1. 1 FOL CS 331/531 Dr M M Awais Composition Find Overall Substitutions Given two or more sets of substitutions

2. 2 FOL CS 331/531 Dr M M Awais Composition Domain: Variables in the substitution together form the domain (denominators) S1={a/X1, b/X2, …z/Xn} Then Dom(S1)={X1, X2, …Xn} St={a/B} B is the domain or Dom(S1) Replacements: Terms are the replacements (numerators) Rep(S1)={a,b,…z} St={a/B}a is the replacement or Rep(St)

3. 3 FOL CS 331/531 Dr M M Awais Composition Rule Two subtitutions S1 and S2 can compose in two ways: S1 composed with S2S12 or S1.S2 S2 composed with S1S21 or S2.S1 S12 is obtained by applying S2 to S1 Adding S2 to the set obtained in step 1 Removing substitutions from S2 that have Dom(S2)=Dom(S1)

4. 4 FOL CS 331/531 Dr M M Awais Step 1: Applying S2 to S1 Finding Dom(S2) Finding Rep(S1) If elements of Dom(S2) = elements of Rep(S1) S2 applies to S1 otherwise does not. Example: S1={W/A,e/D}, S2={s/W,g/K} Dom(S2)={W, K}, Rep(S1)={W,e}, W matches hence W/A & s/W generate s/A Hence the new set obtained after applying S2 to S1 is: S12={s/A,e/D}

5. 5 FOL CS 331/531 Dr M M Awais Step 2&3: Add+Drop Add S2 to S12={s/A,e/D} S12={s/A,e/D, s/W,g/K} Drop any where Dom(S2)=Dom(S12) from substitutions of S2 Nothing gets dropped Final Answer: S12={s/A,e/D, s/W,g/K}

6. 6 FOL CS 331/531 Dr M M Awais Composition S1={A/D} S2={D/X} D is in the domain in S1, and replacement in S2 A is the replacement in S1 X is the domain in S2 S1 can be applied to S2, as the domain of S1(D) is the replacement in S2. Hence S2 modifies as {A/X} S2 cannot be applied to S1, as the domain of S2 is not a replacement in S1, S1 remains{A/D}

7. 7 FOL CS 331/531 Dr M M Awais Composition S1={A/K} replacements=A, domain=K S2={a/A} replacements=a, domain=A Apply S1 to S2 Cannot be applied as K is not a replacement in S2.{a/A} Non Composable Apply S2 t S1 Can be applied, as A is a replacement in S1, {a/K} Composable

8. 8 FOL CS 331/531 Dr M M Awais Composition S1={A/K,f/M} replacements=A,f domain=K,M S2={a/A,j/R} replacements=a,j domain=A,R Apply S1 to S2 A/K Cannot be applied as K is not a replacement in S2. F/M Cannot be applied as M is not a replacement in S2 {a/A,j/R} Non Composable Apply S2 t S1 a/A can be applied, as A is a replacement in S1, {a/K} J/R cannot be applied as R is not a replacement in S1 {a/K, f/M} Composable

9. 9 FOL CS 331/531 Dr M M Awais Example S1={A/K} S2={a/A, H/J} S12={a/K, a/A,H/J} Algorithm 1. Apply S2 to S1 (find overall substitution) and generate a modified S1 2. Add to the answer in Step 1 all the substitutions of S2 (combined +uncombined) 3. Drop all substitutions in S2 which have common denominators with S12

10. 10 FOL CS 331/531 Dr M M Awais Example: S12 S1={A/K} S2={a/A, H/J} Algorithm for S12: Apply S2 to S1 See if A and J are present among the replacements of S1 if Yes replace A with a and J with H This would generate {a/K} Add S2 to S1 (Modified S1 in step 1+ all of S2) Modified S1={a/K} + {a/A,H/J} Drop all substitutions in S2 which have common denominators with S1 nothing gets dropped, hence the answer is S12={a/K,a/A,H/J}

