Introduction iii
- 2. Agents that reason logically(Logical agents) • A Knowledge based Agent • The Wumpus world environment • Representation, Reasoning and Logic • Logics – An Introduction • Propositional Logic • An Agent for the wumpus world – Propositional Logic 2
- 3. Abilities KB agent • Agent must be able to: • Represent states and actions, • Incorporate new percepts • Update internal representation of the world • Deduce hidden properties of the world • Deduce appropriate actions 3
- 4. Agents that Reason Logically • Logical agents have knowledge base, from which they draw conclusions • TELL: provide new facts to agent • ASK: decide on appropriate action 4
- 5. Sample: Wumpus World • Show original wumpus game • goal is to shoot wumpus • example of logical reasoning • Our version: • Find gold, avoid wumpus, climb back out of cave 5
- 6. A Wumpus Agent • Agent does not perceive its own location (unlike sample game), but it can keep track of where it has been • Percepts: • Stench – wumpus is nearby • Breeze – pit is nearby • Glitter – gold is here • Bump – agent has just bumped against a wall • Scream – agent has heard another player die 6
- 7. Wumpus Agent • Actuators: • Forward, Turn Left, Turn Right • Grab (gold) • Shoot (shoots arrow forward until hits wumpus or wall) • agent only has one arrow • Climb (exit the cave) • Environment: • 4x4 grid, start at (1,1) facing right 7
- 8. Wumpus Agent • Death • Agent dies if it enters a pit or square with wumpus • Goal: get gold and climb back out. Don’t die. • 1000 points for climbing out of cave with gold • 1 point penalty for each action taken • 10,000 point penalty for death 8
- 9. Some complex reasoning examples • Start in (1,1) • Breeze in (1,2) and (2,1) • Probably a pit in (2,2) • Smell in (1,1) – where can you go? • Pick a direction – shoot • Walk in that direction • Know where wumpus is 9
- 10. Another example solution No perception 1,2 and 2,1 OK B in 2,1 2,2 or 3,1 P? Move to 2,1 1,1 V no P in 1,1 Move to 1,2 (only option) 10
- 11. Example solution S and No S when in 2,1 1,3 or 1,2 has W 1,2 OK 1,3 W No B in 1,2 2,2 OK & 3,1 P 11
- 12. AI Models and Pr opositional Logic •The Role of a Model •Represent The Environment •Assimilate Knowledge •Learning •Simulation •Inference 12
- 13. Models • If we are going to create programs that are intelligent, then we need to figure out how to represent models • They allow us to predict certain things about the future. 13
- 14. The Role of a Model • Represent the environment • Provide a structure for the assimilation of new knowledge 14
- 15. Represent The Environment • Features in the environment must be represented as features in the model • They should be able to act in the model just as they do in the environment • Needs to be able to represent both long term qualities of the environment and short term states. 15
- 16. Assimilate Knowledge • A model of the world allows an agent to organize new information in the context of what it already knows and draw conclusions. 16
- 17. Learning • An AI agent may start ready programmed with knowledge, or it may have to learn it from experience • The model may change in response to new experiences. 17
- 18. Simulation • Simulate the real environment to test potential actions. • The model needs to accept simulated sensory input and it needs to feed simulated actions back in without actually making those actions in reality • It needs an imagination 18
- 19. Inference • Inference is the process of deriving a conclusion based on what is known 19
- 20. Representation, Reasoning and logic 20
- 21. Logic in general 21
- 22. Ontological and Epistemological Assumptions • ontological assumption :It is understood in connection to the logic of functioning of the agent. - question is: “What does the agent do?” This means discussing both what the agent is and what its behavior constitutes of. • Epistemological assumptions: It consider the nature of knowledge. - question is: “On what knowledge does the agent base its actions?” It is important to discuss the origins of knowledge as well as concepts such as learning and memory. 22
- 23. Types of logic 23
