Explorations in Artificial Intelligence

1. Explorations in Artificial Intelligence Prof. Carla P. Gomes gomes@cs.cornell.edu Module 3-1-2 Logic Based Reasoning Proof Methods

2. Proofs Methods

3. Current module we’ve talked about this approach Next modules Proof methods • Proof methods divide into (roughly) two kinds: • Application of inference rules • Legitimate (sound) generation of new sentences from old • Proof = a sequence of inference rule applications Can use inference rules as operators in a standard search algorithm • Different types of proofs • Model checking • truth table enumeration (always exponential in n) • improved backtracking, e.g., Davis--Putnam-Logemann-Loveland (DPLL) (including some inference rules) • heuristic search in model space (sound but incomplete) e.g., min-conflicts-like hill-climbing algorithms

4. Proof • The sequence of wffs (w1, w2, …, wn) is called a proof (or deduction) of wn from • a set of wffs Δ iff each wi in the sequence is either in Δ or can be inferred from a • wff (or wffs) earlier in the sequence by using a valid rule of inference. • If there is a proof of wn from Δ, we say that wn is a theorem of the set Δ. • Δ├ wn • (read: wn can be proved or inferred from Δ) • The concept of proof is relative to a particular set of inference rules used. If we • denote the set of inference rules used by R, we can write the fact that wn can be • derived fromΔ using the set of inference rules in R: • Δ├ R wn • (read: wn can be proved from Δ using the inference rules in R)

5. Propositional logic: Rules of Inference or Methods of Proof • How to produce additional wffs (sentences) from other ones? What steps can we • perform to show that a conclusion follows logically from a set of hypotheses? • Example • Modus Ponens P P  Q ______________  Q The hypotheses (premises) are written in a column and the conclusions below the bar The symbol  denotes “therefore”. Given the hypotheses, the conclusion follows. The basis for this rule of inference is the tautology (P  (P  Q))  Q) [aside: check tautology with truth table to make sure] • In words: when P and P  Q are True, then Q must be True also. (meaning of • second implication)

6. Propositional logic: Rules of Inference or Methods of Proof • Example • Modus Ponens If you study the CS 372 material  You will pass You study the CS372 material ______________  you will pass • Nothing “deep”, but again remember the formal reason is that • ((P ^ (P  Q))  Q is a tautology.

7. Propositional logic: Rules of Inference or Method of Proof

8. Valid Arguments • An argument is a sequence of propositions. The final proposition is called the conclusion of the argument while the other proposition are called the premises or hypotheses of the argument. • An argument is valid whenever the truth of all its premises implies the truth of its conclusion. • How to show that q logically follows from the hypotheses (p1 p2  …pn)? Show that (p1  p2  …pn)  q is a tautology One can use the rules of inference to show the validity of an argument.

9. Proof Tree • Proofs can also be based on partial orders – we can represent them using a tree structure: • Each node in the proof tree is labeled by a wff, corresponding to a wff in the original set of hypotheses or be inferable from its parents in the tree using one of the rules of inference; • The labeled tree is a proof of the label of the root node. • Example: • Given the set of wffs: • P, R, PQ • Give a proof of Q  R

10. P PQ R Q Q  R Tree Proof P, P Q, Q, R, Q  R MP Conj. What rules of inference did we use?

11. Length of Proofs • Why bother with inference rules? We could always use a truth table • to check the validity of a conclusion from a set of premises. But, resulting proof can be much shorter than truth table method. Consider premises: p_1, p_1  p_2, p_2  p_3 … p_(n-1)  p_n To prove conclusion: p_n Inference rules: Truth table: n-1 MP steps 2n Key open question: Is there always a short proof for any valid conclusion? Probably not. The NP vs. co-NP question.

