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Chapter 2 & 3 (Part 2): A Representation & Reasoning System & Using Definite Knowledge

Chapter 2 & 3 (Part 2): A Representation & Reasoning System & Using Definite Knowledge. Proofs (2.7) (How to compute logical consequences of a KB rather that testing whether a given formula is a KB logical consequence?) Functions Symbols: Language Extension (2.8)

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Chapter 2 & 3 (Part 2): A Representation & Reasoning System & Using Definite Knowledge

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  1. Chapter 2 & 3 (Part 2):A Representation & Reasoning System & Using Definite Knowledge Proofs (2.7) (How to compute logical consequences of a KB rather that testing whether a given formula is a KB logical consequence?) Functions Symbols: Language Extension (2.8) Representing Abstract Concepts (3.5) D. Poole, A. Mackworth, and R. Goebel, Computational Intelligence: A Logical Approach, Oxford University Press, January 1998

  2. Proofs (2.7) • A proof is a mechanically derivable demonstration that a formula logically follows from a knowledge base. • Given a proof procedure, KB g means g can be derived from knowledge base KB. • Recall KBg means g is true in all models of KB. • A proof procedure is sound (w.r.t. a semantics) if KB g implies KB g (every thing that can be derived from KB is a logical consequence of KB). • A proof procedure is complete (w.r.t a semantics) if there is a proof of each logical consequence of KB: That is, if KB g implies KB g (there is a proof of g). CS 426 Artificial Intelligence, Ch. 2 & 3:A representation & reasoning system, Using definite knowledge

  3. Proof (2.7) (cont.) • Bottom-up Ground Proof Procedure • One rule of derivation, a generalized form of modus ponens: • If “h  b1 …  bm”is a clause in the knowledge base, and each bi has been derived from KB, then h can be derived from KB. • You are forward chaining on this clause (going forward from what is known but not backward from the query!). • (This rule also covers the case when m = 0. CS 426 Artificial Intelligence, Ch. 2 & 3:A representation & reasoning system, Using definite knowledge

  4. (C: consequence set) Proof (2.7) (cont.) • Bottom-up proof procedure If g C then g is proven CS 426 Artificial Intelligence, Ch. 2 & 3:A representation & reasoning system, Using definite knowledge

  5. Proof (2.7) (cont.) • Example a b ^ c. b d ^ e. b g ^ e. c e. d. e. fa ^ g. CS 426 Artificial Intelligence, Ch. 2 & 3:A representation & reasoning system, Using definite knowledge

  6. Proof (2.7) (cont.) • Example (cont.) {} {d} {e,d} {c,e,d} {b,c,e,d} {a,b,c,e,d} CS 426 Artificial Intelligence, Ch. 2 & 3:A representation & reasoning system, Using definite knowledge

  7. Proof (2.7) (cont.) • Soundness of bottom-up proof procedureIf KB g then KB g. (every atom in C is a logical consequence of KB) Suppose there is a g (atom) such that KB g and KB g. Let g be the first atom added to C that’s not true in every model of KB. Suppose g isn’t true in model Iof KB.There must be a clause in KB of form: g  b1 …  bm where: Each bi is true in I (since g is the first) and g is false in I. So this clause is false in I. Therefore I isn’t a model of KB. Contradiction: thus no such g exists ! CS 426 Artificial Intelligence, Ch. 2 & 3:A representation & reasoning system, Using definite knowledge

  8. Proof (2.7) (cont.) • Fixed Point • The C generated at the end of the bottom-up algorithm is called a fixed point (any further application of the rule won’t change C). • Let I be the interpretation in which every element of the fixed point is true and every other atom not in C is false Then I is a model of KB. Proof: suppose h  b1 …  bm in KB is false in I. The only way this could happen is if b1,b2…,bm C and h C. Thus h can be added to C. Contradiction to C being the fixed point. In conclusion, there can be no clause in KB that is false in the interpretation I (e.g., I is a model) • I is called a Minimal Model. CS 426 Artificial Intelligence, Ch. 2 & 3:A representation & reasoning system, Using definite knowledge

  9. Proof (2.7) (cont.) • Completeness of the bottom-up proof • If KB g then KB g. Suppose KB g. Then g is true in all models of KB. Thus g is true in the minimal model (defined by the fixed point). Thus g is generated by the bottom up algorithm. Thus KB g. • Complexity of the bottom-up proof: • The bottom-up proof algorithm halts and is linear w.r.t. to the size of the knowledge base. CS 426 Artificial Intelligence, Ch. 2 & 3:A representation & reasoning system, Using definite knowledge

  10. Proof (2.7) (cont.) • Top-down Proof Procedure • Idea: search backward from a query to determine if it is a logical consequence of KB. • An answer clause is of the form: yes a1a2 … am • The SLD (Selector Function on Definite Clauses) Resolution of this answer clause on atom aiwith the clause: aib1…bp is the answer clause yes a1…ai-1b1…bpai+1…am CS 426 Artificial Intelligence, Ch. 2 & 3:A representation & reasoning system, Using definite knowledge