11. 11 FOL CS 331/531 Dr M M Awais Example: S21 S1={A/K} S2={a/A, H/J} Algorithm for S21: Apply S1 to S2 See if K is present among the replacements of S2 if Yes replace K with A (no change in S2) This would generate {a/A, H/J} Add S1 to S2 (Modified S2 in step 1+ all of S1) Modified S1={a/A,H/J}+{A/K} Drop all substitutions in S1 which have common denominators with S2 nothing gets dropped, hence the answer is S12={a/A,H/J,A/K}

12. 12 FOL CS 331/531 Dr M M Awais Example S1={D/A} S2={a/D} S12 Step 1: Combining (S2 to S1)S12={a/A} Step 2:Addition (Add S2)S12={a/A}+{a/D} Step 3: DropS12={a/A,a/D} S1={D/A,S/G,n/N} S2={x/X, f/D, q/S,w/A} S12 Step 1:CombiningS12={f/A,q/G,n/N} Step 2:Addition S12={f/A,q/G,n/N}+{x/X, f/D, q/S,w/A} Step 3: DropS12={f/A,q/G,n/N,x/X,f/D,q/S}

13. 13 FOL CS 331/531 Dr M M Awais Is S12 same as S21 S1={D/A} S2={a/D} S21 Step 1: Combine (S1 to S2)S2={a/D} Combination cannot possible because no A is present in S2, so cannot replace with D, output is same as S2 Step 2: AddS21={a/D}+{D/A} Step 3: DropS21={a/D,D/A} S1={D/A,S/G,n/N} S2={x/X, f/D, q/S,w/A} S21 Step 1: CombineS2={ x/X, f/D, q/S,w/A } Step 2: AddS2={ x/X, f/D, q/S,w/A }+{D/A,S/G,n/N} Step 3: DropS21={ x/X, f/D, q/S,w/A, S/G,n/N}

14. 14 FOL CS 331/531 Dr M M Awais Is S12 same as S21 S1={D/A} S2={X/A} S12={D/A} Combine:S12={D/A} Add:S12={D/A}+{X/A} Drop:S12={D/A} X/A gets dropped S21={X/A} Combine:S21={X/A} Add:S21={X/A}+{D/A} Drop:S21={X/A} D/A gets dropped

15. 15 FOL CS 331/531 Dr M M Awais Example S1={D/A, d/F,S/E} S2={g/D} S12={g/A, d/F, S/E, g/D} S12={g/A, d/F, S/E, g/D} (nothing gets dropped) S21={g/D, D/A, d/F, S/E} S21={g/D, D/A,d/F,S/E} (nothing gets dropped)

16. 16 FOL CS 331/531 Dr M M Awais Example S1={a/A,b/B,c/C} S2={A/X,B/Y,S/Q} S12={a/A,b/B,c/C} + {A/X,B/Y,S/Q} S12={a/A, b/B, c/C, A/X, B/Y, S/Q} S21= {a/X, b/Y, S/Q} + {a/A,b/B,c/C} S21={a/X, b/Y, S/Q, a/A, b/B, c/C}

17. 17 FOL CS 331/531 Dr M M Awais Substitution: Discussion Domain: Variables in the substitution together form the domain (denominators) Replacements: Terms are the replacements (numerators) Pure Vs Impure Substitutions: A pure substitution is if all replacements are free of the variables in the domain of the substitution. Otherwise impure {s/X, t/I, j/G} (pure) The composition of pure substitutions may be impure and vice versa Composability: A substitution S1 and S2 are composable if and only if the variables in the domain of S1 do not appear among the replacements of S2, otherwise they are noncomposable. S1={U/X} S2={d/U}composable S1={U/X} S2={X/G}noncomposable

18. 18 FOL CS 331/531 Dr M M Awais Application Example Knowledge base: knows(ali,sana) knows(X,shahid)knows(X,Y) We also know knows(ali,X)  hates(ali,X) Unify the conditions to find ali hates whom unify(knows(ali,X), knows(ali,sana)) = {sana/X} Using the binding and the rule: ali hates sana unify(knows(ali,X), knows(X,shahid)) = {ali/X,shahid/X} This would fail as X cannot have two values at the same time As the composition of {ali/X} with {shahid/X} or vice versa will result in only one valid substitution unify(knows(ali,X), knows(X,Y)) = {ali/X,X/Y} This would mean that ali knows everyone including himself, therefore hates everyone.