Dec1999 CO230 : Cognitive Science (Question 3)Suggested Solutions

(a)

With the aid of an example,define the term “predicate ”.

A predicate is an expression that allows the use of variables for inference (1). A predicate has a truth-value that can either be true or false,but this is not known until the variables in the expression are substituted by constants from their domains (1). Example:mother(X,Y)represents that X is the mother of Y.(1)

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(b)

Make use of the following predicates and rules for this part of the question:

Employees ={jo,tom}
Department={sales,purchases}

manager(X,Y)X is the manager of department Y
works in(X,Y)X works in department Y

(i)Restate the predicate expression ~Y: Employees works_in(Y,sales) as an English sentence. [1 mark ]

Award marks for other acceptable answers with slight variations in phrasing Not all the employees work in the sales department (1).

(ii)Restate the predicate expression ~Y:Employees ~ works_in(Y,sales)as an English sentence which contains the phrase “At least one ”.[1 mark ]

Award marks for other acceptable answers with slight variations in phrasing At least one employee works in sales department (1).

(iii)Restate the predicate expression ~Y:Department ~ works_in(tom,Y) as an English sentence which contains the word “all ”.[1 mark ]

Award marks for other acceptable answers with slight variations in phrasing Tom is working in all departments.(1).

(iv)Restate the predicate expression X:Employees Y:Department
works in(X,Y)as an English sentence which contains the word “all ”and the phrase “at least one ”.[2 marks]

Award marks for other acceptable answers with slight variations in phrasing All employees must work (1)in at least one department (1).

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(c)

Make use of the following predicates and rules for this part of the question:

BallGame ={basketball,hockey,soccer,football}
Team ={myteam,localteam,regionalteam}

plays(X,Y)team X plays game Y
good(X,Y)team X is good at game Y
lose(X,Y)team X loses game Y

Express the following natural language sentences as quantified predicate expressions:

(i)There is a BallGame that my team plays that it is good at.[1 mark ]

Y:BallGame plays(myteam,Y) good(myteam,Y)(1)
In this,and subsequent parts,logically equivalent answers should also receive credit.

(ii)My team will not lose a ball game that it is good at.[1 mark ]

Y:BallGame good(myteam,Y) =>~lose(myteam,Y)(1)

(iii)My team plays all BallGames but it is good at only some of them.
[2 marks ]

XY :BallGame plays(myteam,X)good(myteam,Y)(2)
Award one mark for an answer that has most of the correct components,but is incomplete or has assembled them wrongly.

(iv)If we introduce another predicate win(X,Y)(team X wins game Y),state if the sentences “My team will not lose the BallGames it is good at ”and “My team will win the BallGames it is good at ”are logical equivalents.Explain your answer, stating any reasonable assumptions that are made.(Note:no marks will be awarded if justification for the answer is not given)[3 marks ]

The set of games defined by BallGame have been carefully selected based on the characteristic that all are time based. Thus,the likely outcomes are win,lose or draw,unless there is a penalty shootout in the event of a draw,in which case there are only two outcomes:win or lose. Multiple games have been chosen so that there is a high likelihood that all students would know at least one game.

•case 1: the candidate assumes that there are three outcomes to the game:win, lose and draw -the sentences are not logically equivalent because losing does not mean not winning (and vice versa).(1)for correctness,(2)for explanation.

•case 2: the candidate assumes that there are two outcomes to the game (win and lose):the sentences are not logically equivalent because no logical relationship has been stated between the predicates win and lose.The candidate does not consider the possibility of a draw.(0)for correctness,(2)for explanation.

•case 3:the candidate assumes that there are two outcomes to the game (win and lose),and assumes that there is a penalty shootout in the event of a draw: the sentences are not logically equivalent because no logical relationship has been stated between the predicates win and lose.(1)for correctness,(2)for explanation.

•case 4:the candidate assumes that there are two outcomes to the game (win and lose),and assumes that the following logical relationships between win and lose are stated:

X:TeamY:BallGame win(X,Y) =>~lose(X,Y)
X:TeamY:BallGame lose(X,Y) =>~win(X,Y)

The candidate does not consider the possibility of a draw.The sentences are logically equivalent.(0)for correctness,(2)for explanation.

•case 5:the candidate assumes that there are two outcomes to the game (win and lose),assumes that there is a penalty shootout in the event of a draw,and assumes that the following logical relationships between win and lose are stated:

X:TeamY:BallGame win(X,Y)=>~lose(X,Y)
X:TeamY:BallGame lose(X,Y)=>~win(X,Y)

The sentences are logically equivalent.(1)for correctness,(2)for explanation.

•case 6:the candidate makes no assumptions:the sentences are not logically equivalent because no logical relationship has been stated between the predicates win and lose.(1)for correctness,(2)for explanation.

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