Artificial Intelligence: Methods and Applications Lecture 3: Review of FOPL

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1 Artificial Intelligence: Methods and Applications Lecture 3: Review of FOPL Henrik Björklund Ruvan Weerasinghe Umeå University What I d be doing Topics in Knowledge Representation 12 th Nov (Tue) Revision of FOL 15 th Nov (Fri) Reasoning with categories 2 nd Dec (Tue) Reasoning with uncertainty 6 th Dec (Fri) Probabilistic reasoning

2 Representing Knowledge Explicit vs Tacit Declarative vs Procedural Facts and Rules Need to Represent and Reason (do inference) Representing Knowledge Natural language Database schema Propositional logic Restrictions on representation (syntax) Well defined inferencing (semantics) First order (predicate) logic Overcomes representational restriction Maintains inferencing mechanism

3 Desirable Properties of a KR System Representational adequacy Inferential adequacy Inferential efficiency Acquisitional efficiency Hard to optimize all in one formalism Main issue: knowledge acquisition bottleneck Knowledge Engineering Identify the questions of interest Gather the knowledge (acquisition) Define predicates, constants (the ontology) Encode the axioms of the domain Encode the specific knowledge of the task Use the system Debug/maintain the system

4 Example Finding out if snakes are dangerous! color, head shape, length, movement, marking Turns out only some ~10% of the species are actually venomous Pythons can still maim or kill (without being venomous)! Find out the snake taxonomy Related to their dangerousness Example Representing a snake called slithery snake_slithery / slithery_snake snake(slithery) / slithery(snake) called_slithery(snake)! isa(snake, slithery) / isa(slithery, snake) isa or instance? Subjects and Predicates transform (usually) Slithery is not poisonous

5 Example The general knowledge in the domain? Snakes can be venomous Snakes can bite Snakes can constrict (strangulate) Venom, constriction can be fatal Other implicit knowledge? By default snakes are not harmful? If someone dies, they stay dead! Example Specific domain knowledge Snake habitats Snake motion Snake color and marking Snake length Snake food When and why do snakes sting, bite, constrict

6 Exercise If it doesn t rain tomorrow Carl will play tennis Stephan is not a graduate but is a good student All basketball players are tall Some people like durian If promising was a virtue Peter would be a saint Nobody likes taxes Everyone is loyal to someone Common Mistake 1 Typically, is the main connective with Common mistake: using as the main connective with : x At(x,UU) Smart(x) means Everyone is at UU and everyone is smart How do you say Everyone at UU is smart?

7 Common Mistake 2 Typically, is the main connective with Common mistake: using as the main connective with : x At(x,UU) Smart(x) is true if there is anyone who is not at UU! How do you say Someone at UU is smart? Summary of Propositional Logic PL is declarative PL allows partial/disjunctive/negated information (unlike most data structures and databases) PL is compositional: meaning of B P is derived from meaning of B and of P Meaning in PL is context-independent (unlike natural language, where meaning depends on context) PL has very limited expressive power E.g., cannot say all basketball players are tall except by writing one sentence for each basketball player!

8 Why First Order (Predicate) Logic? Weaknesses of Propositional Logic Cannot represent/reason at sub-sentence level e.g. P = It is raining Q = It is not raining But ~P Q Cannot generalise (does not allow variables) e.g. P = Socrates is mortal Q = Plato is mortal How to represent All men are mortal? Laws of FOPL If S is a sentence so is ~S If S 1 & S 2 are sentences so is S 1 S 2 If S 1 & S 2 are sentences so is S 1 S 2 If S 1 & S 2 are sentences so is S 1 S 2 If S 1 & S 2 are sentences so is S 1 S 2 If X is a variable and S a sentence then (a) X S is a sentence (b) X S is a sentence

9 Laws of FOPL (contd.) (for all) is the universal quantifier means true for all values of the variable (there exists) is the existential quantifier means true for some values of the variable e.g. Y X mother(x, Y) X Y father(x, Y) mother(x, Y) parent(x, Y) Relationships of Quantifiers ~ X p(x) = X ~p(x) ~ X p(x) = X ~p(x) X (p(x) q(x)) = X p(x) Y q(y) X (p(x) q(x)) = X p(x) Y q(y) NB: X p(x) = Y p(y) and X p(x) = Y p(y) Ex.: Why is X (p(x) q(x)) X p(x) Y q(y)? and X (p(x) q(x)) X p(x) Y q(y)?