  11. Proof (2.7) (cont.) • Top-down Ground Proof Procedure (cont.) • Derivations • An answer is an answer clause with m = 0. That is, it is the answer clause yes  . • A derivation of query “?q1…qk” from KB is a sequence of answer clauses y0; y1, …, ynsuch that • y0 is the answer clause yes q1 … qk, (query) • yi is obtained by resolving yi-1 with a clause in KB, and • ynis an answer. CS 426 Artificial Intelligence, Ch. 2 & 3:A representation & reasoning system, Using definite knowledge

  12. Proof (2.7) (cont.) • Top-down ground proof procedure (cont.) • Search backward from a query to determine if it is a logical consequence of the given clauses Solve(q1^ ^… ^qk)ac:=“yes q1^ ^… ^qk”repeatselect a conjunct ai from the body of acchoose clause C from KB with ai as head replace ai in the body of ac by the body of Cuntil ac is an answer CS 426 Artificial Intelligence, Ch. 2 & 3:A representation & reasoning system, Using definite knowledge

  13. Proof (2.7) (cont.) • Example a b ^c. bd ^e. bg^e. ce. d. e. fa ^g. ?a. (query) CS 426 Artificial Intelligence, Ch. 2 & 3:A representation & reasoning system, Using definite knowledge

  14. Proof (2.7) (cont.) • Example (cont.) ac:=yes a. yesb ^c. (select b and choose b d ^e) yesd ^ e^c. (or g ^e: failure: no rule chosen headed by g!) yese ^c. yesc. yese. yes. “The two operations involved are “select” (What to Do ?) and “choose” (How to Do ?). CS 426 Artificial Intelligence, Ch. 2 & 3:A representation & reasoning system, Using definite knowledge

  15. Proof (2.7) (cont.) • Nondeterministic Choice • “Don’t-care nondeterminism”: If one selection doesn’t lead to a solution, there is no point in trying other alternatives. Select [no resources available: requests should not be kept waiting! No backtracking] • “Don’t-know nondeterminism”: If one choice doesn’t lead to a solution, other choices may, but this search may be prohibitive. Choose (NP≠ P: no polynomial oracle does exist!) CS 426 Artificial Intelligence, Ch. 2 & 3:A representation & reasoning system, Using definite knowledge

  16. Proof (2.7) (cont.) • Reasoning with Variables • An instance of an atom or a clause is obtained by uniformly substituting terms for variables. • A substitution is a finite set of the form {V1/ t1, …, Vn/ tn}, where each Viis a distinct variable and each tiis a term. • The application of a substitution  ={V1/ t1, …, Vn / tn} to an atom or clause e, written e, is the instance of e with every occurrence of Vireplaced by ti. CS 426 Artificial Intelligence, Ch. 2 & 3:A representation & reasoning system, Using definite knowledge

  17. Proof (2.7) (cont.) Application Examples Z CS 426 Artificial Intelligence, Ch. 2 & 3:A representation & reasoning system, Using definite knowledge

  18. Proof (2.7) (cont.) • Unifiers • Substitution is a unifier of e1 and e2 if e1 = e2. • Substitution is a most general unifier (mgu) of e1 and e2 if •  is a unifier of e1 and e2; and • if substitution ’ also unifies e1 and e2, then e’is an instance of efor all atoms e. • If two atoms have a unifier, they have a most general unifier. CS 426 Artificial Intelligence, Ch. 2 & 3:A representation & reasoning system, Using definite knowledge

  19. Proof (2.7) (cont.) • Unification Example CS 426 Artificial Intelligence, Ch. 2 & 3:A representation & reasoning system, Using definite knowledge

  20. Proof (2.7) (cont.) • Most General Unifiers • Example: Atoms e1= p(X,Y) and e2=p(Z,Z) have as unifiers: 1={X/b, Y/b, Z/b}, 2={X/c, Y/c, Z/c} and 3={X/Z, Y/Z}. The 3 unifier is mgu, since: • 3 unifies e1 and e2 • If 1= ’then e1’=p(b,b)is an instance of e13=p(Z,Z) Similarly, e2’=p(b,b)is an instance of e23=p(Z,Z) If 2= ’then e1’=p(c,c)is an instance of e13=p(Z,Z) Similarly, e2’=p(c,c)is an instance of e23=p(Z,Z) CS 426 Artificial Intelligence, Ch. 2 & 3:A representation & reasoning system, Using definite knowledge

  21. Proof (2.7) (cont.) • Proofs using Bottom-up Procedure with Variables • A ground instance of a clause is obtained by uniformly substituting constants appearing in a KB or in any query for the variables in the clause CS 426 Artificial Intelligence, Ch. 2 & 3:A representation & reasoning system, Using definite knowledge

  22. Proof (2.7) (cont.) • Example q(a). q(b). r(a). s(W) r(W). p(X,Y) q(X) ^ s(Y). The set of all ground instances is: q(a). q(b). r(a). s(a) r(a) s(b) r(b) CS 426 Artificial Intelligence, Ch. 2 & 3:A representation & reasoning system, Using definite knowledge

