CS 416 Artificial Intelligence Lecture 9 Logical Agents Chapter 7 Lecture 9 Logical Agents Chapter 7

Assignment due Monday I’ll release binaries that provide more output Explanation for why your output doesn’t exactly match the provided solution’s We request “expand nodes in following order”We request “expand nodes in following order” –Could mean nodes are put on fringe in requested order –Could mean things are pulled from fringe in requested order We’ll be flexible in the gradingWe’ll be flexible in the grading I’ll release binaries that provide more output Explanation for why your output doesn’t exactly match the provided solution’s We request “expand nodes in following order”We request “expand nodes in following order” –Could mean nodes are put on fringe in requested order –Could mean things are pulled from fringe in requested order We’ll be flexible in the gradingWe’ll be flexible in the grading

Where are we? We’ve studied classes of search problems Find the best sequence of actionsFind the best sequence of actions –Uninformed search (BFS, Iterative Deepening) –Informed search (A*) Find the best value of something (possibly a sequence)Find the best value of something (possibly a sequence) –Simulated annealing, genetic algorithms, hill climbing Finding the best action in an adversarial settingFinding the best action in an adversarial setting We’ve studied classes of search problems Find the best sequence of actionsFind the best sequence of actions –Uninformed search (BFS, Iterative Deepening) –Informed search (A*) Find the best value of something (possibly a sequence)Find the best value of something (possibly a sequence) –Simulated annealing, genetic algorithms, hill climbing Finding the best action in an adversarial settingFinding the best action in an adversarial setting

Logical Agents What are we talking about, “logical?” Aren’t search-based chess programs logicalAren’t search-based chess programs logical –Yes, but knowledge is used in a very specific way  Win the game  Not useful for extracting strategies or understanding other aspects of chess We want to develop more general-purpose knowledge systems that support a variety of logical analysesWe want to develop more general-purpose knowledge systems that support a variety of logical analyses What are we talking about, “logical?” Aren’t search-based chess programs logicalAren’t search-based chess programs logical –Yes, but knowledge is used in a very specific way  Win the game  Not useful for extracting strategies or understanding other aspects of chess We want to develop more general-purpose knowledge systems that support a variety of logical analysesWe want to develop more general-purpose knowledge systems that support a variety of logical analyses

Why study knowledge-based agents Partially observable environments combine available information (percepts) with general knowledge to select actionscombine available information (percepts) with general knowledge to select actions Natural Language Language is too complex and ambiguous. Problem-solving agents are impeded by high branching factor.Language is too complex and ambiguous. Problem-solving agents are impeded by high branching factor.Flexibility Knowledge can be reused for novel tasks. New knowledge can be added to improve future performance.Knowledge can be reused for novel tasks. New knowledge can be added to improve future performance. Partially observable environments combine available information (percepts) with general knowledge to select actionscombine available information (percepts) with general knowledge to select actions Natural Language Language is too complex and ambiguous. Problem-solving agents are impeded by high branching factor.Language is too complex and ambiguous. Problem-solving agents are impeded by high branching factor.Flexibility Knowledge can be reused for novel tasks. New knowledge can be added to improve future performance.Knowledge can be reused for novel tasks. New knowledge can be added to improve future performance.

Components of knowledge-based agent Knowledge Base Store informationStore information –knowledge representation language Add information (Tell)Add information (Tell) Retrieve information (Ask)Retrieve information (Ask) Perform inferencePerform inference –derive new sentences (knowledge) from existing sentences Knowledge Base Store informationStore information –knowledge representation language Add information (Tell)Add information (Tell) Retrieve information (Ask)Retrieve information (Ask) Perform inferencePerform inference –derive new sentences (knowledge) from existing sentences

The wumpus world A scary world, indeed A maze in a caveA maze in a cave A wumpus who will eat youA wumpus who will eat you One arrow that can kill the wumpusOne arrow that can kill the wumpus Pits that can entrap you (but not the wumpus for it is too large to fall in)Pits that can entrap you (but not the wumpus for it is too large to fall in) A heap of gold somewhereA heap of gold somewhere A scary world, indeed A maze in a caveA maze in a cave A wumpus who will eat youA wumpus who will eat you One arrow that can kill the wumpusOne arrow that can kill the wumpus Pits that can entrap you (but not the wumpus for it is too large to fall in)Pits that can entrap you (but not the wumpus for it is too large to fall in) A heap of gold somewhereA heap of gold somewhere

