Propositional Logic

Propositional Logic Russell and Norvig Chapter 7

sensors environment ? agent actuators Knowledge base Knowledge-Based Agent

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

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

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

Performance measure gold +1000, death -1000 -1 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 Wumpus World PEAS description

Wumpus world characterization • FullyObservable 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

Exploring a wumpus world

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

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

Examples of Logics • Propositional calculusA  B  C • First-order predicate calculus( x)( y) Mother(y,x) • Logic of BeliefB(John,Father(Zeus,Cronus))

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.

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.

valid sentenceor tautology Satisfiability of a KB A KB is satisfiable iff it admits at least one model; otherwise it is unsatisfiable KB1 = {P, QR} is satisfiableKB2 = {PP} is satisfiable KB3 = {P, P} is unsatisfiable

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

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 

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

{  , }   {, }  Example: Modus Ponens From Battery-OK  Bulbs-OK  Headlights-Work Battery-OK  Bulbs-OK Infer Headlights-Work

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

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)  

Is each of the following a sound inference rule? {   ,}  {   ,}   

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

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

Proof The proof of a sentence  from a set of sentences KB is the derivation of by applying a series of sound inference rules Battery-OK  Bulbs-OK  Headlights-Work Battery-OK  Starter-OK Empty-Gas-Tank  Engine-Starts Engine-Starts Flat-Tire  Car-OK Headlights-Work Battery-OK Starter-OK Empty-Gas-Tank Car-OK Battery-OK  Starter-OK by 5,6 Battery-OK  Starter-OK Empty-Gas-Tank by 9,7 Engine-Starts by 2,10 Engine-Starts  Flat-Tire by 3,8 Flat-Tire by 11,12

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

Deduction vs. Satisfiability Test KB  iff {KB,} is unsatisfiable • 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

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

Unit Resolution Rule • Given two sentences:L1 …  Lp and Mwhere Li,…, Lp and M are all literals, and M and Li are complementary literals • Infer:L1 … Li-1Li+1 …  Lp

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

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

Full Resolution Rule • Given two clauses:L1 …  LpandM1 …  MqwhereLi andMj are complementrary • Infer the clause:L1…Li-1Li+1…LkM1…Mj-1Mj+1…Mk

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

Example From: P  Q ( P  Q) Q  R ( Q  R) Infer: P  R ( P  R)

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

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

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

Example Battery-OK Bulbs-OK  Headlights-Work Battery-OK Starter-OK  Empty-Gas-Tank  Engine-Starts Engine-Starts  Flat-Tire  Car-OK Headlights-Work Battery-OK Starter-OK Empty-Gas-Tank Car-OK 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?

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}

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

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

The DPLL algorithm Determine if an input propositional logic sentence (in CNF) is satisfiable. Improvements over truth table enumeration: • Early termination A clause is true if any literal is true. A sentence is false if any clause is false. • 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. • Unit clause heuristic Unit clause: only one literal in the clause The only literal in a unit clause must be true.

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 α1 …  αnβ Query: A Horn clause without positive literal α1 … αn I.e. ( α1 …  αn ) Horn clause logic is the basis for Logic Programming

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

Propositional Logic