CPSC 433 Artificial Intelligence Unification & Resolution Examples Andrew Kuipers Please include [CPSC433] in the subject line of any s regarding this course.

CPSC 433 Artificial Intelligence Unification – A Quick Review Unification is needed to compute mgu (most general unifier) Yet another set-based search problem: States: set of term equations u  v, with  indicating failure Transitions: Extension Rules to generate new term equations Goal condition: all equations in the state have form x  t and Occurcheck and Substitute are not applicable

CPSC 433 Artificial Intelligence Unification – A Quick Review Delete:E  {t  t} E Decompose:E  {f(t 1,…,t n )  f(s 1,…,s n )} E  {t 1  s 1,…,t n  s n } Orient:E  {t  x} (if t not a var) E  {x  t}

CPSC 433 Artificial Intelligence Unification – A Quick Review Substitute:E  {x  t, t'  s'} E  {x  t, t'[x  t]  s'[x  t]} Occurcheck:E  {x  t}  Clash:E  {f(t 1,…,t n )  g(s 1,…,s n )}  if x not in t s[x  t] : replace all x’s in s with t if x in t if f  g

CPSC 433 Artificial Intelligence Unification Example 1: mgu( f(g(x,y),c), f(g(f(d,x),z),c) )

CPSC 433 Artificial Intelligence Unification Example 1: mgu( f(g(x,y),c), f(g(f(d,x),z),c) ) E = { f(g(x, y), c) ≈ f(g(f(d, x), z), c) }

CPSC 433 Artificial Intelligence Unification Example 1: mgu( f(g(x,y),c), f(g(f(d,x),z),c) ) E = { f(g(x, y), c) ≈ f(g(f(d, x), z), c) } 1.Decompose E  {f(t 1,…,t n )  f(s 1,…,s n )} E  {t 1  s 1,…,t n  s n } E = { g(x, y) ≈ g(f(d,x),z), c ≈ c }

CPSC 433 Artificial Intelligence Unification Example 1: mgu( f(g(x,y),c), f(g(f(d,x),z),c) ) E = { g(x, y) ≈ g(f(d,x),z), c ≈ c }

CPSC 433 Artificial Intelligence Unification Example 1: mgu( f(g(x,y),c), f(g(f(d,x),z),c) ) E = { g(x, y) ≈ g(f(d,x),z), c ≈ c } 2. Delete E  {t  t} E E = { g(x, y) ≈ g(f(d,x),z) }

CPSC 433 Artificial Intelligence Unification Example 1: mgu( f(g(x,y),c), f(g(f(d,x),z),c) ) E = { g(x, y) ≈ g(f(d,x),z) }

CPSC 433 Artificial Intelligence Unification Example 1: mgu( f(g(x,y),c), f(g(f(d,x),z),c) ) E = { g(x, y) ≈ g(f(d,x),z) } 3. Decompose E  {f(t 1,…,t n )  f(s 1,…,s n )} E  {t 1  s 1,…,t n  s n } E = { x ≈ f(d, x), y ≈ z }

CPSC 433 Artificial Intelligence Unification Example 1: mgu( f(g(x,y),c), f(g(f(d,x),z),c) ) E = { x ≈ f(d, x), y ≈ z }

CPSC 433 Artificial Intelligence Unification Example 1: mgu( f(g(x,y),c), f(g(f(d,x),z),c) ) E = { x ≈ f(d, x), y ≈ z } 4. OccurcheckE  {x  t} if x in t  E = , f(g(x,y), c) and f(g(f(d,x),z),c) not unifiable

CPSC 433 Artificial Intelligence Unification Example 2: mgu( h(c,d,g(x,y)), h(z,d,g(g(a,y),z)) )

CPSC 433 Artificial Intelligence Unification Example 2: mgu( h(c,d,g(x,y)), h(z,d,g(g(a,y),z)) ) E = { h(c,d,g(x,y)) ≈ h(z,d,g(g(a,y),z)) }

