CS621 : Artificial Intelligence Pushpak Bhattacharyya CSE Dept., IIT Bombay Lecture 34 Nonmonotonic reasoning

Predicate Calculus deals with complete and consistent knowledge Introduction through an example (Zohar Manna, 1974): Problem: A, B and C belong to the Himalayan club. Every member in the club is either a mountain climber or a skier or both. A likes whatever B dislikes and dislikes whatever B likes. A likes rain and snow. No mountain climber likes rain. Every skier likes snow. Is there a member who is a mountain climber and not a skier? Given knowledge has: Facts Rules

Predicate Calculus: Example contd. Let mc denote mountain climber and sk denotes skier. Knowledge representation in the given problem is as follows: member(A) member(B) member(C) ∀x[member(x) → (mc(x) ∨ sk(x))] ∀x[mc(x) → ~like(x,rain)] ∀x[sk(x) → like(x, snow)] ∀x[like(B, x) → ~like(A, x)] ∀x[~like(B, x) → like(A, x)] like(A, rain) like(A, snow) Question: ∃x[member(x) ∧ mc(x) ∧ ~sk(x)] We have to infer the 11th expression from the given 10. Done through Resolution Refutation.

10 7 12 5 4 13 14 2 11 15 16 13 2 17

Characteristic of classical reasoning Monotonicity: every inference is valid, since the axioms and inference rules are valid Cannot deal with pieces of assertion which can be both true and false in the current state of knowledge

Questions How can the knowledge base be extended to allow inferences to be made on the basis of lack of knowledge as well as on the presence of it? How can the knowledge base be updated properly when a new fact is added to the system or when an old one is removed? How can knowledge be used to help resolve conflicts when there are several inconsistent nonmonotonic inferences that couldbe drawn?

“Interpretation of WFFs” Fix a domain D Assign for each constant, an element of D Assign for each n-ary function, a mapping from Dn  D Assign for each m-ary predicate, a mapping from Dm  {T, F}

Interpretation in Logic Logical expressions or formulae are “FORMS” (placeholders) for whom contents are created through interpretation. Example: This is a Second Order Predicate Calculus formula. Quantification on ‘F’ which is a function.

Examples Interpretation:1 D=N (natural numbers) a = 0 and b = 1 x ∈ N P(x) stands for x > 0 g(m,n) stands for (m x n) h(x) stands for (x – 1) Above interpretation defines Factorial

Examples (contd.) Interpretation:2 D={strings) a = b = λ P(x) stands for “x is a non empty string” g(m, n) stands for “append head of m to n” h(x) stands for tail(x) Above interpretation defines “reversing a string”

More Examples ∀x [ P(x) → Q(x)] Following interpretations conform to above expression: man(x) → mortal(x) dog(x) → mammal(x) prime(x) → 2_or_odd(x) CS(x) → bad_hand_writing(x)

Structure of Interpretation All interpretations begin with a domain ‘D’, constants (0-order functions) and functions pick values from there. With respect to - ∀x [ P(x) → Q(x)] D = {living beings} P: D → {T, F} Q: D → {T, F} P can be looked upon as a table shown here: Elements of D i.e. x P(x) Ram T Pushpak Virus1201 F

Interpretation of Factorial in the structure D = {0, 1, 2, …, ∞} a = 0, b = 1 g(m, n) = m x n and g: D x D → D h(x) = (x – 1) and h(0) = 0, h: D →D P(x) is x > 0 and P: D → {T, F}

Multiple worlds in NMR For the given set of WFFs, have a model When a new set of WFFS arrive, have a new/modified model and examine which existing WFFs need to be retracted

Operator M For all x and y, if x and y are related and if the fact that x gets along with y is consistent with everything else that is believed, then conclude that x will defend y

How to prove consistency: Negation by Failure Used in Prolog We attempt to prove ~p If we fail, we assume ~p to be false Then we call p consistent Decidable in Prolog, because prolog works with Horn clauses (only one literal on the RHS of a rule)