Logical Agents

1. Logical Agents Chapter 7

2. Outline • Knowledge-based agents • Wumpus world • Logic in general - models and entailment • Propositional (Boolean) logic • Equivalence, validity, satisfiability • Inference rules and theorem proving • forward chaining • backward chaining • resolution

3. Knowledge bases • Knowledge base = set of sentences in a formal language • Declarative approach to building an agent (or other system): • Tell it what it needs to know • Then it can Ask itself what to do - answers should follow from the KB • Agents can be viewed at the knowledge level i.e., what they know, regardless of how implemented • Or at the implementation level • i.e., data structures in KB and algorithms that manipulate them

4. A simple knowledge-based agent • The agent must be able to: • Represent states, actions, etc. • Incorporate new percepts • Update internal representations of the world • Deduce hidden properties of the world • Deduce appropriate actions

5. Performance measure gold +1000, death -1000 -1 per step, -10 for using the arrow Environment Squares adjacent to wumpus are smelly Squares adjacent to pit are breezy Glitter iff gold is in the same square Shooting kills wumpus if you are facing it Shooting uses up the only arrow Grabbing picks up gold if in same square Releasing drops the gold in same square Sensors: Stench, Breeze, Glitter, Bump, Scream Actuators: Left turn, Right turn, Forward, Grab, Release, Shoot Wumpus World PEAS description

6. Wumpus world characterization • FullyObservable No – only local perception • Deterministic Yes – outcomes exactly specified • Episodic No – sequential at the level of actions • Static Yes – Wumpus and Pits do not move • Discrete Yes • Single-agent? Yes – Wumpus is essentially a natural feature

7. Logic in general • Logics are formal languages for representing information such that conclusions can be drawn • Syntax defines the sentences in the language • Semantics define the "meaning" of sentences; • i.e., define truth of a sentence in a world • E.g., the language of arithmetic • x+2 ≥ y is a sentence; x2+y > {} is not a sentence • x+2 ≥ y is true iff the number x+2 is no less than the number y • x+2 ≥ y is true in a world where x = 7, y = 1 • x+2 ≥ y is false in a world where x = 0, y = 6

8. Entailment • Entailment means that one thing follows from another: KB ╞α • Knowledge base KB entails sentence α if and only if α is true in all worlds where KB is true • E.g., the KB containing “the Giants won” and “the Reds won” entails “Either the Giants won or the Reds won” • E.g., x+y = 4 entails 4 = x+y • Entailment is a relationship between sentences (i.e., syntax) that is based on semantics

9. Models • Logicians typically think in terms of models, which are formally structured worlds with respect to which truth can be evaluated • We say mis a model of a sentence α if α is true in m • M(α) is the set of all models of α • Then KB ╞ α iff M(KB)  M(α) • E.g. KB = Giants won and Redswon α = Giants won

10. Entailment in the wumpus world Situation after detecting nothing in [1,1], moving right, breeze in [2,1] Consider possible models for KB assuming only pits 3 Boolean choices  8 possible models

11. Wumpus models

12. Wumpus models • KB = wumpus-world rules + observations

13. Wumpus models • KB = wumpus-world rules + observations • α1 = "[1,2] is safe", KB ╞ α1, proved by model checking

14. Wumpus models • KB = wumpus-world rules + observations

15. Wumpus models • KB = wumpus-world rules + observations • α2 = "[2,2] is safe", KB ╞ α2

16. Inference • KB ├i α = sentence α can be derived from KB by procedure (inference algorithm) i • Soundness: i is sound if whenever KB ├i α, it is also true that KB╞ α (aka Truth Preserving) • Completeness: i is complete if whenever KB╞ α, it is also true that KB ├i α • Preview: we will define a logic (first-order logic) which is expressive enough to say almost anything of interest, and for which (in some cases) there exists a sound and complete inference procedure. • That is, the procedure will answer any question whose answer follows from what is known by the KB.

17. Propositional Logic A proposition is a declarative sentence that is either TRUE or FALSE (not both). • Examples: • The Earth is flat • 3 + 2 = 5 • I am older than my mother • Tallahassee is the capital of Florida • 5 + 3 = 9 • Athens is the capital of Georgia

18. Propositional Logic A proposition is a declarative sentence that is either TRUE or FALSE (not both). • Which of these are propositions? • What time is it? • Christmas is celebrated on December 25th • Tomorrow is my birthday • There are 12 inches in a foot • Ford manufactures the world’s best automobiles • x + y = 2 • Grass is green

19. Propositional Logic • Compound propositions : built up from simpler propositions using logical operators • Frequently corresponds with compound English sentences. Example: Given p: Jack is older than Jill q: Jill is female We can build up r: Jack is older than Jill and Jill is female (p  q) s: Jack is older than Jill or Jill is female (p  q) t: Jack is older than Jill and it is not the case that Jill is female (p q)

20. Propositional logic: Syntax Let symbols S1, S2 represent propositions, also called sentences If S is a proposition, S is a proposition (negation) If S1 and S2 are propositions, S1 S2 is a proposition (conjunction) If S1 and S2 are propositions, S1 S2 is a proposition (disjunction) If S1 and S2 are propositions, S1 S2 is a proposition (implication) (might sometimes see ) If S1 and S2 are propositions, S1 S2 is a proposition (biconditional) (might sometimes see )

21. Propositional Logic - negation Let p be a proposition. The negation of pis written p and has meaning: “It is not the case that p.” Truth table for negation:

22. Propositional Logic - conjunction • Conjunction operator “” (AND): • corresponds to English “and.” • is a binary operator in that it operates on two propositions when creating compound proposition • Def. Let p and q be two arbitrary propositions, the conjunction of p and q, denoted p  q, is true if both p and qare true, and false otherwise.