CPSC 433 Artificial Intelligence Unification Example 2: mgu( h(c,d,g(x,y)), h(z,d,g(g(a,y),z)) ) E = { h(c,d,g(x,y)) ≈ h(z,d,g(g(a,y),z)) } 1. Decompose E  {f(t 1,…,t n )  f(s 1,…,s n )} E  {t 1  s 1,…,t n  s n } E = { c ≈ z, d ≈ d, g(x, y) ≈ g(g(a, y), z) }

CPSC 433 Artificial Intelligence Unification Example 2: mgu( h(c,d,g(x,y)), h(z,d,g(g(a,y),z)) ) E = { c ≈ z, d ≈ d, g(x, y) ≈ g(g(a, y), z) }

CPSC 433 Artificial Intelligence Unification Example 2: mgu( h(c,d,g(x,y)), h(z,d,g(g(a,y),z)) ) E = { c ≈ z, d ≈ d, g(x, y) ≈ g(g(a, y), z) } 2. Delete E  {t  t} E E = { c ≈ z, g(x, y) ≈ g(g(a, y), z) }

CPSC 433 Artificial Intelligence Unification Example 2: mgu( h(c,d,g(x,y)), h(z,d,g(g(a,y),z)) ) E = { c ≈ z, g(x, y) ≈ g(g(a, y), z) }

CPSC 433 Artificial Intelligence Unification Example 2: mgu( h(c,d,g(x,y)), h(z,d,g(g(a,y),z)) ) E = { c ≈ z, g(x, y) ≈ g(g(a, y), z) } 3. Decompose E  {f(t 1,…,t n )  f(s 1,…,s n )} E  {t 1  s 1,…,t n  s n } E = { c ≈ z, x ≈ g(a, y), y ≈ z }

CPSC 433 Artificial Intelligence Unification Example 2: mgu( h(c,d,g(x,y)), h(z,d,g(g(a,y),z)) ) E = { c ≈ z, x ≈ g(a, y), y ≈ z }

CPSC 433 Artificial Intelligence Unification Example 2: mgu( h(c,d,g(x,y)), h(z,d,g(g(a,y),z)) ) E = { c ≈ z, x ≈ g(a, y), y ≈ z } 4. Substitute E  {x  t, t'  s'} E  {x  t, t'[x  t]  s'[x  t]} E = { c ≈ z, x ≈ g(a, z), y ≈ z }

CPSC 433 Artificial Intelligence Unification Example 2: mgu( h(c,d,g(x,y)), h(z,d,g(g(a,y),z)) ) E = { c ≈ z, x ≈ g(a, z), y ≈ z }

CPSC 433 Artificial Intelligence Unification Example 2: mgu( h(c,d,g(x,y)), h(z,d,g(g(a,y),z)) ) E = { c ≈ z, x ≈ g(a, z), y ≈ z } 5. Orient E  {t  x} E  {x  t} E = { z ≈ c, x ≈ g(a, z), y ≈ z }

CPSC 433 Artificial Intelligence Unification Example 2: mgu( h(c,d,g(x,y)), h(z,d,g(g(a,y),z)) ) E = { z ≈ c, x ≈ g(a, z), y ≈ z }

CPSC 433 Artificial Intelligence Unification Example 2: mgu( h(c,d,g(x,y)), h(z,d,g(g(a,y),z)) ) E = { z ≈ c, x ≈ g(a, z), y ≈ z } 6. Substitute E  {x  t, t'  s'} E  {x  t, t'[x  t]  s'[x  t]} E = { z ≈ c, x ≈ g(a, c), y ≈ c }

CPSC 433 Artificial Intelligence Unification Example 2: mgu( h(c,d,g(x,y)), h(z,d,g(g(a,y),z)) ) E = { z ≈ c, x ≈ g(a, c), y ≈ c }

CPSC 433 Artificial Intelligence Unification Example 2: mgu( h(c,d,g(x,y)), h(z,d,g(g(a,y),z)) ) E = { z ≈ c, x ≈ g(a, c), y ≈ c } No more applicable extensions!

