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Logical Agents

Logical Agents. CS 171/271 (Chapter 7) Some text and images in these slides were drawn from Russel & Norvig’s published material. Logic and Knowledge Bases. Logic: means of representation and reasoning Knowledge Base (KB): set of sentences (expressed in some language)

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Logical Agents

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  1. Logical Agents CS 171/271 (Chapter 7) Some text and images in these slides were drawn fromRussel & Norvig’s published material

  2. Logic and Knowledge Bases • Logic: means of representation and reasoning • Knowledge Base (KB): set of sentences (expressed in some language) • Inference: deriving new sentences from sentences in the KB

  3. Knowledge-BasedAgent Function TELL: adds a sentence to the KBASK: queries the KB

  4. Example: Wumpus World • 4 by 4 grid of rooms • A room may contain: Agent, Wumpus, Pit, Gold • Agent can perceive pit or wumpus from neighboring squares • Agent starts in lower left corner, can move to neighboring squares, or shoot an arrow N,E,W, or S

  5. Wumpus WorldPEAS Description • Performance measure: • gold +1000 • death –1000 • -1 per step • -10 for using up arrow • Environment: • 4 by 4 grid of rooms • one room contains the agent (initially at [1,1] facing right) • one room (not [1,1]) contains the wumpus (and it stays there) • one room contains the gold • the other rooms may contain a pit

  6. PEAS Description, continued • Actuators: Left turn, Right turn, Forward, Grab, Shoot • Shooting kills wumpus if you are facing it • Shooting uses up the only arrow • Grabbing picks up gold if in same square • Agent dies when it enters a room containing pit/live wumpus • Sensors: Stench, Breeze, Glitter, Bump, Scream • Squares adjacent to wumpus are smelly • Squares adjacent to a pit are breezy • Glitter perceived in square containing gold • Bump perceived when agent hits a wall • Scream perceived everywhere when wumpus is hit

  7. Wumpus World and Knowledge • State of knowledge • What is known about the rooms at time t • Associate one or more values to each room, when known: A, B, G, OK, P, S, V, W(use ? to indicate possibility) • Contrast against what are actually in the rooms • A move and resulting percept allow agent to update the state of knowledge • Next move would depend on what is known

  8. Example: Initial Stateand First Move [None,None,None,None,None] [None,Breeze,None,None,None]

  9. Sample Action Sequence: forward, turn around, forward,turn right, forward, turn right, forward, turn left, forward

  10. Later Moves Actions: forward, turn around, forward, turn right,forward, turn right, forward, turn left, forward

  11. Inference • Agent can infer that there is a wumpus in [1,3] • Stench in [1,2] means wumpus is in [1,1], [1,3], or [2,2] • Wumpus not in [1,1] by the rules of the game • Wumpus not in [2,2] because [2,1] had no stench • Agent can also infer that there is a pit in [3,1] (how?)

  12. Logic • Representation • Syntax: how well-formed sentences are specified • Semantics: “meaning” of the sentences; truth with respect to each possible world (model) • Reasoning • Entailment: sentence following from another sentence ( a ╞ b )

  13. Models and Entailment • Logicians typically think in terms of models, with respect to which truth can be evaluated • model: a possible world • 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 = I am smart and you are prettyα = I am smart

  14. Models and Entailmentin 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

  15. Wumpus Models

  16. Wumpus Models KB = wumpus-world rules + observations

  17. Wumpus Models α1 = "[1,2] is safe", KB ╞ α1 proved by model checking

  18. Wumpus Models α2 = "[2,2] is safe", KB ╞ α2

  19. Inference Algorithm • An inference algorithm i is a procedure that derives sentences from a knowledge base: KB ├i s • i is sound if it derives only entailed sentences • i is complete if it can derive anysentence that is entailed

  20. Propositional Logic (PL) • PL: logic that consists of proposition symbols and connectives • Each symbol is either true or false • Syntax: describes how the symbols and connectives form sentences • Semantics: describes rules for determining the truth of a sentence wrt to a model

  21. Syntax • A sentence in Propositional Logic is either Atomic or Complex • Atomic Sentence • Symbol: e.g., P, Q, R, … • True • False • Complex Sentence • Let S and T be sentences (atomic or complex) • The following are also sentences:S, S  T, S  T, S  T, S  T

  22. Connectives • S: negation • if P is a symbol, P and P are called literals • S  T: conjunction • S and T are called conjuncts • S  T: disjunction • S and T are called disjuncts • S  T: implication • S is called the premise, T is called the conclusion • S  T: biconditional

  23. Back to the Wumpus World • Start with a vocabulary of proposition symbols, for example: • Pi,j: there is a pit in room [i,j] • Bi,j: there is a breeze in room [i,j] • Sample sentences (could be true or false) • P1,2 • B2,2P2,3 • P4,3 B3,3  B4,2  B4,4 • P3,4 B1,3 • Note issue of precedence with connectives

  24. Semantics • Truth of symbols are specified in the model • Truth of complex sentences can be determined using truth tables

  25. Knowledge Base forthe Wumpus World • Rules constitute the initial KB and can be expressed in PL; for example: • P1,1 • P4,4 B3,4  B4,3 • As the agent progresses, it can perceive other facts and incorporate it in its KB; for example: • B1,1 if it doesn’t perceive a breeze in room [1,1] • B2,1 if it perceives a breeze in room [2,1] • Can view the KB as a conjunction of all sentences asserted as true so far

  26. Inference in theWumpus World • We want to decide on the existence of pits in the rooms; i.e. does KB╞ Pi,j ? • Suppose we have already perceivedB1,1 andB2,1 • KB contains the rules and these facts • What can we say about: • P1,1, P1,2, P2,1, P2,2, P3,1 ?

  27. Truth Table Depicting128 Possible Models

  28. Inference Examples • KB is true when the rules hold—only for three rows in the table • The three rows are models of KB • Consider the value of P1,2 for these 3 rows • P1,2 is false in all rows(the rows are models of α1 = P1,2) • Thus, there is no pit in room [1,2] • Consider the value of P2,2 for these 3 rows • P1,2 false in one row, true for 2 rows • Thus, there may be a pit in room [2,2]

  29. P1,2

  30. P2,2

  31. Inference by Enumeration • We want an algorithm that determines whether KB entails some sentence α • Strategy: • Enumerate all possible models (true-false combinations of symbols in KB) • Consider only those models of KB (models where KB is true) • Return true if α is true for all such models

  32. Inference by Enumeration

  33. Analysis • Inference by Enumeration is sound and complete • By definition of sound and complete  • Runs in exponential time - O(2n) • Requires linear space - O(n)

  34. To be continued… What’s next? • Other Logical Inference Algorithms: can’t really do better than exponential, but there are algorithms that do reasonably better in practice • First-order Logic (FOL):deals with a world of objects, functions, and relations, rather than just facts (PL)

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