23. Propositional Logic - conjunction Conjunction operator p  q is true when p and q are both true. Truth table for conjunction:

24. Propositional Logic - disjunction Disjunction operator  (or): • loosely corresponds to English “or.” • binary operator Def.: Let p and q be two arbitrary propositions, the disjunction of p and q, denoted p  q is false when both p and q are false, and true otherwise. •  is also called inclusiveor • Observe thatp  qis true when p is true, or q is true, or both p and q are true.

25. Propositional Logic - disjunction Disjunction operator p  q is true when p or q (or both) is true. Truth table for conjunction:

26. Propositional Logic - XOR Exclusive Or operator (): • corresponds to English “either…or…” (exclusive form of or) • binary operator Def.: Let p and q be two arbitrary propositions, the exclusive or of p and q, denoted p  q is true when either por q (but not both) is true.

27. Propositional Logic - XOR Exclusive Or: p  q is true when p or q (not both) is true. Truth table for exclusive or: F T T F

28. Propositional Logic- Implication Implication operator(): • binary operator • similar to the English usage of “if…then…”, “implies”, and many other English phrases Def.: Let p and q be two arbitrary propositions, the implication pqis false when p is true and q is false, and true otherwise. p  q is true when p is true and q is true, q is true, or p is false. p  q is false when p is true and q is false. Example: r : “The dog is barking.” s : “The dog is awake.” r  s : “If the dog is barking then the dog is awake.”

29. Propositional Logic- Implication Truth table for implication: T F

30. Propositional Logic- Implication Truth table for implication: T F T T • If the temperature is below 10 F, then water freezes.

31. Propositional Logic- Implication Some terminology, for an implication p  q: • Its converseis:q  p. • Its inverseis:¬p  ¬ q. • Itscontrapositive is: ¬q  ¬p. One of these has the same meaning (same truth table) as p q. Which one ?

32. Propositional Logic- Biconditional Biconditional operator (): • Binary operator • Partly similar to the English usage of “If and only if Def.: Let p and q be two arbitrary propositions. Thebiconditional p  q is true when qand p have the same truth values and false otherwise. Example: p : “The dog plays fetch.” q :“The dog is outside.” p  q:“The plays fetchif and only if it is outside.”

33. Propositional Logic- Biconditional Truth table for biconditional: T F F T

34. Nested Propositions • Use parentheses to group sub-expressions in a compound proposition: “I’m sick, and I’m going to the doctor or I’m staying home.” = p (q  s) • (p  q)  s would mean something different

35. Propositional Logic: Precedence By convention… Examples:  p  q  r is equivalent to (( p)  q)  r p  q  r  s is equivalent to p  (q  (r  s))

36. Propositional logic: Semantics Each model specifies true/false for each proposition symbol E.g. P1,2 P2,2 P3,1 false true false With these symbols, 8 possible models, can be enumerated automatically. Rules for evaluating truth with respect to a model m: S is true iff S is false S1 S2 is true iff S1 is true and S2 is true S1 S2 is true iff S1 is true or S2 is true (or both) S1 S2 is true iff S1 is false or S2 is true i.e., is false iff S1 is true and S2 is false (only case) S1 S2 is true iff S1S2 is true andS2S1 is true Simple recursive process evaluates an arbitrary sentence, e.g., P1,2  (P2,2 P3,1) = true (true  false) = true true = true

37. Truth tables for connectives Not the preferred form of a Truth Table (right, this one is upside down)

38. converse contrapositive inverse ® Ø Ø ® ® Ø Ø q p p q q p T T T T F T F F T T T T Propositional Logic Proving the equivalence using truth tables ® p q p q T T T F T F T F T T F F

39. Wumpus world sentences Let Pi,j be true if there is a pit in [i, j]. Let Bi,j be true if there is a breeze in [i, j].  P1,1 B1,1 B2,1 • "Pits cause breezes in adjacent squares" B1,1 (P1,2 P2,1) B2,1  (P1,1 P2,2  P3,1)

40. Truth tables for inference Awk!

41. Inference by enumeration • Depth-first enumeration of all models is sound and complete • For n symbols, time complexity is O(2n), space complexity is O(n)

42. Logical equivalence • Two sentences are logically equivalent if they are true in the same set of models. Also, α ≡ ß iff α╞ βand β╞ α

43. p Øp p Ø p p p Øp Øp p Øp p Øp T T T F F T T F T F F T F T F T F F Propositional Equivalences • A tautologyis a proposition that is always true. • Ex.: p Øp • A contradictionis a proposition that is always false. • Ex.: p Ø p • A contingency is a proposition that is neither a tautology nor a contradiction. • Ex.: pØp

44. Propositional Logic: Logical Equivalence • If p and q are propositions, then p is logically equivalent toq if their truth tables are the same. • “p is equivalent to q.” is denoted by p q • p, q are logically equivalent if their biconditional p q is a tautology.

45. Propositional Logic: Logical Equivalence How do we prove that two compound propositions are logically equivalent ? • Construct the truth table of both compound propositions • Check if their truth-values are the same whenever the truth value of their propositions agree.

46. P p  p T F T F T F The equivalence holds since these two columns are the same. Propositional Logic: Logical Equivalence • p  p

47. Propositional Logic: Logical Equivalence • p  q  p  q

48. Propositional Logic: Logical Equivalence • Is p (qr)  (p q)  (pr) ?

49. Propositional Logic: Logical Equivalences • Identity p  T  p p  F  p • Domination p  T  T p  F  F • Idempotence pp p pp p • Double negation p p

50. Propositional Logic: Logical Equivalences • Commutativity: p  q  q  p p  q  q  p • Associativity: (p  q)  r  p  ( q  r ) (p  q)  r  p  ( q  r )