CPSC 433 Artificial Intelligence Resolution – A Quick Review Used to prove a consequence from a set of logical formulae States: Sets of clauses L 1 (t 11,…,t 1,n1 )  …  L m (t m1,…,t m,nm ) - L i is a predicate symbol or its negation, - t ij terms out of function symbols and variables (x,y…) - variables in different clauses are disjunct Initially: The consequence we want to prove is negated and added to the set of clauses representing the initial state Goal: Derive the empty clause (  ), ie: reductio ad absurdum

CPSC 433 Artificial Intelligence Resolution – A Quick Review Transitions: New clauses generated by Resolution or Factorization Resolution:C  P, D   P’  (C  D) Factorization: C  P  P'  (C  P) Note: clauses are commutative: p  q  q  p, p  q  r same as r  p  q where  = mgu(P,P’)

CPSC 433 Artificial Intelligence Resolution Example 3: 1.¬P(x)  P(f(x)) 2.¬Q(a, y)  ¬R(y, x)  P(x) 3.R(b, g(a, z)) 4.Q(a, b) Goal: P(f(g(a, c)))

CPSC 433 Artificial Intelligence Resolution Example 3: Negate goal, add to folmulas 1. ¬P(x)  P(f(x)) 2. ¬Q(a, y)  ¬R(y, x)  P(x) 3. R(b, g(a, z)) 4. Q(a, b)

CPSC 433 Artificial Intelligence Resolution Example 3: Negate goal, add to folmulas 1. ¬P(x)  P(f(x)) 2. ¬Q(a, y)  ¬R(y, x)  P(x) 3. R(b, g(a, z)) 4. Q(a, b) 5. ¬P(f(g(a, c)))

CPSC 433 Artificial Intelligence Resolution Example 3: Negate goal, add to folmulas Resolve (4) and (2) 1. ¬P(x)  P(f(x)) 2. ¬Q(a, y)  ¬R(y, x)  P(x) 3. R(b, g(a, z)) 4. Q(a, b) 5. ¬P(f(g(a, c)))

CPSC 433 Artificial Intelligence Resolution Example 3: Negate goal, add to folmulas Resolve (4) and (2) 1. ¬P(x)  P(f(x)) 2. ¬Q(a, y)  ¬R(y, x)  P(x) 3. R(b, g(a, z)) 4. Q(a, b) 5. ¬P(f(g(a, c))) ¬Q(a, y)  ¬R(y, x)  P(x), Q(a, b) ¬R(b, x)  P(x) σ = { y ≈ b }

CPSC 433 Artificial Intelligence Resolution Example 3: Negate goal, add to folmulas Resolve (4) and (2) 1. ¬P(x)  P(f(x)) 2. ¬Q(a, y)  ¬R(y, x)  P(x) 3. R(b, g(a, z)) 4. Q(a, b) 5. ¬P(f(g(a, c))) 6. ¬R(b, x)  P(x) ¬Q(a, y)  ¬R(y, x)  P(x), Q(a, b) ¬R(b, x)  P(x) σ = { y ≈ b }

CPSC 433 Artificial Intelligence Resolution Example 3: Negate goal, add to folmulas Resolve (4) and (2) Resolve (6) and (3) 1. ¬P(x)  P(f(x)) 2. ¬Q(a, y)  ¬R(y, x)  P(x) 3. R(b, g(a, z)) 4. Q(a, b) 5. ¬P(f(g(a, c))) 6. ¬R(b, x)  P(x)

CPSC 433 Artificial Intelligence Resolution Example 3: Negate goal, add to folmulas Resolve (4) and (2) Resolve (6) and (3) 1. ¬P(x)  P(f(x)) 2. ¬Q(a, y)  ¬R(y, x)  P(x) 3. R(b, g(a, z)) 4. Q(a, b) 5. ¬P(f(g(a, c))) 6. ¬R(b, x)  P(x) ¬R(b, x)  P(x), R(b, g(a, z)) P(g(a, z)) σ = { x ≈ g(a,z) }

