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Logical Agents

Logical Agents. Chapter 7 AIMA 2 nd Ed. Outline. Knowledge-Based Agents Wumpus World Logic in general – models and entailment Propositional (Boolean) logic Equivalence, validity and satisfiability Inference rules and theorem proving forward chaining backward chaining resolution.

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Logical Agents

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  1. Logical Agents Chapter 7 AIMA 2nd Ed.

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

  3. Inference engine Knowledge base Knowledge bases domain-independent algorithms • Knowledge base = set of sentences in a formal language • Declarative approach to build an agent (or other system): Tell 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 they’re implemented • Or at the implementation level: • i.e., data structures in KB and algorithms that manipulate them domain-specific content

  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. WUMPUS WORLD stench pit Wumpus breeze Gold Agent

  6. Wumpus World: PEAS description • Performance: • pick up gold: +1000 • fall into a pit or eaten by wumpus: -1000 • each action taken: -1 • using up the arrow: -10 • Enviroment: • 4 x 4 grid rooms. Agent start at bottom left (square [1,1]), facing right. Gold and wumpus locations randomly chosen. Each square other than start can be a pit with probability 0.2

  7. Wumpus World: PEAS description • Actuators: • Forward,Turn Left 900, Turn Right 900 • Grab : grab object in the same square as the agent. • Shoot: fire an arrow in a straight line in the direction the agent is facing. The arrow continues until it hits (and kills) the wumpus or hits a wall. The agent only has one arrow  only the first shoot action has any effect. • The agent dies if it enters a square containing a pit or a live wumpus. (It is safe to enter a square with a dead wumpus).

  8. Wumpus World: PEAS description • Sensors: five sensors represented with an ordered pair with five members, each contains a single bit of information. • In the square containing the wumpus and in the directly (not diagonally) adjacent squares, it (the agent) will perceive a stench. • In the squares directly adjacent to a pit, it will perceive a breeze. • In the square where the gold is, it’ll perceive a glitter. • When an agent walks into a wall, it’ll perceive a bump. • When the wumpus is killed, it emits a woeful screamthat can be perceived anywhere in the cave. • E.g.: if there’s a stench and a breeze, but no glitter, bump or scream, the agent will receive the percept [Stench, Breeze, None, None, None].

  9. Wumpus world characterization Observable?? 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

  10. Wumpus world: Initial State A = Agent B = Breeze G = Glitter, Gold OK = Safe square P = Pit S = stench V = visited W = wumpus PERCEPT: [None, None, None, None, None]

  11. Wumpus world: After one move A = Agent B = Breeze G = Glitter, Gold OK = Safe square P = Pit S = stench V = visited W = wumpus Percept: [None, Breeze, None, None, None]

  12. Wumpus world: After third move A = Agent B = Breeze G = Glitter, Gold OK = Safe square P = Pit S = stench V = visited W = wumpus Percept: [Stench, None, None, None, None]

  13. Wumpus world: After fifth move A = Agent B = Breeze G = Glitter, Gold OK = Safe square P = Pit S = stench V = visited W = wumpus Percept: [Stench, Breeze, Glitter, None, None]

  14. Wumpus world: Example tight spots Breeze in (1,2) and (2,1)  no safe actions. You have to compute the probability of a pit in each of (3,1), (2,2) and (1,3) to decide the most “OK” room.

  15. Wumpus world: Example tight spots • Stench in (1,1)  cannot move. • Can use strategy of coercion: • shoot straight ahead • wumpus was there  dead  safe • wumpus wasn’t there  safe

  16. Logic in general • Logics are formal languages for representing information such that conclusions can be drawn. • Syntax defines the sentence 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 true in a world where x = 0, y = 6

  17. Entailment • Entailment means that one thing follows from another: KB=  • Knowledge base KB entails sentence  if and only if  is true in all worlds where KB is true. • E.g., the KB containing “Milan won” and “Roma won” entails “Either Milan won or Roma won”. • E.g., x + y = 4 entails 4 = x + y • Entailment is a relationship between sentences (i.e., syntax) that is based on semantics. • Note: brains process syntax (of some sort).

  18. 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  if  is true in m • M() is the set of all models of  • Then KB=  if and only if M(KB)  M() • E.g. KB = Milan won and Roma won  = Milan won

  19. Entailment in wumpus world • Situation after detecting nothing in [1,1], moving right, breeze in [2,1] • Consider possible models for this! (assuming only pits) • 3 boolean choices  8 possible models

  20. Wumpus world models

  21. Wumpus world models KB = wumpus-world rules + observations

  22. Wumpus world models KB = wumpus-world rules + observations 1 = “[1,2] is safe” KB= 1 , proved by model checking

  23. Wumpus world models KB = wumpus-world rules + observations

  24. Wumpus world models KB = wumpus-world rules + observations 2 = “[2,2] is safe” KB= 2

  25. Inference • KB i = sentence  can be derived from KB by procedure i • Set of all consequences of KB is a haystack;  is a needle. Entailment = needle being in the haystack; inference = finding it. • Soundness: i is sound if whenever KBi, it also true that KB=  • Completeness: i is complete if whenever KB = , it is also true that KBi • an unsound procedure essentially makes things up as it goes along – it announces the discovery a nonexistent needle. • an incomplete procedure cannot derive some of entailed sentence in the KB – we know that a particular needle exists in the haystack but the procedure is unable to find that needle.

  26. Sentence Sentences Entails Semantics Semantics Representation World Aspect of the real world Aspect of the real world Follows Correspondence between World and Representation If a KB is true in the real world, then any sentence α derived from KB by a sound inference procedure is also true in the real world.

  27. Logic • Grounding: the connection, if any, between logical reasoning process and the real environment in which the agent exists. • “How do we know that KB is true in the real world?”  philosophical question  many discussions  see chapter 26. • Simple answer: the agent’s sensors create the connection. • The meaning and truth of percept sentences are defined by the processes of sensing and sentence construction. • Some part of knowledge is not a direct representation of a single percept, but a general rule derived, perhaps, from perceptual experience but not identical to a statement of that experience. This kind of general rule are produced by a sentence construction process called learning.

  28. Propositional Logic: Syntax • The proposition symbols P1, P2, etc. are sentences • If S is a sentence, S is a sentence (negation) • If S1 and S2 are sentences, S1  S2 is a sentence (conjunction) • If S1 and S2 are sentences, S1  S2 is a sentence (disjunction) • If S1 and S2 are sentences, S1S2 is a sentence (implication) • If S1 and S2 are sentences, S1S2 is a sentence (biconditional)

  29. Propositional Logic: Semantics • Each model specifies true/false for each proposition symbol • E.g., P1,2P2,2P3,1 8 possible models falsefalsetrue • Truth evaluation rules with respect to a model m: • S is true iff S is false • S1  S2 is true iff S1 is true andS2 is true • S1  S2 is true iff S1 is true orS2 is true • S1S2 is true iff S1 is falseorS2 is true i.e., is false iff S1 is true andS2 is false • 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) = false (false true) = true  (falsetrue) = true  true = true

  30. Truth tables for connectives

  31. 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] • There is no pit in [1,1]: • R1: P1,1 • A square is breezy if and only if there is an adjacent pit: • R2: B1,1 (P1,2P2,1) • R3: B2,1 (P1,1P2,2P3,1) • Include the breeze percepts for the first two squares visited: • R4: B1,1 • R5: B2,1

  32. Truth table for the knowledge base Is KB entails 1 ( “there is no pit in [1,2]”) ?  1: P1,1 KB is true in 3 out of 128 possible models. Since 1 is also true in those 3 models, then KB entails 1.

  33. Inference by enumeration • Depth-first enumeration of all models is sound and complete • O(2n) for n symbols; problem is co-NP-complete. • PL-True? returns true if a sentence holds within a model • Extend(P, true, model) returns a new partial model in which P has the value true

  34. Logical Equivalence • Two sentences are logically equivalent if and only if true in same models: •  ≡  if and only if =  and =     ≡    commutativity of     ≡    commutativity of  ((  ) γ)≡ (  ( γ)) associativity of  ((  ) γ)≡ (  ( γ)) associativity of  () ≡  double-negation elimination (  ) ≡ ( ) contraposition (  ) ≡ (  ) implication elimination (  ) ≡ ((  ) (  )) biconditional elimination (  ) ≡ ( ) de Morgan (  ) ≡ ( ) de Morgan (  ( γ))≡ ((  )  ( γ)) distributivity of  over  (  ( γ))≡ ((  )  ( γ)) distributivity of over 

  35. Validity and Satisfiability • A sentence is valid if it is true in all models, • e.g., True, A A, (A  (A  B))  B • Validity is connected to inference via Deduction Theorem: • KB=  if and only if (KB) is valid • A sentence is satisfiable if it is true in some model • e.g., A  B, C • A sentence is unsatisfiable if it is true in no models • e.g., A A • Satisfiability is connected to inference via the following: • KB=  if and only if (KB) is unsatisfiable • i.e. prove  by reductio ad absurdum (contradiction)

  36. Proof methods Proof methods divide into (roughly) two kinds: • Application of inference rules • Legitimate (sound) generation of new sentences from old ones • Proof = a sequence of inference rule applications. Can use inference rules as operators in a standard search alg. • Typically require translation of sentences into a normal form • Model checking • Truth table enumeration (always exponential in n) • Improved backtracking, e.g., Davis-Putnam-Longemann-Loveland • Heuristic search in model space (sound but incomplete), e.g., min-conflicts-like hill-climbing algorithms

  37. Resolution • Resolution is one of inference rules; other rules include Modus Ponens, And-Elimination, etc. • Conjunctive Normal Form (CNF – universal) • conjunction of disjunctions of literals • clauses • e.g., (A B)  (B C D)

  38. Resolution • Resolution inference rule (for CNF): complete for propositional logic l1 … lk, m1 … mn l1 … li-1 li+1 … lk m1 … mj-1mj+1 … mn • where li and mj are complementary literals. P1,3P2,2,P2,2, P1,3 • Resolution is sound and complete for PL

  39. Conversion to CNF B1,1 (P1,2P2,1) • Eliminate , replacing with () (). (B1,1 (P1,2P2,1)) ((P1,2P2,1)B1,1) • Eliminate , replacing  with . (B1,1P1,2P2,1) ((P1,2P2,1)B1,1) • Move  inwards using de Morgan’s rules and double negation. (B1,1P1,2P2,1) ((P1,2P2,1)B1,1) • Apply distributivity law ( over ) and flatten. (B1,1P1,2P2,1) (P1,2B1,1)  (P2,1 B1,1)

  40. Resolution algorithm • Proof by contradiction, i.e., show KB  unsatisfiable. PL-Resolve returns the set of all possible clauses obtained by resolving its two inputs.

  41. Resolution example • KB = (B1,1 (P1,2P2,1)) B1,1 •  =  P1,2 P2,1 B1,1 P1,2 P1,2B1,1 B1,1 B1,1P1,2P2,1 P2,1 B1,1P2,1B1,1 B1,1P1,2B1,1 P1,2 P1,2P2,1 P2,1 P1,2P2,1 P1,2

  42. Horn form • Horn form (restricted) • Real world KB often contain only clauses of restricted kind called Horn clauses • KB = conjunction of Horn clauses • Horn clause: • disjunction of literals of which at most one is positive • e.g., C B  A can be written as (C  B)  A • Horn clause with exactly one positive literal are called definite clause • The positive literal  head; the negatives  the body • Horn clause with no positive literal can be written as an implication whose conclusion is FALSE.

  43. Forward and backward chaining • Modus Ponens (for Horn form): complete for Horn KBs: • Can be used with forward chaining or backward chaining. These algorithms run in linear time in the size of KB.

  44. Forward chaining • Idea: fire any rule whose premises are satisfied in the KB, add its conclusion to the KB, until query is found

  45. Simple (inefficient?) forward chaining algorithm

  46. Forward chaining example Q 0 1 P 2 1 0 M 2 0 1 L 0 2 1 1 0 2 A B

  47. FC: Proof of completeness FC derives every atomic sentence that is entailed by KB • FC reaches a fixed point where no new atomic sentences are derived • Consider the final state as a model m, assigning true/false to symbols • Every clause in the original KB is true in m • Proof: Suppose a clause a1 …  ak  b is false in m. Then a1 …  ak is true in m and b is false in m. Therefore the algorithm has not reached a fixed point! • Hence m is a model of KB • If KB = q, q is true in every model of KB, including m

  48. Backward chaining • Idea: work backwards from the query q; To prove q by BC: • check if q is known already, or • prove by BC all premises of some rule concluding q • Avoid loops: check if new subgoal is already on the goal stack • Avoid repeated work: check if new subgoal: • has already been proved true, or • has already failed

  49. Backward chaining example Q P M L B A

  50. Forward vs. backward chaining • FC is data-driven, appropriate for automatic, unconscious processing, • e.g., object recognition, routine decisions • may do lots of work that is irrelevant to the goal • BC is goal-driven, appropriate for problems solving • e.g., Where are my keys? How do I get into Fasilkom UI? • Complexity of BC can be much less than linear in size of KB

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