In forming an analogy between two situations, objects are mapped between analogues most often because they share a common role in their contexts rather than sharing superficial attributes. For example an analogy between the solar system and an atom can be made because planets REVOLVE AROUND the sun, in a similar way to how electrons REVOLVE AROUND the nucleus, rather than there being similarities between planets and electrons etc.
Many computational models of analogy making such as the Structure Mapping Engine (Falkenhainer, Forbus & Gentner, 1990) and the Analogical Constraint Mapping Engine (Holyoak & Thagard, 1989) start with fixed representations of each of the situations and create mappings based on similarities in their relational frameworks. However, as argued by Hofstadter (1995), how we perceive many situations (i.e. the representations that are formed in our working memories) is highly dependent on the mappings that we are making. For example, if we are drawing an analogy between DNA and a zipper, we are likely to form an active representation of DNA which highlights its physical, base-paired structure. In contrast, comparing DNA to source code would more likely bring about a representation that highlights its information-carrying abilities. Thus, rather than analogy making being a result of the perception of a situation followed by a mapping process, these two processes seem to be codependent and are not temporally separable (as supposed by models such as the SME and the ACME).
Exercise 1: Describe two analogies (both with the same source, but different targets as above), that illustrate the point that what properties come to mind about a situation or object is dependent on the analogy being made and mention how the properties that differ in your analogies.
Copycat differs from many traditional computational models of analogy in the sense that representation formation and mapping occur concurrently. As mentioned in the previous section, it builds these structures in the Workspace (Copycat’s "working memory"). The structures that are built in this area represent the perceived relationship between adjacent letters and groups of letters, as well as noting what objects are thought to play the same role between the strings. From the correspondences built between the initial and modified strings, a rule is built that describes the perceived transformation between these strings. This rule is adapted to fit the target string, by looking at what conceptual slippages have been noted to occur within the correspondences between the initial and target strings. This rule is then applied to the target string to form a solution. For example, a possible solution to the problem abc : abd, iijjkk : ? is the string iijjll. In deriving this solution, Copycat modifies the initial transformation rule "replace the rightmost letter by its successor" by substituting the concept of letter for group, which is the conceptual slippage that must occur to perceive the letter c in the initial string and the letter group kk in the target string as "playing the same role."
This section describes types of structures that are built in the Workspace, starting with the lower-level structures upon which the higher level structures can be built. The structures that can be built include descriptions (which describe attributes about each object), bonds (which denote relationships between adjacent objects in each string), groups (which are collections of adjacent objects that are linked by common bonds), correspondences (which map objects between strings that are perceived as "playing the same role"), a rule (which describes the transformation between the initial and modified strings), and a translated rule (which is the rule describing how the target string should be modified to form a solution).
Before any correspondences can be built between letters or groups of letters between the strings, certain attributes of each of the objects needs to be perceived. These attributes fall under certain categories, of which an object can only be assigned one attribute for each category. For example, in the classification of a letter category, an object cannot be perceived as being both the letter A and the letter C.
When a problem is initially set up in Copycat, certain descriptions are immediately assigned to each of the letters. The categories of these descriptions are object-category (in which each object is perceived as being either a letter or a group of letters), alphabetic-position-category (which denotes what letter in the alphabet each object is), and string-position-category (which indicates for certain obvious positions, where the object is located, such as leftmost).
Exercise 2. From the "File" pulldown menu, select "new". This will allow you to run a new problem. Type in the strings abc, abd and iijjkk in the initial string box, the modified string box and the target string box respectively, and click on OK. This will set up the problem abc : abd, iijjkk : ?.
Question 2.2 Certain descriptions given to these letters can distinguish them from the other letters in the string that it is contained within. What are the distinguishing descriptions given to the letter "c" in the above problem?
Using the attributes that are assigned to each of the objects in the Workspace, relationships between adjacent objects in each of the strings can be perceived. These perceived relationships are represented by bonds in the Workspace.
Exercise 3: Given that they are adjacent letters, how would you describe the relationship between:
As humans are used to scanning words and letters in one direction (e.g. from left to right for native English speakers), the relationships that we perceive are usually biased in this direction. For example, it is likely that you perceived the relationship between a and b in the string abc as being a successor relationship (i.e. they are successive letters in the alphabet). However, Copycat has been implemented so that no such bias exists, so that it is equally likely that the letters a and b in the string abc is perceived as being a successor relationship to the right, or a predecessor relationship to the left. Furthermore, Copycat is restricted to perceiving only three types of relationships between adjacent objects; a successor relationship, a predecessor relationship, and a sameness relationship (i.e. the objects have the same letter-category).
In the Workspace, once a relationship has been perceived between two adjacent letters in a string, a bond is built between these letters. Each bond has a type (i.e. successor, predecessor and sameness) and a direction in which the relationship was noted (ie. left or right). Sameness bonds however are not assigned a direction, as it does not matter which direction you scan the letters from, the relationship is the same.