CPSC 433 Artificial Intelligence Resolution Example 3: Negate goal, add to folmulas Resolve (4) and (2) Resolve (6) and (3) 1. ¬P(x)  P(f(x)) 2. ¬Q(a, y)  ¬R(y, x)  P(x) 3. R(b, g(a, z)) 4. Q(a, b) 5. ¬P(f(g(a, c))) 6. ¬R(b, x)  P(x) 7. P(g(a, z)) ¬R(b, x)  P(x), R(b, g(a, z)) P(g(a, z)) σ = { x ≈ g(a,z) }

CPSC 433 Artificial Intelligence Resolution Example 3: Negate goal, add to folmulas Resolve (4) and (2) Resolve (6) and (3) Resolve (7) and (1) 1. ¬P(x)  P(f(x)) 2. ¬Q(a, y)  ¬R(y, x)  P(x) 3. R(b, g(a, z)) 4. Q(a, b) 5. ¬P(f(g(a, c))) 6. ¬R(b, x)  P(x) 7. P(g(a, z))

CPSC 433 Artificial Intelligence Resolution Example 3: Negate goal, add to folmulas Resolve (4) and (2) Resolve (6) and (3) Resolve (7) and (1) 1. ¬P(x)  P(f(x)) 2. ¬Q(a, y)  ¬R(y, x)  P(x) 3. R(b, g(a, z)) 4. Q(a, b) 5. ¬P(f(g(a, c))) 6. ¬R(b, x)  P(x) 7. P(g(a, z)) P(g(a, z)), ¬P(x)  P(f(x)) P(f(g(a,z))) σ = { x ≈ g(a,z) }

CPSC 433 Artificial Intelligence Resolution Example 3: Negate goal, add to folmulas Resolve (4) and (2) Resolve (6) and (3) Resolve (7) and (1) 1. ¬P(x)  P(f(x)) 2. ¬Q(a, y)  ¬R(y, x)  P(x) 3. R(b, g(a, z)) 4. Q(a, b) 5. ¬P(f(g(a, c))) 6. ¬R(b, x)  P(x) 7. P(g(a, z)) 8. P(f(g(a, z))) P(g(a, z)), ¬P(x)  P(f(x)) P(f(g(a,z))) σ = { x ≈ g(a,z) }

CPSC 433 Artificial Intelligence Resolution Example 3: Negate goal, add to folmulas Resolve (4) and (2) Resolve (6) and (3) Resolve (7) and (1) Resolve (8) and (5) 1. ¬P(x)  P(f(x)) 2. ¬Q(a, y)  ¬R(y, x)  P(x) 3. R(b, g(a, z)) 4. Q(a, b) 5. ¬P(f(g(a, c))) 6. ¬R(b, x)  P(x) 7. P(g(a, z)) 8. P(f(g(a, z)))

CPSC 433 Artificial Intelligence Resolution Example 3: Negate goal, add to folmulas Resolve (4) and (2) Resolve (6) and (3) Resolve (7) and (1) Resolve (8) and (5) 1. ¬P(x)  P(f(x)) 2. ¬Q(a, y)  ¬R(y, x)  P(x) 3. R(b, g(a, z)) 4. Q(a, b) 5. ¬P(f(g(a, c))) 6. ¬R(b, x)  P(x) 7. P(g(a, z)) 8. P(f(g(a, z))) P(f(g(a, z))), ¬P(f(g(a, c))) □ σ = { z ≈ c }

CPSC 433 Artificial Intelligence Resolution Example 3: Negate goal, add to folmulas Resolve (4) and (2) Resolve (6) and (3) Resolve (7) and (1) Resolve (8) and (5) Search complete, goal proved 1. ¬P(x)  P(f(x)) 2. ¬Q(a, y)  ¬R(y, x)  P(x) 3. R(b, g(a, z)) 4. Q(a, b) 5. ¬P(f(g(a, c))) 6. ¬R(b, x)  P(x) 7. P(g(a, z)) 8. P(f(g(a, z))) 9. □