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Propositional Logic Russell and Norvig Chapter 7

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Knowledge-Based Agent environment agent ? sensors actuators Knowledge base

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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

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Types of Knowledge Procedural, e.g.: functions Such knowledge can only be used in one way -- by executing it Declarative, e.g.: constraints It can be used to perform many different sorts of inferences

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Logic Logic is a declarative language to: Assert sentences representing facts that hold in a world W (these sentences are given the value true) Deduce the true/false values to sentences representing other aspects of W

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Wumpus World PEAS description Performance measure gold +1000, death 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

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Wumpus world characterization Fully 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

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Exploring a wumpus world

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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

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Connection World-Representation World W Conceptualization Facts about W hold Sentences represent Facts about W represent Sentences entail

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Examples of Logics Propositional calculus A B C First-order predicate calculus ( x)( y) Mother(y,x) Logic of Belief B(John,Father(Zeus,Cronus))

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Model A model of a sentence is an assignment of a truth value – true or false – to every atomic sentence such that the sentence evaluates to true.

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Model of a KB Let KB be a set of sentences A model m is a model of KB iff it is a model of all sentences in KB, that is, all sentences in KB are true in m.

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Satisfiability of a KB A KB is satisfiable iff it admits at least one model; otherwise it is unsatisfiable KB1 = {P, Q R} is satisfiable KB2 = { P P} is satisfiable KB3 = {P, P} is unsatisfiable valid sentence or tautology

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Logical Entailment KB : set of sentences : arbitrary sentence KB entails – written KB – iff every model of KB is also a model of Alternatively, KB iff {KB, } is unsatisfiable KB is valid

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Inference Rule An inference rule { , } consists of 2 sentence patterns and called the conditions and one sentence pattern called the conclusion If and match two sentences of KB then the corresponding can be inferred according to the rule

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Inference I: Set of inference rules KB: Set of sentences Inference is the process of applying successive inference rules from I to KB, each rule adding its conclusion to KB

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Example: Modus Ponens From Battery-OK Bulbs-OK Headlights-Work Battery-OK Bulbs-OK Infer Headlights-Work { , } { , }

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Connective symbol (implication) Logical entailment Inference KB iff KB is valid

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Soundness An inference rule is sound if it generates only entailed sentences All inference rules previously given are sound, e.g.: modus ponens: { , } The following rule: { , } is unsound, which does not mean it is useless (an inference rule for abduction, outside scope of this course)

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Is each of the following a sound inference rule? { , } { , }

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Completeness A set of inference rules is complete if every entailed sentences can be obtained by applying some finite succession of these rules Modus ponens alone is not complete, e.g.: from A B and B, we cannot get A

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Proof The proof of a sentence from a set of sentences KB is the derivation of by applying a series of sound inference rules

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Proof 1. Battery-OK Bulbs-OK Headlights-Work 2. Battery-OK Starter-OK Empty-Gas-Tank Engine-Starts 3. Engine-Starts Flat-Tire Car-OK 4. Headlights-Work 5. Battery-OK 6. Starter-OK 7. Empty-Gas-Tank 8. Car-OK 9. Battery-OK Starter-OK by 5,6 10. Battery-OK Starter-OK Empty-Gas-Tank by 9,7 11. Engine-Starts by 2, Engine-Starts Flat-Tire by 3,8 13. Flat-Tire by 11,12

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Inference Problem Given: KB: a set of sentence : a sentence Answer: KB ?

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KB iff {KB, } is unsatisfiable Deduction vs. Satisfiability Test Hence: Deciding whether a set of sentences entails another sentence, or not Testing whether a set of sentences is satisfiable, or not are closely related problems

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Complementary Literals A literal is a either an atomic sentence or the negated atomic sentence, e.g.: P, P Two literals are complementary if one is the negation of the other, e.g.: P and P

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Unit Resolution Rule Given two sentences: L 1 … L p and M where L i,…, L p and M are all literals, and M and L i are complementary literals Infer: L 1 … L i-1 L i+1 … L p

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Examples From: Engine-Starts Car-OK Engine-Starts Infer: Car-OK From: Engine-Starts Car-OK Car-OK Infer: Engine-Starts Modus ponens Modus tollens Engine-Starts Car-OK

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Shortcoming of Unit Resolution From: Engine-Starts Flat-Tire Car-OK Engine-Starts Empty-Gas-Tank we can infer nothing!