In the Workspace in the current implementation, a Bond is denoted by an arc between the two letters, with an arrowhead denoting the direction (if it is not a sameness bond). From this representation, the type of bond can easily be seen. For example a possible successor bond is as follows:
This representation can be interpreted as being a successor relationship as
the relationship between the joined letters to the right (i.e. the direction
indicated by the arrowhead) is successor.
Exercise 4. load the problem abc : abd, kji : ? by using the File Pulldown menu, and selecting New and typing in the appropriate strings. Set the random seed of the problem to 68, by pulling down the "Run" menu and selecting Set Random Seed. Run the problem (by pressing Play from the Main Control window), and try to identify the kinds of bonds that are appearing as Copycat formulates a solution. You can press Pause at any time to pause the system, and Play to resume running.
Note down the solution string, and the types of bonds (note the type and the direction) that are formed between the letters in the abc string and the kji string in the final solution.
Once relationships between adjacent letters have been perceived, these letters that are bonded by the same type of bond (with the same direction) can be perceptually grouped. For example, the string abc can be thought of as a group of successive letters to the right, or the letters jj can be perceived as a "sameness group" in the string iijjkk.
These groups themselves, like letters, are given descriptions. These descriptions denote such attributes as the bond-type (i.e. the type of bonds that link the objects contained within the group), the direction (i.e. the direction in which the bond relationship holds), and the letter-category (if all the letters contained within the group are of the same letter-category).
Furthermore, the description "whole" can be given to the group if the it spans the whole string, or the descriptions "leftmost", "middle" and "rightmost" can be given if the object is perceived to be the leftmost, the middle or the rightmost object in the string.
In the current Copycat implementation, if a collection of letters are perceived to be a group, the letters are enclosed by a thick box, with the descriptions for this group overwriting the descriptions of the letters that are contained within it. For example:
Exercise 5. Press "Stop" to reset the Workspace and then run the problem abc : abd, kji : ? using 68 as a random seed, and try to identify the kinds of groups that are appearing as Copycat formulates a solution. Note down the solution string, and the descriptions that are allocated to the groups abc, and kji.
Exercise 6. Rerun the previous problem, this time using a random seed of 177. Note down the solution string, and the descriptions that are allocated to the groups abc, and kji.
Because sameness groups are capable of acquiring descriptions regarding their letter-category, these groups themselves can be bonded to other objects, and can form high-level groups.
Exercise 7. Run the problem abc : abd, iijjkk : ? using 46 as a random seed. Explain how the string iijjkk has been perceived as a successor group.
As stated earlier, there are two processes that interact deeply in the formation of an analogy; representation formation and mapping. The previous structures that we have seen are all the structures that Copycat uses in forming a representation for each of the letter strings. From these structures, correspondences between objects that are perceived to be "playing the same role" can be formed.
In the current implementation of Copycat, correspondences are represented by lines joining the equated objects. Normally these lines run from the middle of the bottom of the object in the initial string, to the middle of the top of the object in the target string. However, if both objects span their corresponding strings, the correspondence is displayed as a horseshoe shaped link. For example:
These correspondences are based on similarities in the descriptions attached to each of the objects. For example, the letters c and k are equated above because they share the description "rightmost". However, the descriptions that these correspondences are based upon are the distinguishing descriptions only (i.e. a description that sets it apart from the other objects in the same string). If this were not the case, a correspondence could be formed between the letters c and j because they share the description "letter".
Exercise 8. Run the
problem abc : abd, cba : ? using 147 as a random seed and identify which descriptions
allow the mapping of the two c’s
together (i.e. what distinguishing
descriptor(s) do these two objects have in common?).
As you will have seen, in the mapping of the two c’s, there is a mismatch in the string-position-category descriptions (i.e. the letter c is the rightmost letter in the initial string, but is the leftmost letter in the target string). Another way of putting it, is the rightmost letter in the initial string, is playing the same role as the leftmost letter in the target string. As put by Hofstadter, there is a conceptual slippage between the concepts rightmost and leftmost. That is, under the pressures exerted by the problem, the concept rightmost in the initial string has been equated with (or has "slipped into") the concept leftmost in the target string.
In forming a letter-string analogy, an initial RULE is used to explain the transformation in the source domain. This rule is then adapted to fit the target domain, taking into account the conceptual slippages that occur across the domains. The initial Rule is in the form:
Replace the letter category of X by Y
where X is the description attached to the modified letter in the initial string that uniquely identifies it (e.g. the letter-category of the letter (a..z), or the position-category (leftmost, rightmost etc)), and Y indicates what the letter should be replaced with (i.e. either the letter, or the relationship that has been noted to occur in the Replacement , such as successor or predecessor).
If forming a solution, Copycat adapts the initial rule (i.e. the rule that explains the transformation noted in the source analogue), by applying all the conceptual slippages that have occurred in the Workspace. This modified rule is then applied to the target string to form the solution string.
Exercise 10. Run the problem abc : abd, kji : ? using a random seed value of 19, and note down the initial and the translated Rules. (The translated rule appears at the bottom of the Workspace at the end of a problem).
What correspondence contains the slippage that has been applied to the initial rule to derive the translated rule, and how was this correspondence allowed to form?
By the end of this section you should know: