Analogy

 

as an essential of perception and cognition

Analogy as an essential of perception and cognition

 

A cooperative project by Radomir Zugic, Russel Todd Scheinberg, Anne Lämmel, Kalia Dogbevi, Eugenie Giesbrecht

 

 

in the course of the lecture „Functional Programming“ at University Osnabrück

 held by Ute Schmid

 

Abstract. The ability to resolve analogical tasks is a fundamental skill of most intelligent organisms. Analogies are emminent in intelligence tests and serve as  criterium and measure for cognitive abilities. Enlarging the perspective to the realms of neurobiology, we realize that in every sensory system, analogical reasoning is a basis and necessary prerequisite for perception. Analogical reasoning can thus be seen as an essential of perception and cognition as such.  How can such a principle be modeled? Taking the nervous system itself as analogy and number sequences from intelligence tests as framework, we tried to design an algorithmic concept that reflects possible elements from the simple to more complex cognition on the field of analogy.

 

 

 

 



Introduction

The task was to find an algorithmic concept to deal with  analogies. Number sequence tasks from intelligence tests were taken as framework: in a given sequence, the underlying principle is to be detected and the right consequent number is to be found.

Example:

2 4 6 8 10       -> 12     (+2)

Usually, searches through a mathematical problem space are used to generate hypotheses and check them on each subsequent number. Approaches have been made by using iterative deepening, polynomial equation solving, grammar generation and anti-unification.

We wanted to examine this issue with an interdisciplinary component and from a cognitive sciene point of view. When you take a look at the title picture by Salvador Dalí, you can get an impression of how many analogic inferences you must have performed in less than a second in order to “see” the picture. Similarly complex tasks have to be performed in any sensory system instantaneously, not to speak of music and the like.    The idea of interpreting the sequence of numbers as simplified sensory input indicated that already small organism should be able to detect such analogies in order to survive. We set out to model the way this could happen, and chose current, but very basic knowledge about the visual system and the architecture of the brain as guideline for our algorithm.

 

1 Intelligence Tests

What is an analogy?

          As  there are many controversial interpretations of the term ‘analogy’, we would like to refer to the reference section for further definitions. From what we could see most sources agree on, an analogy detecting system should have a sense of equality, a measure for relations, an ability for grouping, and an internal representation or language that is homogeneous and thereby allows comparison (abstraction).

Why number sequences?

What are the characteristics of the analogical principle lying behind number sequences?

1.  Analogy of the result: The numbers themselves are in some constant relation that can be continued.

2.  Analogy of the argument: Rather than the numbers themselves, the arguments of the operation that yields them provide the analogy. Example: +1*2-3:4 and the like.

3.  Controversial point: symbolic versus mathematical analogy. Do we care about the mathematical relationship resulting of +1*2-3:4 or do we just group 1234 and +*-: as a collection of symbols?

Number sequences require basic mathematical knowledge, but this is not what is being tested in the IQ-tests: it is a prerequisite for performing the actual analogic reasoning task. Given the subject can perform the required actions, its ability to derive analogies is taken as a measure for its intelligence. Along with the designers of these tests we state that analogy is a fundamental principle of cognition as such and want to find an overall principle modeling this fact.

3 Solutions

Inferential Approaches

A frequent approach is to draw inferences subsequently from one number to the next and find a representation for the underlying principle. This has the advantage of being mathematically/logically correct and the power of completeness and determinability.

Inference determines the space of correct actions following from a given state, whereas analogy is a way of perceiving a state (and only secondarily to perform an action). The abstracted information drawn from detecting an analogy does not have to be representationally as exact as the input, which makes it subjective on the one hand and applicable for other cross-systems-use on the other. Thus, inferential approaches usually have to deal with an immense amount of possible correct actions and correct representations. To stop and ponder over the general relation of the whole and only then come to a conclusion is usually neither intended nor conceptually  desired.

Our Approach

We set out to design an algorithmic scheme that incorporates the most basic principles of some of the features that neural systems are supposed to possess generally. We accepted, that thereby mathematical correctness might be lost in favour of our theoretical framework.

- The modules in our program are named with biological terms just to clarify their function. They do not imply that our program actually ‘perceive’ or ‘act’ in the usual reading of the words. -

4 The Program

Main aspects

The main routines are classified by perception, detection and action. Unlike mathematical approaches, action (i.e. the solution) does not necessarily have to correlate with perception. The neuronal principles of recurrency, competition of responses, discarding irrelevant information, generating new one and the lack of explicit representations have been considered.

Input

As stated above, the number sequences are interpreted as sensory input. For example, this could be – in the visual system - either light dots of variable intensity, or (i.e. for higher neurons) bars of variable length or orientation.

Perception

Prerequisites are operators (senses) and connections to the stimuli (links). While connections in the nervous system are hard-wired and optimalized (->amacrine cells), we chose to take all layers of neighbourhood as connection pattern for our operators.

The operators can be compared to our ability to add or multiply, or more profoundly, to perceive the length or orientation of a bar. It turned out that a restriction to operators that are held as simple as possible and uniformly output natural numbers with –1 as a bottom element opened a way for general analogy perception and cooperation among operators.

As reaction to a stimulus an operator can construct a percept map, in which the results of the operator application are listed according to their relation by the connections specified. Layers thus go from direct neighbours to first and last. (Examples: see appendix I). The theory is that whatever relationship was emminent in the stimulus pattern is passed on to the percept map.

Detection

Some constellation in the percept map must now trigger a response(reflex).

The response can result in a direct action or be used for object recognition.

For simplicity, we took ‘equality’ as a triggering criterium, referring to the ability of the nervous system to enhance simultaneous firing. The triggering reflex is coded as a triple of Strength, Quality and Layer (on which it occurred). A measure of sureness (relevance) can be determined by the ratio of strength of reflex and strength of stimulus.

If nothing relevant is found in the percept map, the map itself is used as input again, which can be compared to  processing in higher areas or in a recurrent fashion. Thus, only what triggers a response survives: we loose information and generate new at the same time.

Action

The reflex can now elicit an action of arbitrary kind. In simple organisms, a constellation of light bursts causes not a light reaction, but muscle contractions! In our case, the reflex should generate another number, which essentially means to prolong the ‘object’ by one step. It should be noted that - at this point - to merely detect an analogy is a task different from finding the next member!

Same actions are combined, the strongest activation being kept. Different actions compete by intensity, but stay present all the time: what has been perceived once can not be overwritten completely.

 

Example

In 1 2 3 4  the operator ‘minus’ yields 1 1 1 in layer Nr.2 of the percept map, which triggers Strength(3+1), Value(1), Layer(2). Strength is equal to length of stimulus-> 100% sureness. Action: Prolong Layer 2 by Value(1) and propagate it upward(add (1) to 4). Result: 5.

5 Interactions

One feature of higher cognition is to combine the input of several operators. We do this simplifiedly by designing a loop which passes percept maps around the operators.

With the fundamentals set, we can now design a recombination of them in order to model object grouping, which appeared to be necessary for analogical tasks.(see appendix II)

The reflex detection worked by detecting equals on a whole layer. But we can detect equals also only in part of the layer and thereby segment the stimulus pattern. This results again in a triple of Strength, Quality and List of Positions(instead of Layer). This triple can be seen as a rudimentary object description. The List of Positions can serve as input again, so that analogies in its distribution can be detected.(see appendix III).

   The general restriction to natural numbers and –1 allows for cooperation: one operator can perceive an uncertainty-object (coherent positions of –1) and trigger a search for coherent responses at these positions in percept maps of other operators, thus combining to one response by two operators. Example: In a pattern governed by +2*2, the ‘*’-operator might read regularly –1,2, –1,2 etc..which triggers a search to look for a companion for the regular uncertainties (which he would find in ‘+’).   

(see appendix IV)

6 Summary of the procedure

The stimulus is given as input to a number of operators. Which parts of the stimulus they perceive is determined by the connectivity specified for them. Thus, each operator constructs a percept map. This map is ordered by the structure of the specified connectivity. If this percept map contains characteristic constellations, it can trigger a reflex which can lead to an appropriate action (i.e. finding the next number). At the same time, the map can serve as input for the same and other operators. While every triggered reaction is kept, they compete by strength and occupation of work space, which results in a qualitative segmentation of the latter.

