This page explains some aspects about the theory of PRISM.

The Premises

Definition
A premise represents a spatial relation between two objects. An object is described by letters or a word. The syntactic form of a premise is Object1 Relation Object2 which represents statements like The car is left to the church for example. The premises are sequently processed by PRISM to construct a intital model which represents the spatial statements of all premises. The procession of the premises is based on the following rules:
Processing Order
If both objects of the premise are not part of the mental model yet, the left object of the premise is always processed first. If one object of the premise is already part of the mental model, it is located by the Focus before the other object of the premise is inserted into the model. If both objects are already part of the mental model the left object of the premise is located first by the focus.
Insertion principle
If one object of the premise is already part of the mental model, the new object is inserted by the First Free Fit principle. The Focus locates the object which is already part of the mental model and then moves along the direction of the premise's relation until it reaches a free field, in which the new object is placed.
Discontinuous premises
If both objects of the premise are not part of the mental model yet, PRISM adds a new layer to the working memory to insert the objects.
Relocation principle
If both objects of the premise are already part of the mental model, but the relation between them is not fulfilled in the model, then the objects have to be relocated. If they are in the same layer, the focus locates the nearest object and moves it along the direction of the premise's relation to the first free position which fulfills the relation.
Merging layers
If the objects are in different layers, PRISM groups the objects of the newer layer and insert the whole group into the older layer, so that the relation is fulfilled.

The Conclusion

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Verification
The conclusion is defined like a single premise. It conatins two objects and relation between them. Instead of adding the conclusion to the mental modell, PRISM will check if the statement specified in the conclusion is valid in the constructed model. Therefor the focus locates the nearest objetc of the conlusion. Then it moves to the other object of the conclusion and checks if the direction of its movement is correspond to the relation of the conclusion.
Alternative models
During the verification of the conclusion, PRISM will not search for alternative models. See the following section about the reasoner's response for the existance of alternative models and the validity of the conclusion.
Generating relations
Instead of verifying a conclusion, the user can select the 'generate' option for two objects o1 and o2. The program will check which conlusion holds between o1 and o2 by performing the following steps:

1. Move focus to the object which is closer to the current focus position.
2. Move focus to the other object.

The relation will be derived from the movement done in the second two and can be 'left', 'right', 'front', 'behind' or a combination of 'left'/'right' with 'front'/'behind'.

Reasoner's Response and Logical Evaluation

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Simulating human performance
The idea behind PRSIM is to simulated human performance in spatial reasoning tasks. If the user specifies a conclusion which will be asked after the procession of the premises, PRISM answers this conclusion by using the constructed preferred mental model without taking possible alternative models account. This behavior may lead to logical incorrect answers and fits well to typical errors and performance rates of human subjects. This section descibes the relationship between the ambiguity of models, PRISM's response simulating the performance of a human subject and the logical correctness. The relationship differs for models which are consistent or inconsistent.
Reasoner's Response
If a conclusion is specified, PRISM answers
- VALID: If the conclusion is explicitly represented in the constructed model.
- INVALID: If the conclusion is not explicitly represented in the constructed model.
Logical correctness
The answer of the reasoner is evaluated according to its logical correctness.
The conclusion does
- strictly FOLLOW from the given premises if it holds in all possible models which can be constructed from the premises.
- NOT FOLLOW from the given premises if there is at least one model which constructs a contradiction to the conclusion.

The logical correctness of the answer is derived from:

Case Validity Conclusion Answer Reasoner Logical Variation
1 FOLLOWS VALID CORRECT No
2* FOLLOWS INVALID INCORRECT No
3 NOT FOLLOWS VALID INCORRECT Yes
4 NOT FOLLOWS INVALID CORRECT No

* Case 2 does not occur in the program.

Inconsistent Premises
If the specified premises are inconsistent, the conclusion does FOLLOW from the given premises, because any derivation from an inconsistent set of premises is logical correct.

If a conclusion is specified, PRISM ansers
- VALID: If the conclusion is explicitly represented in the constructed model.
- INVALID: If the conclusion is not explicitly represented in the constructed model.