12. Beyond Propositional Logic:Predicates and Quantifiers

13. Predicates • Propositional logic assumes the world contains facts that are true or false. • But let’s consider a statement containing a variable: • x > 3 since we don’t know the value of x we cannot say whether the expression is true or false • x > 3 which corresponds to “x is greater than 3” Predicate, i.e. a property of x

14. “x is greater than 3” can be represented as P(x), where P denotes “greater than 3” • In general a statement involving n variables x1, x2, … xn can be denoted by • P(x1, x2, … xn ) • P is called a predicate or the propositional function P at the n-tuple (x1, x2, … xn ).

15. When all the variables in a predicate are assigned values  Proposition, with a certain truth value. Predicate: On(x,y) Propositions: ON(A,B) is False (in figure) ON(B,A) is True Clear(B) is True

16. Variables and Quantification • How would we say that every block in the world has a property – say “clear”  We would have to say: • Clear(A); Clear(B); … for all the blocks… (it may be long or worse we may have an infinite number of blocks…) What we need is: Quantifiers:  Universal quantifier x P(x) - P(x) is true for all the values x in the universe of discourse  Existential quantifier x P(x) - there exists an element x in the universe of discourse such that P(x) is true.

17. Universal quantification • Everyone at Cornell is smart: • x At(x,Cornell)  Smart(x) • Implicity equivalent to the conjunction of instantiations of P At(Mary,Cornell)  Smart(Mary)  At(Richard,Cornell)  Smart(Richard)  At(John,Cornell)  Smart(John)  …

18. A common mistake to avoid • Typically,  is the main connective with  • Common mistake: using  as the main connective with : x At(x,Cornell)  Smart(x) means “Everyone is at Cornell and everyone is smart”

19. Existential quantification • Someone at Cornell is smart: • x (At(x,Cornell)  Smart(x)) • xP(x) “ There exists an element x in the universe of discourse such that P(x) is true” • Equivalent to the disjunction of instantiations of P (At(John,Cornell)  Smart(John))  (At(Mary,Cornell)  Smart(Mary))  (At(Richard,Cornell)  Smart(Richard))  ...

20. Another common mistake to avoid • Typically,  is the main connective with • Common mistake: using  as the main connective with : • x At(x,Cornell)  Smart(x) • when is this true? is true if there is anyone who is not at Cornell!

21. Quantified formulas • If ω is a wff and x is a variable symbol, then both x ωand x ω are • wffs. • x is the variable quantified over • ω is said to be within the scope of the quantifier • if all the variables in ω are quantified over in ω, we say that we have a closed wff or closed sentence. • Examples: • x [P(x)  R(x)] • x [P(x)(y [R(x,y)  S(x))]]

22. Properties of quantifiers • x y is the same as yx • x y is the same as yx • x y is not the same as yx • xy Loves(x,y) • “Everyone in the world loves at least one person” • y x Loves(x,y) • Quantifier duality: each can be expressed using the other • x Likes(x,IceCream) x Likes(x,IceCream) • x Likes(x,Broccoli) xLikes(x,Broccoli) • “There is a person who is loved by everyone in the world” 

24. Negation

25. Love Affairs Loves(x,y) x loves y • Everybody loves Jerry • x Loves (x, Jerry) • Everybody loves somebody • x yLoves (x, y) • There is somebody whom somebody loves • yxLoves (x, y) • Nobody loves everybody •  x yLoves (x, y) ≡ xyLoves (x, y) • There is somebody whom Lydia doesn’t love • yLoves (Lydia, y) Note: flipping quantifiers when ¬ moves in.

26. Love Affairscontinued… • There is somebody whom no one loves • y xLoves (x, y) • There is exactly one person whom everybody loves (uniqueness) • y(x Loves(x,y)  z((w Loves (w ,z) z=y)) • There are exactly two people whom Lynn Loves • x y ((xy)  Loves(Lynn,x) Loves(Lynn,y)  • z( Loves (Lynn ,z) (z=x  z=y))) • Everybody loves himself or herself • x Loves(x,x) • There is someone who loves no one besides herself or himself • x y Loves(x,y) (x=y) (note biconditional )

27. Let Q(x,y) denote “xy =0”; consider the domain of discourse the real • numbers • What is the truth value of • a) y x Q(x,y)? • b) x y Q(x,y)? False True (additive inverse)