  23. Proof (2.7) (cont.) • Example (cont’d) p(a,a)  q(a) ^ s(a). p(a,b)  q(a) ^ s(b). p(b,a)  q(b) ^ s(a). p(b,b)  q(b) ^ s(b) the bottom-up proof procedure will derive: q(a), q(b), r(a), s(a), p(a,a) and p(b,a). CS 426 Artificial Intelligence, Ch. 2 & 3:A representation & reasoning system, Using definite knowledge

  24. Proof (2.7) (cont.) • Definite Resolution with Variables: Top-down Derivation • A generalized answer clause is of the form yes(t1,…,tk) a1 a2… am where t1, …, tk are terms and a1, …, am are atoms. • The SLD resolution of this generalized answer clause on ai with the clause a  b1… bp • where ai and a have most general unifier , is the clause: (yes (t1, …, tk)  a1… a i-1 b1… bp a i+1… am CS 426 Artificial Intelligence, Ch. 2 & 3:A representation & reasoning system, Using definite knowledge

  25. : Proof (2.7) (cont.) • To solve query ?B with variables V1, …, Vk Set ac to generalized answer clause yes(V1, …, Vk) B; while ac is not an answer do \\ (yes .) is false Suppose ac is yes(t1, …, tk)  a1 a2 …  am Select atom ai in the body of ac; Choose clause a  b1 …  bp in KB; Rename all variables in a  b1 … bp to have new names; Let  be the most general unifier of ai and a. Fail if they don’t unify; Set ac to (yes(t1, …, tk)  a1 …  ai-1 b1 … bp ai+1 … am) end while. CS 426 Artificial Intelligence, Ch. 2 & 3:A representation & reasoning system, Using definite knowledge

  26. Proof (2.7) (cont.) • Example1 Consider the database of Figure 2.3 (page 44) and the query: You want to know: Which room is two doors east of r107 ? You translate the query ? two_doors_east(R,r107) into: yes(R) (class exercise). two_doors_east(R,r107) CS 426 Artificial Intelligence, Ch. 2 & 3:A representation & reasoning system, Using definite knowledge

  27. Proof (2.7) (cont.) Solution: ?two_doors_east(R, r107). yes(R)two_doors_east(R, r107) % this is ac % resolve with two_doors_east(E1,W1)imm_east(E1,M1) ^ imm_east(M1,W1) % substitution: {E1/R, W1/r107} yes(R)imm_east(R,M1) ^ imm_east(M1, r107) % select leftmost conjunct and resolve with imm_east(E2,W2)imm_west(W2,E2) % substitution : {E2/R, W2/M1} ( mgu for ai and a) yes(R)imm_west(M1, R) ^ imm_east(M1, r107) % select leftmost conjunct and resolve with imm_west(r109,r111) ***** % substitution: {M1/r109, R/r111} yes(r111)imm_east(r109, r107) % resolve with imm_east(E3,W3)imm_west(W3,E3) % substitution: {E3/r109, W3/r107} yes(r111)imm_west(r107, r109) % resolve with imm_west(r107, r109) and substitution: {} yes(r111). CS 426 Artificial Intelligence, Ch. 2 & 3:A representation & reasoning system, Using definite knowledge

  28. Example2 (Class Exercise) Resolve and Substitute: {Y1/A} Resolve with connected_to(w6,w5) CS 426 Artificial Intelligence, Ch. 2 & 3:A representation & reasoning system, Using definite knowledge

  29. Function Symbols: Language Extension (2.8) • Often we want to refer to individuals in terms of components. • Examples:You may want to say that some course is held at 11:30am. You do not want to have to choose a constant for each different time. • We extend the notion of term . So that a term can be f (t1, …, tn) where f is a function symbol and the tiare terms. • In an interpretation and with a variable assignment, term f (t1, …, tn) denotes an individual in the domain. • With one function symbol and one constant we can refer to infinitely many individuals. Example:before(T1,T2), true if T1 is before T2 in a day. CS 426 Artificial Intelligence, Ch. 2 & 3:A representation & reasoning system, Using definite knowledge

  30. Representing Abstract Concept: Lists (3.5) • A list is an ordered sequence of elements. • Let’s use the constant nil to denote the empty list, and the function cons(H, T) to denote the list with first element H and rest-of-list T. These are not built-in. • The list containing david, alan and randy is Cons(david, cons(alan, cons(randy, nil))) • append(X, Y, Z) is true if list Z contains the elements of X followed by the elements of Y append(nil, Z, Z). append(cons(A, X), Y, cons(A, Z))  append(X, Y, Z). Example: X=abc Y=123 Z=abc123 if append(X,Y,Z) is true cons(A,X)=[A,abc] cons(A,Z)=[A,abc123] cons(A,Z)=cons(A,X)Y if Append(X,Y,Z) is true which means: [A,abc123]=[A,abc123] CS 426 Artificial Intelligence, Ch. 2 & 3:A representation & reasoning system, Using definite knowledge

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