1. Resolution Refutation Formal Aspects of Computer Science - Week 9 Resolution Refutation, Relation with Prolog Lee McCluskey, room 2/07 Email lee@hud.ac.uklee@hud.ac.uk http://scom.hud.ac.uk/scomtlm/cia2326/

2. Resolution Refutation Recap Last Week we covered: -Resolution -Resolution Refutation This week -Relationship with Prolog -How to get answers from Resolution Refutation (and Prolog) -Another Example

3. Resolution Refutation Back to Student Example Again ! Put it into Prolog S = student, D = academic, T = teaches Ax ( S(x)=>D(x) ) ; Ax ( (Ey (T(x,y) & D(y) ) => D(x) ) S(Fred) ; T(Jeff,Fred) CLAUSAL FORM: 1.~S(z) V D(z) d(Z) :- s(Z). 2. ~T(x,y) V ~D(y) V D(x) d(X) :- t(X,Y),d(Y). 3. S(Fred) s(fred). 4. T(Jeff,Fred) t(jeff,fred). Query ------------------------- ?d(jeff)

4. Resolution Refutation How does Prolog Work? d(Z) :- s(Z). d(X) :- t(X,Y),d(Y). s(fred). t(jeff,fred). ? d(jeff) ~d(jeff)d(Z) V ~s(Z). s(jeff)? d(X) V ~t(X,Y) V ~d(Y). ~d(jeff) ~t(jeff,Y) V ~d(Y)t(jeff,fred) ~d(fred)d(Z) V ~s(Z). ~s(fred) dead end Prolog Execution Tree s(fred) contradiction)

5. Resolution Refutation Example 1. ~S(x) V D(x)3. S(Fred) D(Fred) 2. ~T(x,y) V ~D(y) V D(x) ~T(x,Fred) V D(x)5. ~D(Jeff) ~T(Jeff,Fred)4. T(Jeff,Fred) Subs = Fred / x Subs = Fred / y Subs = Jeff / x..So D(Jeff) follows from our premises

6. Resolution Refutation Answer Extraction Resolution Refutation (and the PL Prolog) uses the unifications (instantiations via matching) to extract an answer from queries of the form Ex Query-Wff(x) ? (eg member(X, [1,2,3]) ?) The idea is as follows: (1)Attach Query-Wff(x) to the negated query and all its ancestors down the tree to the null clause. (2)Apply all the substitutions to w that were used in the proof tree. This will give an instance of x, say A, such that wff[A/x] follows from W. (3)Of course, if it is possible to reach a contradiction (the null clause) using more than one proof tree then more than one instantiation will be obtained.

7. Resolution Refutation One More (hard) Example The Martians Problem … Imagine Deep Space 1 travels to Mars and observes many things about the Martians, including the fact that some seem very hostile towards humans. Concrete observations are as follows: (a) All green Martians have antennae. (b) A Martian is friendly to humans if all of its children have antennae. (c) A Martian is green if at least one of its parents is green. On its way back from Mars the robot is hotly pursued by a spacecraft containing green Martians only. Should the robot suspect it is being attacked? Or can the robot reason with its observations to answer the question: `Are all green Martians friendly?'' and hence avert an inter-planetary conflict.

8. Resolution Refutation Summary Prolog uses a form of resolution within a constrained “search strategy” to work: this is called “SLD” resolution – single literal, depth first”. The refutation procedure is therefore NOT complete. To obtain answers from queries of the form Ex wff(x), we use the answer extraction technique.