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Extra info for Abstract Set Theory (Notre Dame Mathematical Lectures 8)

Example text

Alternatively, reasoning can be started with something to be proved. In this case we look for an implication with its consequence part containing the predicate to be proved. Thereafter we prove the predicates in the condition part of this implication. This is called backward reasoning, because it uses modus ponens backward. In case of both directions a reasoning path, that is a chain of rules can be constructed between the facts and the goal state. This reasoning chain can be seen as a path in the state-space, a sequence of rules leading from one state to another.

Assume, we have a basic tube type with only one valve attached to it, and an "advanced" tube type, where measurement devices are also present. In order to be able to describe instances of both types sharing common attributes and behaviour, we construct the following class hierarchy in the declaration part of our program. {parent class {p-attributes {p-procedure } } } {p-class body } class tube val: procedure ... end . . end; valve; open-valve (error-code); {statements to open} {open-valve} {statements to initialize} {tube} {sub-class } tube class meas-tube {s-attributes } T,v: {s-procedure } procedure ...

In the backward reasoning strategy, rules are used in a reverse direction, from their action part to the condition part. A rule is able to fire when its action part contains the current subgoal needed to prove. Similarly to forward reasoning problems, backward reasoning problems are defined as follows. ) This is a decision task where in the worst case, the whole search tree must be traversed. As the size of the tree (the number of nodes) increases, the number of necessary computation steps increases exponentially, thus the problem is NP-complete.