Outline. Logical Agents. Logical Reasoning. Knowledge Representation. Logical reasoning Propositional Logic Wumpus World Inference
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1 Outline Logical Agents ECE57 Applied Artificial Intelligence Spring 007 Lecture #6 Logical reasoning Propositional Logic Wumpus World Inference Russell & Norvig, chapter 7 ECE57 Applied Artificial Intelligence R. Khoury (007) Page Logical Reasoning Recall: Game-playing with imperfect information Partially-observable environment Need to infer about hidden information wo new challenges How to represent the information we have (knowledge representation) How to use the information we have to infer new information and make decisions (knowledge reasoning) ECE57 Applied Artificial Intelligence R. Khoury (007) Page Knowledge Representation Represent facts about the environment Many ways: ontologies, mathematical functions, Statements that are either true or false Language o write the statements Syntax: symbols (words) and rules to combine them (grammar) Semantics: meaning of the statements Expressiveness vs. efficiency Knowledge base (KB) Contains all the statements Agent can ELL it new statements (update) Agent can ASK it for information (query) ECE57 Applied Artificial Intelligence R. Khoury (007) Page
2 Knowledge Representation Example: Language of arithmetic Syntax describes well-formed formulas (W) X + Y > 7 (W) X Y + (not a W) Semantics describes meanings of formulas X + Y > 7 is true if and only if the value of X and the value of Y summed together is greater than 7 ECE57 Applied Artificial Intelligence R. Khoury (007) Page 5 Knowledge Reasoning Inference Discovering new facts and drawing conclusions based on existing information During ASK or ELL All humans are mortal Socrates is human Entailment A sentence β is inferred from sentences α β is true given that the α are true α entails β α β ECE57 Applied Artificial Intelligence R. Khoury (007) Page 6 Propositional Logic Sometimes called Boolean Logic Sentences are true () or false () Words of the syntax include propositional symbols P, Q, R, P = I m hungry, Q = I have money, R = I m going to a restaurant and logical connectives negation NO conjunction AND disjunction OR implication I-HEN biconditional I AND ONLY I ECE57 Applied Artificial Intelligence R. Khoury (007) Page 7 Propositional Logic Atomic sentences Propositional symbols rue or false Complex sentences Groups of propositional symbols joined with connectives, and parenthesis if needed (P Q) R Well-formed formulas following grammar rules of the syntax ECE57 Applied Artificial Intelligence R. Khoury (007) Page 8
3 ECE57 Applied Artificial Intelligence R. Khoury (007) Page 9 Propositional Logic Complex sentences evaluate to true or false Using truth tables Semantics (P Q) R P Q R Q P ECE57 Applied Artificial Intelligence R. Khoury (007) Page 0 Propositional Logic Semantics P Q P Q P Q P Q P Q P ruth tables for all connectives Given each possible truth value of each propositional symbol, we can get the possible truth values of the expression ECE57 Applied Artificial Intelligence R. Khoury (007) Page Propositional Logic Example Propositional symbols: A = he car has gas B = I can go to the store C = I have money D = I can buy food E = he sun is shining = I have an umbrella G = I can go on a picnic If the car has gas, then I can go to the store A B I can buy food if I can go to the store and I have money (B C) D If I can buy food and either the sun is not shining or I have an umbrella, I can go on a picnic D ( E ) G ECE57 Applied Artificial Intelligence R. Khoury (007) Page D ( E ) G D ( E ) E E G E D
4 Wumpus World Wumpus World D cave divided in rooms Gold Glitters Agent has to pick it up Pits Agent falls in and dies Agent feels breeze near pit Wumpus Agent gets eaten and dies if Wumpus alive Agent can kill Wumpus with arrow Agent smells stench near Wumpus (alive or dead) Initial state: (,) Goal: Get the gold and get back to (,) Actions: urn 90, move forward, shoot arrow, pick up gold Cost: +000 for getting gold, -000 for dying, - per action, -0 for shooting the arrow ECE57 Applied Artificial Intelligence R. Khoury (007) Page ECE57 Applied Artificial Intelligence R. Khoury (007) Page Exploring the Wumpus World Wumpus? Pit? OK OK Wumpus? OK Pit? ECE57 Applied Artificial Intelligence R. Khoury (007) Page 5 Wumpus World Logic Propositional symbols P i,j = there is a pit at (i,j) B i,j = there is a breeze at (i,j) S i,j = there is a stench at (i,j) W i,j = there is a Wumpus at (i,j) K i,j = (i,j) is ok Rules P i,j (B i+,j B i-,j B i,j+ B i,j- ) W i,j (S i+,j S i-,j S i,j+ S i,j- ) B i,j (P i+,j P i-,j P i,j+ P i,j- ) S i,j (W i+,j W i-,j W i,j+ W i,j- ) K i,j ( W i,j P i,j ) Have to be written out for every (i,j) ECE57 Applied Artificial Intelligence R. Khoury (007) Page 6
5 Wumpus World KB. K,. B,. S, a. B, (P, P, ) b. S, (W, W, ) c. K, ( W, P, ) d. K, ( W, P, ) ECE57 Applied Artificial Intelligence R. Khoury (007) Page 7 Wumpus World Inference. K,. S, 5. P,. B,. P, B, P, P, B, P, P, B, (P, P, ) ECE57 Applied Artificial Intelligence R. Khoury (007) Page 8 Wumpus World Inference Wumpus World Inference. K,. S, 5. P, 7. W,. B,. P, 6. W, S, W, W, S, W, W, S, (W, W, ) ECE57 Applied Artificial Intelligence R. Khoury (007) Page 9 P, W,. K,. S, 5. P, 7. W, 9. K,. B,. P, 6. W, 8. K, K, P, W, W, P, K, ( W, P, ) ECE57 Applied Artificial Intelligence R. Khoury (007) Page 0 5
6 Wumpus World KB. K, 0.B,. B.P,,, P,. S. S,,. W,. P,. W, 5. P, 6. W, 7. W, 5. B, 6. P, 7. P, 8.S 8. K,, 9.W, W, 9. K, 0.K, Wumpus? Pit? OK OK Wumpus? OK Pit? ECE57 Applied Artificial Intelligence R. Khoury (007) Page Inference with ruth ables Sound Only infers true conclusions from true premises Complete inds all facts entailed by KB ime complexity = O( n ) Checks all truth values of all symbols Space complexity = O(n) ECE57 Applied Artificial Intelligence R. Khoury (007) Page Inference with Rules Speed up inference by using inference rules Use along with logical equivalences No need to enumerate and evaluate every truth value ECE57 Applied Artificial Intelligence R. Khoury (007) Page Rules and Equivalences Logical equivalences (α β) (β α) (α β) (β α) ((α β) γ) (α (β γ)) ((α β) γ) (α (β γ)) ( α) α (α β) ( β α) (α β) ( α β) (α β) ((α β) (β α)) (α β) ( α β) (α β) ( α β) (α (β γ)) ((α β) (α γ)) (α (β γ)) ((α β) (α γ)) Inference rules (α β), α β (α β) α α, β (α β) (α β), β α (α β), ( β γ) (α γ) ECE57 Applied Artificial Intelligence R. Khoury (007) Page 6
7 Wumpus World & Inference Rules KB: B,. B, (P, P, ) Biconditional elimination. (B, (P, P, )) ((P, P, ) B, ) And elimination. (P, P, ) B, Contraposition. B, (P, P, ) Modus Ponens 5. (P, P, ) De Morgan s Rule 6. P, P, ECE57 Applied Artificial Intelligence R. Khoury (007) Page 5 Resolution Inference with rules is sound, but only complete if we have all the rules Resolution rule is both sound and complete (α β), ( β γ) (α γ) But it only works on disjunctions! Conjunctive normal form (CN). Eliminate biconditionals: (α β) ((α β) (β α)). Eliminate implications: (α β) ( α β). Move/Eliminate negations: ( α) α, (α β) ( α β), (α β) ( α β). Distribute over : (α (β γ)) ((α β) (α γ)) ECE57 Applied Artificial Intelligence R. Khoury (007) Page 6 CN Example. B, (P, P, ) Eliminate biconditionals. (B, (P, P, )) ((P, P, ) B, ) Eliminate implications. ( B, P, P, ) ( (P, P, ) B, ) Move/Eliminate negations. ( B, P, P, ) (( P, P, ) B, ) Distribute over 5. ( B, P, P, ) ( P, B, ) ( P, B, ) Resolution Algorithm Given a KB Need to answer a query α KB α? Proof by contradiction Show that (KB α) is unsatisfiable i.e. leads to a contradiction If (KB α), then (KB α) must be true ECE57 Applied Artificial Intelligence R. Khoury (007) Page 7 ECE57 Applied Artificial Intelligence R. Khoury (007) Page 8 7
8 Resolution Algorithm Convert (KB α) into CN or every pair of clauses that contain complementary symbols Apply resolution to generate a new clause Add new clause to sentence End when Resolution gives the empty clause (KB α) No new clauses can be added (fail) ECE57 Applied Artificial Intelligence R. Khoury (007) Page 9 Wumpus World & Resolution ( B, P, P, ) ( P, B, ) ( P, B, ) CN form of B, (P, P, ) B, Query: P, ( B, P, P, ) ( P, B, ) ( P, B, ) B, P, ( B, P, P, ) ( P, B, ) P, P, ( B, P, P, ) ( P, B, ) Empty clause! KB P, ECE57 Applied Artificial Intelligence R. Khoury (007) Page 0 Resolution Algorithm Sound Complete Not efficient ECE57 Applied Artificial Intelligence R. Khoury (007) Page Horn Clauses Resolution algorithm can be further improved by using Horn clauses Disjunction clause with at most one positive symbol α β γ Can be rewritten as implication (α β) γ Inference in linear time! Using Modus Ponens orward or backward chaining ECE57 Applied Artificial Intelligence R. Khoury (007) Page 8
9 orward Chaining Data-driven reasoning Start with known symbols Infer new symbols and add to KB Use new symbols to infer more new symbols Repeat until query proven or no new symbols can be inferred Work forward from known data, towards proving goal. KB: α, β, δ, ε. (α β) γ. (δ ε) λ. (λ γ) q ECE57 Applied Artificial Intelligence R. Khoury (007) Page Backward Chaining Goal-driven reasoning Start with query, try to infer it If there are unknown symbols in the premise of the query, infer them first If there are unknown symbols in the premise of these symbols, infer those first Repeat until query proven or its premise cannot be inferred Work backwards from goal, to prove needed information. KB: α, β, δ, ε. (λ γ) q. (δ ε) λ. (α β) γ ECE57 Applied Artificial Intelligence R. Khoury (007) Page orward vs. Backward orward chaining Proves everything Goes to work as soon as new information is available Expands the KB a lot Improves understanding of the world ypically used for proving a world model Backward chaining Proves only what is needed for the goal Does nothing until a query is asked Expands the KB as little as needed More efficient ypically used for proofs by contradiction Assumptions Utility-based agent Environment ully observable / Partially observable (approximation) Deterministic / Strategic / Stochastic Sequential Static / Semi-dynamic Discrete / Continuous Single agent / Multi-agent ECE57 Applied Artificial Intelligence R. Khoury (007) Page 5 ECE57 Applied Artificial Intelligence R. Khoury (007) Page 6 9
10 Assumptions Updated Learning agent Environment ully observable / Partially observable Deterministic / Strategic / Stochastic Sequential Static / Semi-dynamic Discrete / Continuous Single agent / Multi-agent ECE57 Applied Artificial Intelligence R. Khoury (007) Page 7 0
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