The logical correctness of the answer is derived from:

Case Validity Conclusion Answer Reasoner Logical Variation
1 FOLLOWS VALID INCORRECT No
2 FOLLOWS INVALID CORRECT No

The Complexity Measure

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Difficulty estimation
PRISM provides a complexity measure which is based on the simulation of human performance in spatial reasoning tasks. Every task which is processed by PRISM gets a value which represents its difficult estimation for human subjects. The higher the value, the more diffiuclt should be the tasks for humans. This section describes the different factors which are taken into account by he complexity measure.
The focus
The focus represents the center of attention of human participants. All operations of the focus which are performed during the construction and verification process are considered for the complexity measure. It is distinguished between move and direction changes. A move operation is the movement of the focus from one field to another. If the previous move operation was performed in a different direction than the current move direction, the direction changes of the focus are raised.
Adding & Relocating
Besides the focus operations, adding and relocating objects operations are taken by the write operations into account. If a object which has to belocated is linked to other objects, the whole group have to be relocated to keep the relations in the model consistent. In this case the grouping costs of n - 1, with n size of the group, are tkaen into account.
Different layers
If the premises are discontinuous a second layer has to be added. In this case additional costs for creating and deleting layers are computed.
Overall costs
After the processing of the task the overall costs are computed by summing up the single values of the move, direction changes, write, grouping and add/del layer operations are summed up. The current correlation between the complexity measure and experimental data is negative and can be seen on the data page.
Costs beside the focus
Some aspects beside the focus operations seem to be relevant and influence the complexity of a task.
Consistency: In case of more complex tasks, invalid premises seems to be easier than valid tasks. People might rather guess 'invalid' than 'valid'
Read direction of premises The number of different relations used in the premises seem to have an influence of the complexity. More different used relations seem to make a task more difficult. Another aspect could be a conflict between the construction and read direction. It might be easier to construct from 'left to right' than from 'right to left'. It might also be easier to construct in the direction of the read premise than in the opposite direction.
For example:
Is there a difference between 'A left B', C right A' and 'A left B, A left C'?

Object attributes

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Mobility or different interpretations
In order to make the objects of your premises more realistic you can assign additional attributes to them. Experiments have shown that attributes like size or mobility of objects influence the way we use them in spatial reasoning. The current version of PRISM supports only the attribute of mobility, but if you like you can interpret this attribute in a different way to test you own assumptions.
Influence
To use the mobility attribute in the current version of PRISM you have to specify an order of the all objects you use in your premises. Comparing to objects, the object with the smaller mobility value seems to be harder to move in real-life than the object with the higher mobility value.
The mobility order influences the model construction in the following way:
In case of an inconsistent premise, the order of mobility is more important than the LO/RO principle: The object with the smaller mobility value is relocated. The object with the bigger mobility value stays fixed.
See the following example for a case where the LO/RO principle is not applied due to the specification of weights: (A left B), (B left C), (C left A) Processing premise 3, PRISM relocates the RO object C, because it has a higher mobility value than LO object A. With any mobility values, LO object A is relcated.

Inconsistent Facts

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Inconsistency detection
It is also possible to define tasks which premises which contradicts each other. If the currently processed premise contradicts the constructed model and it is not possible to resolve this confilct by variating the model, PRISM has detected a logical contradiction in the set of premises you have specified. You can see this detection by the changing color of the focus from blue to red.
Handling
If a inconsistency has been detected, PRISM takes the currently processe as most plausible and starts to rearrange the objects in the model to fulfill the relation of the premise (by necessarily harming previous premises). The rearangement process follows the rules of the LO/RO principle. If no object attributes are given, it is always the LO object of the premise which is relocated until the relation is fulfilled, while the RO object stays fixed.
Consequences
Note that the construction of a inconsistent model changes the logical validity of a possible conclusion. Once the model is inconsistent, logically everything can be followed from its premises and the response of PRISM is evaluated differently. The the section of the Reasoner's Response to see the answer evaluation for inconsistent premises.