R. There exists at least one W # :(R) with W(q)>L W( p) and thus ( p, q) ;(:(R)). Therefore, R # SR(Q) implies R=;(:(R)). Hence, the closed elements concerning ; b : are the elements of SR(Q). Let W QKS(Q, L). By Lemma 2, :(R) # QKS(Q, L) for every binary relation R. Therefore, : b ;(W) # QKS(Q, L) and W{: b ;(W). Let W # QKS(Q, L) and W # W. For every ( p, q) # ;(W) we have W(q) L W( p), which implies W # : b ;(W).
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Assume W W. There exists p, q # Q with W$(q) L W$( p) for each W$ # W and W(q)>L W( p). Thus, ( p, q) # ;(W) and W : b ;(W). Therefore, W : b ;(W) and W= : b ;(W). Hence, the closed elements concerning : b ; are the elements of QKS(Q, L). By Theorem A (see Appendix), (:, ;) defines a one-toone correspondence between the elements of SR(Q) and QKS(Q, L). Example 3. Let Q, L, L and W3 be defined Example 2. Define a surmise relation = C on Q by [(q 1 , q 1 ), (q 2 , q 2 ), (q 3 , q 3 ), (q 1 , q 2 ), (q 1 , q 3 )]. :(C = )=W3 .
as in C == Then
It will be shown in the following theorem that there is a similar one-to-one correspondence between the surmise functions on Q and the knowledge spaces on (Q, L). Define #($)=[W : Q L | \q # Q _K # $(q) \p # K(W(q) L W( p))] {(W)=[(q, [K 1 , ..., K n ]) | _W # W (K i =[ p | W(q) L W( p)]) 7 \W$ # W ([ p | W$(q) L W$( p)] / 3 K)] for all mappings $ : Q ^(^(Q)) and knowledge structures W on (Q, L). SM(Q) is quasi-ordered by the relation R (see Doignon 6 Falmagne, 1985) defined by $ 1 R$ 2 \q # Q \K # $ 2(q)
_K$ # $ 1(q)(K$ K).
Theorem 4. (#, {) establishes a Galois connection between (SM(Q), R ) and (WS(Q, L), ). The closed elements for the closure operators # b { and { b # are the surmise functions on Q respectively the closed knowledge structures on (Q, L). The mappings #, { define a one-to-one correspondence between the surmise functions on Q and the knowledge spaces on (Q, L). Proof. It will be shown that (#, {) has properties (a)(f) of a Galois connection. Since WS(Q, L) is partially ordered by properties (a) and (b) follows immediately. Let $ 1 , $ 2 # SM(Q) with $ 1 R$ 2 and W # #($ 2 ). Therefore, for each q # Q there exists a K # $ 2(q) with W(q) L W( p) for each p # K. The condition $ 1 R$ 2 implies that for each q # Q and each K # $ 2(q) there exists a K$ # $ 1(q) with K$ K. This implies that for each q # Q there is a K$ # $ 1(q) so that W(q) L W( p) for each p # K$. Thus, W # #($ 1 ) and therefore #($ 2 ) #($ 1 ). Hence, (#, {) has property (c).
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MARTIN SCHREPP
Let W1 , W2 # WS(Q, L) with W1 W2 . Let q # Q, {(W1 )=$ 1 and {(W2 )=$ 2 . Let K # $ 1(q). Then there exists W # W1 with K=[ p | W(q) L W( p)] and K is minimal with this property. Since W1 W2 this implies the existence of W # W2 with K=[ p | W(q) L W( p)] and therefore there is a K$ K with K$ # $ 2(q). This implies $ 2 R$ 1 and (#, {) has property (d). Let $ # SM(Q) and q # Q. Let K # {(#($))(q). Therefore, there exists W # #($) with K=[ p | W(q) L W( p)] and K minimal with this property. This implies the existence of K$ # $(q) with W(q) L W( p) for all p # K$, i.e., K$ K. Therefore, $R{(#($)) and (#, {) has property (e). Let W # WS(Q, L) and W # W. For every q # Q there exists C q [ p | W(q) L W( p)] with C q # {(W). For each p # C q we have W(q) L W( p), i.e., W # #({(W)). This implies W #({(W)) and (#, {) has property (f). Therefore, (#, {) is a Galois connection. By Theorem A (see Appendix), # b { is a closure operator on WS(Q, L) and { b # is a closure operator on SM(Q). Again we have to determine the closed elements concerning these closure operators. Assume ${SF(Q). By Lemma 1, {(W) # SF(Q) for all W # WS(Q, L). Therefore, {(#($)) # SF(Q), and ${{ b #($). Assume now $ # SF(Q). Let K # $(q) for q # Q. For every W # #($) this implies that for each q # Q there is a K # $(q) with W(q) L W( p) for all p # K. Since K is a clause of q there exists no W # #($) with [ p | W(q) L W( p)]/K. Let l 1 , l 2 # L with l 1 L l 2 . Then WK :QL defined by WK (p)= l 2 for p # K and W K ( p)=l 1 for p K is an element of #($). This implies K # {(#($))(q). Therefore, K # $(q) implies K # {(#($))(q). Let K $(q) for q # Q. Assume that there exists a W # #($) with K=[ p | W(q) L W( p)]. This implies by definition of #($) that there exists K$ # $(q) with W(q) L W( p) for each p # K$ and therefore K$/K. Thus, K {(#($))(q). In conclusion $ # SF(Q) implies $={(#($)). Hence, the closed elements concerning { b # are the elements of SF(Q). Assume W KS(Q, L). By Lemma 2, #($) # KS(Q, L) for every $ # SM(Q). Therefore, # b {(W) # KS(Q, L) and # b {(W){W. Assume now W # KS(Q, L). Let W # W and q # Q. There exists K # {(W)(q) with K [ p | W(q) L W( p)]. Therefore, for each q # Q there is a K # {(W)(q) with W(q) L W( p) for each p # K. This implies W # #({(W)). Let W W. Since W # KS(Q, L) there exists q # Q so that for each W$ # W there exists a p # Q with W$(q) L W$( p) 7 W(q)>L W( p). Therefore, there is no K # {(W)(q) with K=[ p | W(q) L W( p)] which implies W # b {(W). Thus, W # KS(Q, L) implies W=#({(W)). Hence, the closed elements concerning # b { are the elements of KS(Q, L). By Theorem A (see Appendix), (#, {) define a one-to-one correspondence between SF(Q) and KS(Q, L).