A short guided tour can be activated by the queries ‘?-info(1).’ through ‘?-info(6).’

7 Evalutation and prospects

Problems and prospects:

·    The design of an iteration cycle that can mimic or substitute parallel processing. Although not everything has to be parallelly evaluated, it has to be clear which information is present at one time step, and which has already faded by competition.

·    To switch separation of workspace on or off can yield different results. That is why grouping is done first, solutions extracted, and the whole is perceived again without grouping. The idea is to allow for multiple representations or ‘interpretations’ of the same stimulus so that the strongest can correct the initial impression after some time of computation.

·    The above is an example of a looping technique that is conceptually distinct from the prolog system. The system is hardly able to ‘forget’ unnecessary information on a recursive run and keeps a tremendous amount of superfluous choice points in the stack. After all, prolog is an inference-language; it may be more effective to implement the algorithm in a functional language, as already the concept of “input, function body, output” is compatible with our approach. Further prospects would be to implement structures over maps and objects to define operations on them, the use of functionals for  triggers and the like, or partial functions as representations for ‘partial objects’…While Prolog might be conceptually clearer, it is – from our point of view – unlikely to handle our approach effectively and in consistence with the theory.

·    Large works have been done in the area of computer vision. It was not our goal to design a computer vision system, but to take it as a starting point for abstract research. However, a combination with findings from that area would be favourable.

·    Unfortunately, the algorithm itself could not be finished to our satisfaction because of a time shortage. It will remain as a conceptual basis for further enhanced versions to follow in the future.

 

Comparison to other projects

 

Compared with the other approaches made in the course of this project, we can say that anti-unification focuses on action, while this approach focuses on ‘perception’. Thus, both are, from a computer science point of view, inadequate to solve a simple and specialized mathematical task as to find only the next member of a number series in the most efficient way. This was shown by algorithms ‘simply’ calculating statistical measures, or applying heuristics from subjective experience.

 We can, however, generally claim to have found and specified some of the basic elements that are required in solving this task: each one of the algorithms experienced that 1) operators had to be chosen to be as simple as possible at one point, 2) reasonable answers must be separated from debris, 3) the numbers have to be connected in a specific way, 4) so do operators, etc…. The problems and the successes in the approaches of our colleagues matched at least our theoretical concepts and expectations quite satisfyingly.

 

Appendix

Sample sequences of different complexity with interpretations. Note that a complexity measure should be introduced that relates length of sequence to length of pattern to be detected.

2 5 8 11 14 17 20 23

+3

18 16 19 15 20 14 21 13

-2+3-4+5

5 6 4 6 7 5 7 8

+1-2+2

9 12 16 20 25 30 36 42

+3+4+4+5+5

2 3 6 11 18 27 38 51

+1+3+5+7+9

13 15 18 14 19 25 18 26

+2+3-4+5+6

1 3 6 8 16 18 36 38

+2*2

57 60 30 34 17 22 11 17

+3:2+4:2+5:2

11 8 24 27 9 6 18 21

-3*3+3:3

15 19 22 11 15 18 9 13

+4+3:2

9 6 18 21 7 4 12 15

-3*3+3:3

01011011101111

0 and 1(+)?

Appendix I

Sample Percept Maps with operator ‘minus’ for illustration. However, we preferred splitting it up into  minus positive (increase in value) and minus negative(decrease in value) later.

 

Pattern: 

  1   2   3   4

 

    1   1   1 (<-!)

      2   2    (<-!)

        3       (<-!)

 

 

Pattern:

 1   3   6   8  16  18  36  38

 

    2   3   2   8   2  18   2

     5   5  10  10  20  20

        7  13  12  28  22

         15  15  30  30

           17  33  32

             35  35

               37

 

Pattern:

18  16  19  15  20  14  21

 

   -2   3  -4   5  -6   7

      1  -1   1  -1   1  (-?!)

       -3   4  -5   6 

          2  -2   2 

           -4   5 

              3

 

Pattern:  

  5   6   4   6   7   5   7   8

 

    1  -2   2   1  -2   2   1

     -1   0   3  -1   0   3

        1   1   1   1   1(<-!)

          2  -1   3   2

            0   1   4

              2   2

                3

 

 Pattern

 11  15  18   9  13  16   8  12

 

    4   3  -9   4   3  -8   4

      7  -6  -5   7  -5  -4

       -2  -2  -2  -1  -1

          2   1 -10   3

            5  -7  -6

             -3  -3 (<-!)

                1

 

                Any of these rows can serve as input sequence again. The deeper in the map the triggering occurs, the less its relevance(strength).       

Appendix II

Pattern

 1   0   1   1   0   1   1   1   0   1   1   1  1

 

-1   1   0  -1   1   0   0  -1   1   0   0  0 

    0   1  -1   0   1   0  -1   0   1   0  0

        0   0   0   0   1  -1   0   0   1    0

         -1   1   0   0   0   0   0   0   1

            0   1   0  -1   1   0   0  0

              0   1  -1   0   1   0  0

                0   0   0   0   1  0

                 -1   1   0   0   1

                    0   1   0   0

                      0   1  0

                        0  1

                          0

 

The above pattern is ’easy’, but only recognizable if you perform an act of segmentation or classification of the space. Still, given the immense amount of regularities in the map, a neural system might have found another way to detect them.

     

Appendix III

A conceptual look on grouping.

The last pattern:

 1   0   1   1   0   1   1   1   0   1   1   1  1

 can be interpreted as a collection of patterns that stand in correlation by their relative position on the screen. As the algorithm triggered for whole rows in the previous examples, object recognition would trigger for partial rows in case of frequent equalities:

 1 -> [1,3,4,6,7,8,10,11,12,13]

 0 -> [2,5,9]   

This is a clear segmentation of the workspace. 0 is already defined by a cycle through the minus operator:

2,5,9 -> 3,4 -> 1 next position: 14.   

This partial sureness about the workspace evokes a search for sureness of the rest, which is found in ‘1’.

As this kind of pattern is not common to intelligence tests (‘too easy’…), we didn’t include this mechanism (maybe mistakenly) directly into the algorithm.

Appendix IV

On the basis of the above, maps from different operators can be combined.

 

For the pattern:

3  5  10  12  24  26  52

 

Layer I of mult(positive):

  1.66  2  1.2  2  1.08  2

and the simplified version:

 -1  2  -1  2  -1  2

and Layer I of minus(positive):

  2  5  2  12  2  26

would first result in the separate detection of 2 objects,

2 -> [1,3,5,7],  2 ->[2,4,6].

They can be analyzed by the same techniques, yielding a prolongation of:

2-> [1,3,5,7,9], 2 -> [2,4,6,8]

which would already yield results.

In a further step of combination,

these two objects can again be perceived as one, by inducing a search for the unsureness in [1,3,5,7] of mult(pos), and combine to something like: [7(strength),2,[[1357],[246]]] with operators minus(pos) and mult(pos). As all elements are detachable, they can serve as basis for further combination with information from totally different maps and operators.

This, however, greatly exceeds the requirement of finding only the next number to a sequence…

Reference

Manfred Gotthalmseder,Visuelles Erkennen und Bildschaffen, - Ein Modell der Objekterkennung, sein Bezug zum Bildschaffen und sein Wert für eine allgemeine Theorie des Erkenntnisgewinns,

www.kopcom.com/mg/

Friedrich W. Hesse  Analoges Problemlösen, - eine Analyse kognitiver Prozesse beim analogen Problemlösen, 1991

William H. Calvin The cerebral code – thinking a thought in the mosaics of the mind,1996

David Rose,Vernon G.Dobson(Edition) Models of the Visual Cortex,1985

Stella Vosniadou, A.Ortony (Edition), Similarity and analogical reasoning,1989

Rolf Oerter, Psychologie des Denkens,1971

Richard E. Mayer, Thinking and Problem Solving – an introduction to human cognition and learning,1947


         An artistic analogy by Salvador Dalí: