Deductive Reasoning

Using Artiﬁcial Neural Networks to Simulate Preferential Reasoning

Marco Ragni and Andreas Klein

Center for Cognitive Science, Friedrichstr. 50, 79098 Freiburg, Germany

Keywords:

Knowledge Representation and Reasoning, Preferential Reasoning, Artiﬁcial Neural Networks.

Abstract:

Composition tables are used in AI for knowledge representation and to compute transitive inferences. Most of

these tables are computed by hand, i.e., there is the need to generate them automatically. Furthermore, human

preferred solutions and errors in reasoning can be explained as well based on these tables. First, we will report

brieﬂy psychological results about the preferences in calculi. Then we show that we can train ANNs on a

simple calculus like the point algebra and the trained ANN is able to correctly solve larger calculi such as the

Cardinal Direction Calculus. As human prefer speciﬁc conclusions, we are able to show that based on the

ANN, which is trained on the preferred conclusions of the point algebra alone, is able to reproduce the results

on the larger calculi as well. Finally, we show that humans preferred solutions can be adequately described by

the networks. A brief discussion of the structure of successful ANNs conclude the paper.

1 INTRODUCTION

Consider the following problem:

(1) Ann is smaller than Beth.

Beth is smaller than Cath.

From the given premises, when asked what follows,

it is easy for most people to conclude that Ann is

smaller than Cath. Psychologists have studied prob-

lems like this, typically called three-term-series prob-

lems, for years in an attempt to determine which are

most difﬁcult, and equally as important, why. A typ-

ical resulting dataset from such an experiment can be

thought of as a function that accepts the relationships

between the arguments of the ﬁrst premise and be-

tween the arguments of the second premise and re-

turns the frequency of subjects various responses. In

contrast to the domain of the above problem, which is

one-dimensional, it is also possible to present three-

term-series problems using domains that are multi-

dimensional. Doing so allows for problems with a

much greater degree of underspeciﬁcation between

the domain and codomain, such that there are many

more acceptable answers. It also happens that they

are more ecologically valid. For example, consider

the following (Ragni and Becker, 2010):

(2) Berlin is Northeast of Paris.

Paris is Northwest of Rome.

While it is possible to infer that Berlin must be North

(N) of Rome, it is not possible to determine whether

Berlin is East (E) or West (W) of Rome. As a re-

sult, it is likely that psychological preference plays a

greater role in determining the possible conclusion.

AI researchers have used a similar data structure, the

composition table, in order to efﬁciently represent

knowledge and compute transitive inferences (Cohn,

1997). However, the unfortunate problem with com-

position tables is that populating them with data is of-

ten painstakingly slow and likely to introduce errors,

as more often than not, these computations are done

by hand. It would be of great use to researchers in

both ﬁelds, AI and Psychology, if there were a way

to accurately approximate the values in the ﬁnal table

using methods that were less resource intensive. The

current paper proposes one such method, using Artiﬁ-

cial Neural Networks (ANNs). We begin by showing

that we can train ANNs on the Point Algebra (PA),

a one-dimensional domain, and that these ANNs are

able to correctly solve a larger and complex domain,

the Cardinal Direction (C D). Then we present a psy-

chological experiment using problems from C D and

demonstrate than an ANN trained on human prefer-

ences in PA can reproduce the table of human pref-

erence data collected in C D. Finally, we discuss the

structure of the ANNs.

635

Ragni M. and Klein A..

Deductive Reasoning - Using Artiﬁcial Neural Networks to Simulate Preferential Reasoning.

DOI: 10.5220/0004155106350638

In Proceedings of the 4th International Joint Conference on Computational Intelligence (NCTA-2012), pages 635-638

ISBN: 978-989-8565-33-4

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

Table 1: The preferred relations in reasoning with cardinal direction taken from (Ragni and Becker, 2010). The ﬁrst line in

each cell contains the number of and all possible relations, the second line the preferred relation, the third the percentage of

participants who chose this relation. The grey/white shaded cells are the multiple/single solution cells.

Table 2: Composition table for the Point Algebra (PA),

where < encodes left of, = equal to, and > right of.

< = >

< < < <, =, >

= < = >

> <, =, > > >

2 STATE-OF-THE-ART

For example, consider the domain of possible rela-

tionships between two points in one dimension, e.g.

R. This domain is typically referred to as the Point Al-

gebra (PA). Within PA , it always holds that between

two points, there are only three possible relationships,

at least one of which always holds: A can be either left

of ( <), equal to (=), or right of (>) a second point B.