10 Resolution: An Algorithm for Reasoning with PL STEP 1: Convert PL statements to standard form i.e. into a conjuction of disjuncts STEP 2: Prove by refutation i.e. by proving that the negation of the goal leads to a contradiction Example of Resolution for PL Axiom Clause P P (P Q) R ~P ~Q R (S T) Q ~S Q ~T Q T T

11 Example of Resolution for PL Resolution: Prove R. So we assume ~R ~P ~Q R (2) ~R (G) ~P ~Q P (1) ~T Q (4) ~Q ~T T (5) So, we have proved R. [] (contradiction) Example for FOPL All Romans who know Marcus either hate Caesar or think that anyone who hates anyone is crazy

12 Step I (a) Convert to clausal form e.g. All Romans who know Marcus either hate Caesar or think that anyone who hates anyone is crazy X(roman(X) knows(x, marcus)) hate(x, caesar) ( Y Z hate(y, Z) think_crazy(x, Y)) Step I (b) Convert to conjuctive normal form Eliminate by rule Standardise variables by renaming Move universal quantifiers to the left Eliminate existential quantifiers Convert to conjunction of disjuncts Rename variables taking each conjunct as clause

13 Step I (b) Example Eliminate by rule i.e. a b to be replaced by ~a b further using other rules to convert all to e.g. ~(a b) = ~a ~b we get: X(~roman(X) ~knows(x, marcus)) (hate(x, caesar) ( Y Z ~hate(y, Z) think_crazy(x, Y))) Step I (b) Example (contd) Standardise variables by renaming e.g. Xp(X) Xq(X) = Xp(X) Yq(Y) Move universal quantifiers to left X Y Z(~roman(X) ~knows(x, marcus)) (hate(x,caesar) (~hate(y,z) think_crazy(x, Y)))

14 Step I (b) Example (contd) Eliminate any existential quantifiers X student(x) replaced by student(s1) where S1 is a (skolem) constant Y Z hate(y, Z) would however be replaced by a skolem function S2 in Y hate(s2(y),y) Convert to conjunction of disjuncts substitute (a b) c = (a c) (b c) etc. ~roman(x) ~knows(x, marcus) hate(x, caesar) ~hate(y, Z) think_carzy(x, Y) Step 2 Resolution works by combining two clauses in opposite form (one negated) to produce a resolvent clause by cancelling out the predicate concerned. In cases where such predicates contain arguments that are non-identical, the matching substitution (unification) needs to be applied.

15 Step 2 Example winter summer ~winter cold summer cold man(john) woman(john) ~man(john) male(john) woman(john) male(john) man(john) woman(john) ~man(x) male(x) woman(john) male(john) Unification The terms f(x, a(b,c)) and f(d, a(z, c)) unify. f f d a X a Z c b c The terms are made equal if d is substituted for X, and b is substituted for Z. We also say X is instantiated to d and Z is instantiated to b, or X/d, Z/b.

16 Unification The terms f(x, a(b,c)) and f(z, a(z, c)) unify. f f Z a X a Z c b c Note that Z co-refers within the term. Here, X/b, Z/b. Unification (not!) The terms f(c, a(b,c)) and f(z, a(z, c)) do not unify. f f Z a c a Z c b c No matter how hard you try, these two terms cannot be made identical by substituting terms for variables.

17 Unification Exercise Do terms g(z, f(a, 17, B), A+B, 17) and g(c, f(d, D, E), C, E) unify? g g Z f + 17 C f C E A 17 B A B D D E Example - English [1] Marcus is a man [2] Marcus is a Pompeian [3] All Pompeians are Romans [4] Caesar is a ruler Query: Does Marcus hate Caesar? [5] All Romans are either loyal to Caesar or hate Caesar [6] Everyone is loyal to someone [7] People only try to assassinate rulers they are not loyal to [8] Marcus tries to assassinate Caesar

18 Example - FOPL [1] man(marcus) [2] Pompeian(Marcus) [3] X Pompeian(X) Roman(X) [4] ruler(caesar) [5] X Roman(X) loyalto(x, Caesar) hate(x, Caesar) [6] X Y loyalto(x, Y) [7] X Y man(x) ruler(y) try_assassinate(x, Y) ~loyalto(x, Y) [8] try_assassinate(marcus, Caesar) Example - CNF [1] man(marcus) [2] Pompeian(Marcus) [3] ~Pompeian(X1) Roman(X1) [4] ruler(caesar) [5] ~Roman(X2) loyalto(x2, Caesar) hate(x2, Caesar) [6] loyalto(x3, f1(x3)) [7] ~man(x4) ~ruler(y1) ~ try_assassinate(x4, Y1) ~loyalto(x4, Y1) [8] try_assassinate(marcus, Caesar)

19 Example - Resolution [G] hate(marcus, Caesar)? Assume negation: ~hate(marcus, Caesar) + [5] ~Roman(Marcus) loyalto(marcus, Caesar) + [3] ~Pompeian(Marcus) loyalto(marcus, Caesar) +[2] loyalto(marcus, Caesar) + [7] ~man(marcus) ~ruler(caesar) ~try_assassinate(marcus, Caesar) + [1] ~ruler(caesar) ~try_assassinate(marcus, Caesar) + [4] ~try_assassinate(marcus, Caesar) + [8] [] #contradiction! Therefore, negation is false, and so G is true. Homework Marcus was a man Marcus was a Pompeian Marcus was born in A.D. 40 All men are mortal All Pompeians died when the volcano erupted in A.D. 79 No mortal lives more than 150 years It is now 2013 Alive means not dead If someone dies, then he stays dead Can you find two ways to prove that Marcus is dead?

COMP4418: Knowledge Representation and Reasoning First-Order Logic

COMP4418: Knowledge Representation and Reasoning First-Order Logic Maurice Pagnucco School of Computer Science and Engineering University of New South Wales NSW 2052, AUSTRALIA morri@cse.unsw.edu.au COMP4418