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Full Resolution Rule Given two clauses: L 1 … L p and M 1 … M q where L i and M j are complementrary Infer the clause: L 1 … L i-1 L i+1 … L k M 1 … M j-1 M j+1 … M k

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Example From: Engine-Starts Flat-Tire Car-OK Engine-Starts Empty-Gas-Tank Infer: Empty-Gas-Tank Flat-Tire Car-OK

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Example From: P Q ( P Q) Q R ( Q R) Infer: P R ( P R)

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Not All Inferences are Useful! From: Engine-Starts Flat-Tire Car-OK Engine-Starts Flat-Tire Infer: Flat-Tire Flat-Tire Car-OK

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Not All Inferences are Useful! From: Engine-Starts Flat-Tire Car-OK Engine-Starts Flat-Tire Infer: Flat-Tire Flat-Tire Car-OK tautology

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Not All Inferences are Useful! From: Engine-Starts Flat-Tire Car-OK Engine-Starts Flat-Tire Infer: Flat-Tire Flat-Tire Car-OK True tautology

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Example 1. Battery-OK Bulbs-OK Headlights-Work 2. Battery-OK Starter-OK Empty-Gas-Tank Engine-Starts 3. Engine-Starts Flat-Tire Car-OK 4. Headlights-Work 5. Battery-OK 6. Starter-OK 7. Empty-Gas-Tank 8. Car-OK 9. Flat-Tire We want to show Flat-Tire, given clauses 1-8. Using resolution, we can show that clauses 1-8 along with clause 9 deduce an empty clause. Can you trace the resolution steps?

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Sentence Clause Form Example: (A B) (C D) 1. Eliminate (A B) (C D) 2. Reduce scope of ( A B) (C D) 3. Distribute over ( A (C D)) (B (C D)) ( A C) ( A D) (B C) (B D) Set of clauses: { A C, A D, B C, B D}

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Resolution Refutation Algorithm RESOLUTION-REFUTATION(KB ) clauses set of clauses obtained from KB and new {} Repeat: For each C, C’ in clauses do res RESOLVE(C,C’) If res contains the empty clause then return yes new new U res If new clauses then return no clauses clauses U new

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Efficient Propositional Inference Two families of efficient algorithms for propositional inference: Complete backtracking search algorithms DPLL algorithm (Davis, Putnam, Logemann, Loveland) Incomplete local search algorithms WalkSAT algorithm

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The DPLL algorithm Determine if an input propositional logic sentence (in CNF) is satisfiable. Improvements over truth table enumeration: 1. Early termination A clause is true if any literal is true. A sentence is false if any clause is false. 2. Pure symbol heuristic Pure symbol: always appears with the same "sign" in all clauses. e.g., In the three clauses (A B), ( B C), (C A), A and B are pure, C is impure. Make a pure symbol literal true. 3. Unit clause heuristic Unit clause: only one literal in the clause The only literal in a unit clause must be true.

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Horn Clauses Horn Clause A clause with at most one positive literal. KB: A Horn clause with one positive literal which can be written as α 1 … α n β Query: A Horn clause without positive literal α 1 … α n I.e. ( α 1 … α n ) Horn clause logic is the basis for Logic Programming

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Forward chaining for Horn Clauses Idea: fire any rule whose premises are satisfied in the KB, add its conclusion to the KB, until query is found

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Backward chaining for Horn Clasues 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 1. has already been proved true, or 2. has already failed

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Summary Propositional Logic Model of a KB Logical entailment Inference rules Resolution rule Clause form of a set of sentences Resolution refutation algorithm DPLL algorithm Horn clauses

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