28. The kinship domain: • Brothers are siblings x,y Brother(x,y) Sibling(x,y) • One's mother is one's female parent m,c Mother(c) = m (Female(m) Parent(m,c)) • “Sibling” is symmetric x,y Sibling(x,y) Sibling(y,x)

29. The set domain: • Sets are empty sets or those made by adjoining something to a set • s Set(s)  (s = {} )  (x,s2 Set(s2)  s = {x|s2}) • The empty set has no element adjoined to it • x,s {x|s} = {} • Adjoining an element already in the set has no effect • x,s x  s  s = {x|s} • Only elements have been adjoined to it • x,s x  s  [ y,s2 (s = {y|s2}  (x = y  x  s2))] • Subset • s1,s2 s1 s2 (x x  s1 x  s2) • Equality of sets • s1,s2 (s1 = s2)  (s1 s2 s2 s1) • Intersection • x,s1,s2 x  (s1 s2) (x  s1 x  s2) • Uniion • x,s1,s2 x  (s1 s2)  (x  s1 x  s2)

30. Rules of Inference for Quantified Statements

31. Example: • Let CS372(x) denote: x is taking CS372 class • Let CS(x) denote: x has taken a course in CS • Consider the premises x (CS372(x)  CS(x)) • CS372(Ron) • We can conclude CS(Ron)

32. Arguments • Argument (formal): • Step Reason • 1 x (CS372(x)  CS(x)) premise • 2 CS372(Ron)  CS(Ron) Universal Instantiation • 3 CS372(Ron) Premise • 4 CS(Ron) Modus Ponens (2 and 3)

33. Example • Show that the premises: • 1- A student in this class has not read the textbook; • 2- Everyone in this class passed the first homework • Imply • Someone who has passed the first homework has not read the textbook

34. Example • Solution: • Let C(x) x is in this class; • T(x) x has read the textbook; • P(x) x passed the first homework • Premises: • x (Cx  T(x)) • x (C(x)  P(x)) • Conclusion: we want to show x (P(x)  T(x))

35. Step Reason • 1 x (Cx T(x)) premise • 2 C(a)  T(a) Existential Instantiation from 1 • 3 C(a) Simplification 2 • 4 x (C(x)P(x)) Premise • 5 C(a)  P(a) Universal Instantiation from 4 • 6 P(a) Modus ponens from 3 and 5 • 7 T(a) Simplification from 2 • 8 P(a)  T(a) Conjunction from 6 and 7 • 9 x P(x) T(x) Existential generalization from 8

36. Resolution in Propositional Logic

37. P  Q P  R  Q  R Resolution (for CNF) Very important inference rule – several other inference rules can be seen as special cases of resolution. Soundness of rule (validity of rule): [(P  Q)  (P  R)]  (Q  R) Resolution for CNF – applied to a special type of wffs: conjunction of clauses. Literal – either an atom (e.g., P) or its negation (P). Clause – disjunction of of literals (e.g., (P  Q  R)). Note: Sometimes we use the notation of a set for a clause: e.g. {P,Q,R} corresponds to the clause (PQ R); the empty clause (sometimes written as Nil or {}) is equivalent to False;

38. CNF Conjunctive Normal Form (CNF) A wff is in CNF format when it is a conjunction of disjunctions of literals. (P  Q  R) (S  P  T R) (Q  S) Resolution for CNF – applied to wffs in CNF format. {λ}  Σ1 { λ} Σ2  Σ1 Σ2 Σi- sets of literals i =1 ,2 λ – atom; Resolution Resolvent of the two clauses atom resolved upon

39. R  P P  Q  R  Q R  P P  Q  R Q P P  Q  Q P P  Q  Q can be re-written can be re-written Rule of Inference Chaining Rule of Inference: Modus Ponens Resolution:Notes 1 – Rule of Inference: Chaining 2 – Rule of Inference: Modus Ponens