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Example 4. Let Q, L, L and W2 be defined as in Example 2. Define a surmise function $ on Q by $(q 1 )= [[q 1 ]], $(q 2 )=[[q 1 , q 2 ], [q 2 , q 3 ]], and $(q 3 )=[[q 1 , q 3 ], [q 2 , q 3 ]]. Then #($)=W2 . CONCLUSIONS
It was shown that is possible to generalize the central concepts of knowledge space theory to problem domains in which the answer of a subject to a problem can be classified by an element of a linear scale concerning its quality. For the special case of a linear scale with only two elements the generalized concepts can be identified with the original ones. The connections between the central concepts (surmise function, surmise relation, knowledge space, quasiordinal knowledge space) are valid in our generalization. If a knowledge structure on (Q, L), i.e., the set of all possible response patterns of subjects, fulfills certain closure properties it can be represented by a surmise function or surmise relation. The generalization of the theory of knowledge spaces was motivated by difficulties in testing psychological models of problem solving processes. One such approach is described in Schrepp (1993). In this paper a strictly closed knowledge structure, i.e., a q.-o. knowledge space, was derived from a model on problem solving processes in the field of inductive reasoning. The q.-o. knowledge space describes the set of all patterns of response times to a set of 24 letter series completion problems which are consistent with the model. The q.-o. knowledge space was compared to response patterns of subjects obtained in an experiment. The results show that the model is able to describe the general answer behavior but fails for a special subclass of problems. Thus the model is unable to explain some major influences of the problem structure on the difficulty to solve a letter series completion problem. APPENDIX
Below is a summary of some basic concepts of Galois connections (see Doignon 6 Falmagne, 1985) which are necessary for the understanding of the proofs for Theorems 3 and 4. Definition A. Let ( X, OX ), (Y, OY ) be quasi orders. Let f : X Y and g: Y X be mappings with: (a)
xOX x$ 7 x$OX x f (x)= f (x$)
(b)
yOY y$ 7 y$OY y g( y)= g( y$)
(c)
x OX x$ f (x$) OY f (x)
(d)
yOY y$ g( y$) OX g( y)
(e)
x OX g b f (x)
(f )
y OY f b g( y).
KNOWLEDGE SPACE THEORY
Then ( f, g) is called Galois connection between (X, OX ) and Y( OY ). Notice that conditions (a) and (b) are satisfied if OX or OY is a partial order. Theorem A. Let (X, OX ) and (Y, OY ) be quasi-orders and ( f, g) a Galois connection between (X, OX ) and (Y, OY ). Then: (a)
g b f is a closure operator on ( X, OX ),
(b)
f b g is a closure operator on ( Y, OY ).
Define X 0 =[x | g b f ( g)=x] and Y 0 =[ y | f b g( y)= y]. Then (c) For f 0 = f |X0 : X 0 Y 0 and g 0 = g |Y0 : Y 0 X 0 we have f &1 0 = g0 . REFERENCES Albert, D., Schrepp, M., 6 Held, T. (1994). Construction of knowledge spaces for problem solving in chess. In G. H. Fischer 6 D. Laming
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(Eds.), Contributions to mathematical psychology, psychometrics, and methodology. Springer: BerlinHeidelbergNew York. Birkhoff, G. (1937). Rings of sets. Duke Mathematical Journal, 16, 443454. Doignon, J.-P., 6 Falmagne, J.-C. (1985). Spaces for the assessment of knowledge. International Journal of Man-Machine Studies, 23, 175196. Falmagne, J.-C., 6 Doignon, J.-P. (1988a). A Markovian procedure for assessing the state of a system. Journal of Mathematical Psychology, 32, 232258. Falmagne, J.-C., 6 Doignon, J.-P. (1988b). A class of stochastic procedures for assessing the state of a system. British Journal of Mathematical and Statistical Psychology, 41, 123. Held, T. (1993). Establishment and empirical validation of problem structures based on domain specific skills and textual properties. Unpublished dissertation, University of Heidelberg. Korossy, K. (1993). Modellierung von Wissen als Kompetenz und Performanz [Modeling of knowledge as competence and performance]. Unpublished dissertation, University of Heidelberg. Schrepp, M. (1993). Uber die Beziehung zwischen kognitiven Prozessen und Wissensraumen beim Problemlosen [On the connection between cognitive processes and knowledge spaces in problem solving]. Unpublished dissertation, University of Heidelberg. Schrepp, M. (1995). Modeling interindividual differences in solving letter series completion problems. Zeitschrift fur Psychologie, 203, 173188. Received: March 21, 1997