But you have sensing and action Sensing (each is either on or off – a single bit) wumpus emits a stench in adjacent squareswumpus emits a stench in adjacent squares pits cause a breeze in adjacent squarespits cause a breeze in adjacent squares gold causes glitter you see when in the squaregold causes glitter you see when in the square walking into wall causes a bumpwalking into wall causes a bump death of wumpus can be heard everywhere in worlddeath of wumpus can be heard everywhere in world Sensing (each is either on or off – a single bit) wumpus emits a stench in adjacent squareswumpus emits a stench in adjacent squares pits cause a breeze in adjacent squarespits cause a breeze in adjacent squares gold causes glitter you see when in the squaregold causes glitter you see when in the square walking into wall causes a bumpwalking into wall causes a bump death of wumpus can be heard everywhere in worlddeath of wumpus can be heard everywhere in world

But you have sensing and action Action You can turn left or right 90 degreesYou can turn left or right 90 degrees You can move forwardYou can move forward You can shoot an arrow in your facing directionYou can shoot an arrow in your facing directionAction You can turn left or right 90 degreesYou can turn left or right 90 degrees You can move forwardYou can move forward You can shoot an arrow in your facing directionYou can shoot an arrow in your facing direction

An example

Our agent played well Used inference to relate two different percepts observed from different locationsUsed inference to relate two different percepts observed from different locations Agent is guaranteed to draw correct conclusions if percepts are correctAgent is guaranteed to draw correct conclusions if percepts are correct Used inference to relate two different percepts observed from different locationsUsed inference to relate two different percepts observed from different locations Agent is guaranteed to draw correct conclusions if percepts are correctAgent is guaranteed to draw correct conclusions if percepts are correct

Knowledge Representation Must be syntactically and semantically correct Syntax the formal specification of how information is storedthe formal specification of how information is stored –a + 2 = c (typical mathematical syntax) –a2y += (not legal syntax for infix (regular math) notation) Semantics the meaning of the informationthe meaning of the information –a + 2 = c (c is a number whose value is 2 more than a) –a + 2 = c (the symbol that comes two after ‘a’ in the alphabet is ‘c’) Must be syntactically and semantically correct Syntax the formal specification of how information is storedthe formal specification of how information is stored –a + 2 = c (typical mathematical syntax) –a2y += (not legal syntax for infix (regular math) notation) Semantics the meaning of the informationthe meaning of the information –a + 2 = c (c is a number whose value is 2 more than a) –a + 2 = c (the symbol that comes two after ‘a’ in the alphabet is ‘c’)

Logical Reasoning Entailment one sentence follows logically from anotherone sentence follows logically from another – –the sentence  entails the sentence   entails  ( ) if and only if for every model in which  is true,  is also true  entails  ( ) if and only if for every model in which  is true,  is also true Model: a description of the world where every relevant sentence has been assigned truth or falsehood Entailment one sentence follows logically from anotherone sentence follows logically from another – –the sentence  entails the sentence   entails  ( ) if and only if for every model in which  is true,  is also true  entails  ( ) if and only if for every model in which  is true,  is also true Model: a description of the world where every relevant sentence has been assigned truth or falsehood

An example After one step in wumpus world Knowledge base (KB) isKnowledge base (KB) is –A set of all game states that are possible after being given:  rules of game  percept sequence How does the KB represent the game?How does the KB represent the game? After one step in wumpus world Knowledge base (KB) isKnowledge base (KB) is –A set of all game states that are possible after being given:  rules of game  percept sequence How does the KB represent the game?How does the KB represent the game?

Building the KB Consider a KB intended to represent the presence of pits in a Wumpus world where [1,1] is clear and [2,1] has a breeze There are three cells with two conditions eachThere are three cells with two conditions each 2 3 = Eight possible models2 3 = Eight possible models According to percepts and rules, KB is well definedAccording to percepts and rules, KB is well defined Consider a KB intended to represent the presence of pits in a Wumpus world where [1,1] is clear and [2,1] has a breeze There are three cells with two conditions eachThere are three cells with two conditions each 2 3 = Eight possible models2 3 = Eight possible models According to percepts and rules, KB is well definedAccording to percepts and rules, KB is well defined

Model Checking The agent wishes to check all models of the game in which a pit is in the three candidate spots Enumerate all models where three candidate spots may have pitsEnumerate all models where three candidate spots may have pits The agent wishes to check all models of the game in which a pit is in the three candidate spots Enumerate all models where three candidate spots may have pitsEnumerate all models where three candidate spots may have pits

Checking entailment Can “  1 :There is no pit in [1, 2]” be true? Enumerate all states where  1 is trueEnumerate all states where  1 is true For all models where KB is true,  1 is true alsoFor all models where KB is true,  1 is true also The KB entails  1The KB entails  1 Can “  1 :There is no pit in [1, 2]” be true? Enumerate all states where  1 is trueEnumerate all states where  1 is true For all models where KB is true,  1 is true alsoFor all models where KB is true,  1 is true also The KB entails  1The KB entails  1