- 24. The use of logic • A logic: formal language for representing information, rules for drawing conclusions • Two kinds of logics: • Propositional Logic • Represents facts P∧Q ⇒ R • First Order Logic • Represents facts, objects, and relations ∀x Cat ( x) ⇒ Mammal ( x) 24
- 25. Entailment • One thing follows from another KB |= α (knowledge base entails alpha) • KB entails sentence α if and only if α is true in worlds where KB is true. Ε.g. x+y=4 entails 4=x+y • Entailment is a relationship between sentences that is based on semantics. 25
- 26. Propositional Logic • Represents facts as being either true or false • Formally represented by a letter, e.g. P or Q. • Actually refer to facts about the environment, e.g. the speed limit in town is 30mph • Single facts can be combined into sentences using Boolean operators • These sentences, if true, can become facts in the KB. • A KB is said to entail a sentence if it is true in the KB 26
- 27. Logic consists of • Logical constants: true, false • Proposition symbols: P, Q, R, ... • Logical connectives: & (or ^), ∨, ¬, →, ↔ • Parentheses: ( ) • Propositional logic is an extremely simple representation 27
- 28. Basic symbols • Expressions only evaluate to either “true” or “false.” • P “P is true” • ¬P “P is false” negation • PVQ “either P is true or Q is true or both” disjunction • P^Q “both P and Q are true” conjunction • P => Q “if P is true, the Q is true” implication • PQ “P and Q are either both true or both false” equivalence 28
- 29. For example 29
- 30. Syntax rules for propositional logic • The constants true and false are propositions by themselves. • A proposition symbol such as P or Q is a proposition by itself. • Wrapping parentheses around a proposition produces proposition. 30
- 31. Ambiguity The grammar can be ambiguous, for example: P & Q ∨ R. It is best to use parentheses to eliminate ambiguity. When ambiguity is present, we resolve it with operator precedence: (highest) : ¬ ,& ,∨ ,⇒ , ⇔ (lowest) For example: ¬P ∨ Q & R )⇒ S is equivalent to: ((¬ P) ∨ (Q & R)) ⇒ S 31
- 32. Limitations of Propositional Logic 1. It is too weak, i.e., has very limited expressiveness: • Each rule has to be represented for each situation: e.g., “don’t go forward if the wumpus is in front of you” takes 64 rules 2. It cannot keep track of changes: • If one needs to track changes, e.g., where the agent has been before then we need a timed-version of each rule. To track 100 steps we’ll then need 6400 rules for the previous example. Its hard to write and maintain such a huge rule-base Inference becomes intractable 32
- 33. Inference rules 33
- 34. An Agent for the wumpus world – Propositional Logic 34
- 35. Example of using logic in Wumpus World • KB contains: ¬S1,1 ¬B1,1 ¬S 2,1 B2,1 Stench Agent S1, 2 ¬B1, 2 Start Breeze ¬S 2 , 2 ¬B2, 2 35
- 36. KB also contains knowledge of environment • No stench no wumpus nearby R1 : ¬S1,1 ⇒ ¬W1,1 ∧ ¬W1, 2 ∧ ¬W2,1 R2 : ¬S 2,1 ⇒ ¬W1,1 ∧ ¬W2,1 ∧ ¬W2, 2 ∧ ¬W3,1 R3 : ¬S1, 2 ⇒ ¬W1,1 ∧ ¬W1, 2 ∧ ¬W2, 2 ∧ ¬W1,3 • Stench wumpus nearby R4 : S1, 2 ⇒ W1,1 ∨ W1, 2 ∨ W2, 2 ∨ W1,3 36
- 37. We can determine where wumpus is! • Method 1: Truth table • At least 14 symbols currently: S1,1, S2,1, S1,2, S2,2, W1,1, W2,1, W1,2, W2,2, W3,1, W1,3, B1,1, B2,1, B1,2, B2,2 214 rows, ouch! 37
- 38. We can determine where wumpus is! • Method 2: Inference • Modus Ponens R1 : ¬S1,1 ⇒ ¬W1,1 ∧ ¬W1, 2 ∧ ¬W2,1 ¬W1,1 ∧ ¬W1, 2 ∧ ¬W2,1 • And-Elimination ¬W1,1 ¬W1, 2 ¬W2,1 38
- 39. Inference continued... • Modus Ponens and And-Elimination again: R2 : ¬S 2,1 ⇒ ¬W1,1 ∧ ¬W2,1 ∧ ¬W2, 2 ∧ ¬W3,1 ¬W1,1 ¬W2,1 ¬W2, 2 ¬W3,1 • One more Modus Ponens: R4 : S1, 2 ⇒ W1,1 ∨ W1, 2 ∨ W2, 2 ∨ W1,3 W1,1 ∨ W1, 2 ∨ W2, 2 ∨ W1,3 39
- 40. Inference continued... • Unit Resolution: W1,1 ∨ W1, 2 ∨ W2, 2 ∨ W1,3 ¬W1,1 W1, 2 ∨ W2, 2 ∨ W1,3 ¬W2, 2 W1, 2 ∨ W1,3 ¬W1, 2 Wumpus is in (1,3)!!! W1,3 Shoot it. Shoot where? 40
- 41. Determining action based on knowledge A1, 2 ∧ W1,3 ∧ Forward A ⇒ Shoot A1, 2 ∧ P ,3 ∧ Forward A ⇒ ¬Forward 1 • Propositional logic cannot answer well the question “What action should I take?” • It only answers “Should I take action X?” 41
- 42. Propositional logic seems inefficient • Rule: “Shoot if the wumpus is in front of you” • 16 x 4 = 64 rules for the 4x4 grid • Ditto for pits 42
- 43. First-order logic to the rescue • Uses variables to represent generalities • Can reduce rules significantly 43