The PA, and many other domains like it, can be pre-

sented in the form of a composition table (cf. Table

2). Transitive inferences can be represented in com-

position tables (Bennett et al., 1997). Using a set of

three points a composition table can be constructed,

depicting the relations between them (cf. Table 2).

For instance, point A is left of point B, A < B (left col-

umn), and B is left of point C, B < C (ﬁrst row), then

point A can only be left of point C, A < C (entry in

second column, second row). The cardinal direction

calculus C D consists of 9 base relations (N = north, E

= east, S = south, W = west and combinations such as

NE = north-east). The points in the euclidean plane

can be expressed by relations of PA-algebra: The re-

lation of two points z

can be described by x

= x

and y

> y

for z ∈ R

with z

= (x

, y

). Analogously,

NW by (>,>) and so on. An interesting ﬁnding is

that humans do not consider all possible models but

that there is a so-called preference effect, i.e., in mul-

tiple model cases (nearly always) one preferred model

is constructed from participants and used as a refer-

ence for the deduction process (Rauh et al., 2005). In

a previous experiment (Ragni and Becker, 2010) the

participants received premises (like problem (2) but

with letters instead of real cities) and their task was to

give a relation that holds between the ﬁrst and the last

term. Similar to the point algebra above these prob-

lems can be formally described by the composition

of two base relations and the question for satisﬁable

relations (cf. Table 1). For the above example NE

and NW contains the following three relations: NE,

N, NW. If we omit all one-relation cases (cells with

one entry in Table 1), it results in 40 multiple relation

cases out of the 64 possible compositions. The partic-

ipants in (Ragni and Becker, 2010) were confronted

with all 64 problems and had to infer a conclusion –

showing clear preference effects.

IJCCI2012-InternationalJointConferenceonComputationalIntelligence

636

3 PROVOKING PREFERENCES

In order to make an ANN learn the PA the ﬁrst step

is to ﬁnd a suitable encoding for the speciﬁed rela-

tions and an architecture of the ANN for the consid-

ered kind of problem. Since the problems considered

here are 3-term problems, an architecture with two

nodes in the input layer were used for classifying pos-

sible relations. For the encoding of the premises, −1

was used for the relation <, 0 for =, and 1 for >. For

this kind of problems it is possible, that all three rela-

tions hold between two points. Therefor, three nodes

within the output layer were used. Respectively one

for each possible relation. If a relation holds for given

premises, the corresponding output node returns 1,

else it returns 0. For example, the premises [−1, 0]

with the target output [1, 0, 0] is a suitable pattern for

training the ANN. It represents the three-term-series

problem with the premises a<b, b=c and the solution

a<c. Furthermore, a hidden layer was used and the

number of nodes iteratively increased to ﬁnd a suc-

cessful architecture for the given problem. For train-

ing the ANN, backpropagation was used as learning

algorithm with 1000 training iterations, a learning rate

of .3 and a momentum factor of .1. The tangens hy-

perbolicus was used as sigmoid activation function.

As depicted in Table 3, a suitable architecture requires

six nodes within the hidden layer. Since the ANN

Table 3: Rounded results on training different ANN archi-

tectures for PA. (hn = number of hidden nodes).