40. P P  Q  Q P P  Q  Q Resolution:Notes 3 – Unit Resolution

41. P  Q  R P  W  Q  R  P  R  R  W P  Q  R P  W  Q  R  P  Q  Q  W P  Q  R P  W  Q  R  P  W Resolution:Notes 4 – No duplications in the resolvent set only one instance of Q appears in the resolvent, which is a set! P  Q  R  S P  Q  W  Q  R  S  W 5 – Resolving one pair at a time DO NOT Resolve on Q and R Resolving on Q Resolving on R True

42. {P} {P}  {} Resolution:Notes 6 – Same atom with opposite signs False – any set of wffs containing two contradictory clauses is unsatisfiable. However, a clause {P, P} is True.

43. P  Q P  R  Q  R Soundness of Resolution:Validity of the Resolution Inference Rule resolving on P • Validity (Tautology): (P  Q) (P  R) (Q  R) ;

44. Conversion to CNF • P  (Q  R) • 1.Eliminate , replacing α  β with (α  β)(β  α). (P  (Q  R))  ((Q R)  P) • 2. Eliminate , replacing α  β with α β. (P  Q  R)  ((Q R)  P) • 3. Move  inwards using de Morgan's rules and double-negation: (P Q R)  ((QR)  P) • 4. Apply distributivity law ( over ) and flatten: (P  Q  R)  (Q P)  (R  P)

45. Converting DNF (Disjunctions of conjunctions) into CNF • 1 – create a table – each row corresponds to the literals in each conjunct; • 2 - Select a literal in each row and make a disjunction of these literals; Example: (PQ R ) (S R P) (Q S  P) (P  S  Q)  (P  R  Q) (P  P  Q) (P  S  S) (P  R  S) (P  P  S) (P  P  Q)… How many clauses?

46. Resolution:Wumpus World • P31P2,2, • P2,2 •  P31 P? P?

47. Resolution Refutation • Resolution is sound – but resolution is not complete – e.g., (P R) ╞ (P  R) but • we cannot infer (P  R) using resolution  • we cannot use resolution directly to decide all logical entailments. • Resolution is Refutation Complete: • We can show that a particular wff W is entailed from a given KB how? • Proof by contradiction: • Write the negation of what we are trying to prove (W) as a conjunction of clauses; • Add those clauses (W) to the KB (also a set of clauses), obtaining KB’; prove inconsistency for KB’, i.e., • Apply resolution to the KB’ until: • No more resolvents can be added • Empty clause is obtained • To show that (P  R) ╞Res (P  R) do: (1) negate (P  R), i.e.: (P) (R) ; (2) prove that • (P  R)  (P) (R) is inconsistent ! !

48. Propositional Logic:Proof by refutation or contradiction: • Satisfiability is connected to inference via the following: • KB ╞ α if and only if (KBα) is unsatisfiable • One assumes α and shows that this leads to a contradiction with the facts in KB

49. Resolution:Robot Domain • Example: • BatIsOk • RobotMoves • BatIsOk  BlockLiftable RobotMoves Show that KB ╞ BlockLiftable KB • BatIsOk  BlockLiftable  RobotMoves BlockLiftable • BatIsOk • RobotMoves • BatIsOk  BlockLiftable  RobotMoves BlockLiftable RobotMoves KB’ • BatIsOk  RobotMoves BatIsOk BatIsOk Nil

50. Resolution • Resolution is refutation complete (Completeness of resolution refutation): • If KB╞ W, the resolution refutation procedure, i.e., applying resolution on KB’, will produce the empty clause. • Decidability of propositional calculus by resolution refutation: • If KB is a set of finite clauses and if KB╞ W, then the resolution refutation procedure will terminate without producing the empty clause. • Ground Resolution Theorem • If a set of clauses is not satisfiable, then resolution closure of those clauses contains the empty clause. In general, resolution for propositional logic is exponential ! The resolution closure of a set of clauses W in CNF, RC(W), is the set of all clauses derivable by repeated application of the resolution rule to clauses in W or their derivatives.