Prem 1 hn 2 hn 3 hn 4 hn 5 hn 6 hn 7 hn

<, =, ><, =, ><, =, ><, =, ><, =, ><, =, ><, =, >

-1 -1 1,0,1 1,0,0 1,-1,-1 1,0,0 1,1,0 1,0,0 1,0,0

-1 0 1,0,1 1,0,0 1,1,0 1,0,0 1,1,0 1,0,0 1,0,0

-1 1 1,0,1 1,0,1 1,1,1 1,1,1 1,1,1 1,1,1 1,1,1

0 -1 1,0,1 1,0,0 1,0,0 1,0,0 1,1,0 1,0,0 1,0,0

0 0 1,0,1 1,0,1 1,1,0 1,1,1 0,1,0 0,1,0 0,1,0

0 1 0,0,1 0,0,1 0,0,1 0,0,1 0,0,1 0,0,1 0,0,1

1 -1 1,0,1 1,0,1 1,1,1 1,1,1 1,1,1 1,1,1 1,1,1

1 0 0,0,1 0,0,1 0,0,1 0,0,1 0,0,1 0,0,1 0,0,1

1 1 0,0,0 0,0,1 0,0,1 0,0,1 0,0,1 0,0,1 0,0,1

Errors 3.685 1.980 1.260 0.766 1.503 0.020 0.086

was trained on the complete set of correct patterns of

three-term-series problems of PA, it is some kind of

over-ﬁtted. However, in this case this does not matter,

because the ANN is only used to provide real-valued

outcomes in the complete process to determine possi-

ble sources for preferences. For this purpose, the tar-

get values will be varied from integers to real-values

to reproduce preferences in PA reasoning. The re-

sult shows that PA is learnable by a quite small ANN

with little effort and suggests a good architecture of

an ANN for this kind of task. Human reasoning per-

Table 4: Mapping of C D relations to ANN input. Re-

sults on training the ANN with the varied patterns [[-1,

1],[0.9,1,0.9]] and [[ 1,-1],[0.9,1,0.9]] of PA.

C D PA ANN Learned Target

Rel. Dimension Input Values

x y x y

SW < < -1 -1 0.990 -0.360 0.107

W < = -1 0 0.999 -0.410 0.092

NW < > -1 1 0.902 0.970 0.897

S = < 0 -1 0.987 -0.327 0.040

EQ = = 0 0 -0.003 0.998 0.004

N = > 0 1 0.000 -0.003 1.000

NE > < 1 -1 0.902 0.996 0.884

E > = 1 0 0.000 0.000 1.000

SE > > 1 1 0.000 0.000 1.000

formance is known to be error prone and in cases of

various solutions preferences for particular relations

can be found (Rauh et al., 2005). In the previous sub-

section it was shown that an ANN is basically able to

learn PA . But what happens if the level of believe,

i.e., the target values in the patterns, change? For the

perfect ﬁtting of the previous described ANN the fol-

lowing patterns were used:

[[-1,-1],[1,0,0]],

[[-1, 0],[1,0,0]],

[[-1, 1],[1,1,1]], (1)

[[ 0,-1],[1,0,0]],

[[ 0, 0],[0,1,0]],

[[ 0, 1],[0,0,1]],

[[ 1,-1],[1,1,1]], (2)

[[ 1, 0],[0,0,1]],

and [[ 1, 1],[0,0,1]]. And a variation of some target

values, i.e.

[[-1, 1],[0.9,1,0.9]], (1)

[[ 1,-1],[0.9,1,0.9]], (2)

used in the latter for Cardinal Directions problems,

changes the rounded results shown in Table 3 in the

way the reported preferences of humans suggest. Ta-

ble 4 depicts the outcoming results for both of this

changes by reasoning with PA with the previous de-

scribed ANN with six nodes within the hidden layer.

With a mapping for a two–dimensional C D relation

to two one–dimensional PA relations, the previous re-

sults could be used to pass preferences from one cal-

culus to the other. Considering 3ts-problems in PA

the previously described ANN is used to compute the

possible relation for a given problem. Now, for 3ts-

problems in C D the relations must be split by their x–

and y–dimensions and computed separately. To com-

pute the PA outcome only one ANN is used. Given

1,2

(x, y)z

:= z

1,2

(x)z

∧z

1,2

(y)z

, with q

1,2

∈

C D and r

1,2

∈ PA , and z

2,3

(x, y)z

:= z

2,3

(x)z

∧

2,3

(y)z

, with q

2,3

∈ C D and r

2,3

∈ PA the in-

puts for the ANN are r

1,2

(x) and r

2,3

(x) for the x–

dimensional information speciﬁed in the problem,

and r

1,2

(y) and r

2,3

(y) for the y–dimensional infor-

mation. The mapping of the C D relation is given

in Table 4. The result of the ANN concerning

DeductiveReasoning-UsingArtificialNeuralNetworkstoSimulatePreferentialReasoning

637

Table 5: Mapping from the two ANNs output back to C D .

0,0,1 0,1,0 1,0,0 1,1,1

0,0,1 NE E SE SE,E,NE

0,1,0 N EQ S S,EQ,N

1,0,0 NW W SW SW,W,NW

1,1,1 NW,N,NE W,EQ,E SW,S,SE All

the x-dimension is given by ANN

1,2

(x), r

2,3

(x)) =

1,3

(x)

∨ r

1,3

(x)

∨ r

1,3

(x)

) with r

1,3

(x)

∈ B and

i ∈ {1, 2, 3}, where i = 1 is interpreted as western,

i = 2 as equal to, and i = 3 as eastern if the out-

come for the corresponding index is true. The ana-

log hold for the ANN if concerning the y-dimension,

but the boolean outcome i = 1 is interpreted as

southern, i = 2 as equal to, and i = 3 as north-

ern. In a last step the two PA results are mapped

back to one C D result set by recombining the x–

and y–dimension results. Therefor, the combina-

tion ANN

1,2

(x), r

2,3

(x)) × ANN

1,2

(y), r

2,3

(y)) =

((r

1,3

(x)

∧ r

1,3

(y)

) ∨ (r

1,3

(x)

∧ r

1,3

(y)

) ∨ . . . ∨

1,3

(y)

∨ r

1,3

(x)

)) is computed. Table 5 depicts the

mapping back from the two ANN output sets to C D

relation sets. With the ANN correctly trained for PA

the outcomes (0,0,0), (0,1,1), (1,0,1), and (1,1,0) are

never generated and would reﬂect errors in PA. Using

an ANN trained to PA without variations of training

patterns, as shown above, this procedure reproduces

the composition table for C D perfectly, but only 47

of the human preferences. Using the training patterns

the ANN predicts 56 of the 64 preferred relations (cp.

Table 1 and Table 6).

Table 6: The generated preferences for C D, trained with the

variation [[-1, 1],[0.9,1,0.9]] and [[ 1,-1],[0.9,1,0.9]] on the

PA. 56 out of 64 are correctly predicted.

E N NE NW S SE SW W

E E

(1.0)

(0.99)

(1.0)

N NE

(1.0)

(0.99)

(1.0)

(0.99)

NE NE

(1.0)

NW N

(0.98)

(1.0)

(0.98)

(0.99)

(1.0)

(0.98)

(0.99)

S SE

(1.0)

(0.98)

(0.99)

SE SE

(1.0)

(0.98)

(0.99)

(1.0)

SW S

(0.98)

(0.97)

(0.98)

(0.99)

(0.98)

(0.99)

W EQ

(0.98)

(1.0)

(0.98)

(0.99)

(0.98)

(0.99)

4 CONCLUSIONS

Composition tables are central in the ﬁelds of knowl-

edge representation and reasoning in dealing with in-

ferences in qualitative calculi. They are both impor-

tant for Artiﬁcial Intelligence, which uses them to

check the consistency of a network (Bennett et al.,

1997), and human reasoning, which describes reason-

ing errors by wrong preferred relations (Rauh et al.,

2005). So far about all composition tables had been

generated by hand. The problem becomes relevant

with increasing calculi as the tables become much

more difﬁcult to compute. We could explain human

preferred relations of complex calculi, like cardinal

directions by preferences in the point algebra. The

correct preferences of 56 of 64 (87,5%) of C D could

be correctly reproduced. The problem of selecting

an adequate neural network architecture for a given

problem has become recently more and more in the

research focus (Franco, 2006). Here best ﬁtting neu-

ral networks could be identiﬁed, which reproduce the

correct composition tables and human preferences.

An analysis shows that 6 hidden nodes provide the

best ﬁtting architecture for this approach. The de-

scribed method seems fruitful for both formal and

psychological reasoning.

REFERENCES

Bennett, B., Isli, A., and Cohn, A. G. (1997). When does

a composition table provide a complete and tractable

proof procedure for a relational constraint language?

In Proceedings of the IJCAI97 Workshop on Spatial

and Temporal Reasoning, Nagoya, Japan.

Cohn, A. G. (1997). Qualitative spatial representation and

reasoning techniques. In Brewka, G., Habel, C., and

Nebel, B., editors, KI-97: Advances in Artiﬁcial Intel-

ligence, Berlin, Germany. Springer-Verlag.

Franco, L. (2006). Generalization ability of boolean func-

tions implemented in feedforward neural networks.

Neurocomputing, 70:351–361.

Ragni, M. and Becker, B. (2010). Preferences in cardinal

direction. In Ohlsson, S. and Catrambone, R., edi-

tors, Proc. of the 32nd Cognitive Science Conference,

pages 660–666, Austin, TX.

Rauh, R., Hagen, C., Knauff, M., Kuss, T., Schlieder, C.,

and Strube, G. (2005). Preferred and Alternative Men-

tal Models in Spatial Reasoning. Spatial Cognition

and Computation, 5.

IJCCI2012-InternationalJointConferenceonComputationalIntelligence

638