Checking entailment Can “  2 : There is no pit in [2, 2]” be true? Enumerate all states where  2 is trueEnumerate all states where  2 is true For all models where KB is true,  2 is not always true alsoFor all models where KB is true,  2 is not always true also KB does not entail  2KB does not entail  2 Can “  2 : There is no pit in [2, 2]” be true? Enumerate all states where  2 is trueEnumerate all states where  2 is true For all models where KB is true,  2 is not always true alsoFor all models where KB is true,  2 is not always true also KB does not entail  2KB does not entail  2

Logical inference Entailment permitted logic we inferred new knowledge from entailmentswe inferred new knowledge from entailments Inference algorithms The method of logical inference we demonstrated is called model checking because we enumerated all possibilities to find the inferenceThe method of logical inference we demonstrated is called model checking because we enumerated all possibilities to find the inference Entailment permitted logic we inferred new knowledge from entailmentswe inferred new knowledge from entailments Inference algorithms The method of logical inference we demonstrated is called model checking because we enumerated all possibilities to find the inferenceThe method of logical inference we demonstrated is called model checking because we enumerated all possibilities to find the inference

Inference algorithms There are many inference algorithms If an inference algorithm, i, can derive  from KB, we writeIf an inference algorithm, i, can derive  from KB, we write There are many inference algorithms If an inference algorithm, i, can derive  from KB, we writeIf an inference algorithm, i, can derive  from KB, we write

Inference Algorithms Sound Algorithm derives only entailed sentencesAlgorithm derives only entailed sentences An unsound algorithm derives falsehoodsAn unsound algorithm derives falsehoodsComplete inference algorithm can derive any sentence that is entailedinference algorithm can derive any sentence that is entailed –Means inference algorithm cannot become caught in infinite loop Sound Algorithm derives only entailed sentencesAlgorithm derives only entailed sentences An unsound algorithm derives falsehoodsAn unsound algorithm derives falsehoodsComplete inference algorithm can derive any sentence that is entailedinference algorithm can derive any sentence that is entailed –Means inference algorithm cannot become caught in infinite loop

Propositional (Boolean) Logic A very simple language Propositions are TRUE, FALSE, or COMPLETELY UNKNOWNPropositions are TRUE, FALSE, or COMPLETELY UNKNOWN Syntax of allowable sentencesSyntax of allowable sentences –Atomic sentences –Complex sentences –Backus-Naur Form (BNF) Inference – derive new sentences from old onesInference – derive new sentences from old ones Reasoning – patterns of sound inference that find proofsReasoning – patterns of sound inference that find proofs A very simple language Propositions are TRUE, FALSE, or COMPLETELY UNKNOWNPropositions are TRUE, FALSE, or COMPLETELY UNKNOWN Syntax of allowable sentencesSyntax of allowable sentences –Atomic sentences –Complex sentences –Backus-Naur Form (BNF) Inference – derive new sentences from old onesInference – derive new sentences from old ones Reasoning – patterns of sound inference that find proofsReasoning – patterns of sound inference that find proofs

Atomic sentences Syntax indivisible syntactic elements (also called literals)indivisible syntactic elements (also called literals) Use uppercase letters to represent a proposition that can be true or false (W 1,2 means there is a wumpus in [1, 2])Use uppercase letters to represent a proposition that can be true or false (W 1,2 means there is a wumpus in [1, 2]) True and False are predefined propositions where True means always true and False means always falseTrue and False are predefined propositions where True means always true and False means always falseSyntax indivisible syntactic elements (also called literals)indivisible syntactic elements (also called literals) Use uppercase letters to represent a proposition that can be true or false (W 1,2 means there is a wumpus in [1, 2])Use uppercase letters to represent a proposition that can be true or false (W 1,2 means there is a wumpus in [1, 2]) True and False are predefined propositions where True means always true and False means always falseTrue and False are predefined propositions where True means always true and False means always false

Complex sentences Formed from atomic sentences using connectives ~ (or = not): the negation~ (or = not): the negation ^ (and): the conjunction^ (and): the conjunction V (or): the disjunctionV (or): the disjunction => (or = implies): the implication=> (or = implies): the implication  (if and only if): the biconditional  (if and only if): the biconditional Formed from atomic sentences using connectives ~ (or = not): the negation~ (or = not): the negation ^ (and): the conjunction^ (and): the conjunction V (or): the disjunctionV (or): the disjunction => (or = implies): the implication=> (or = implies): the implication  (if and only if): the biconditional  (if and only if): the biconditional (A and ~A are called literals)

Backus-Naur Form (BNF) Order of precedence is: ~, ^, V, =>, 

Propositional (Boolean) Logic Semantics given a particular model (situation), what are the rules that determine the truth of a sentence?given a particular model (situation), what are the rules that determine the truth of a sentence? –m 1 = {P 1,2 =false, P 2,2 =false, P 3,1 =true} –What is ~P 1,2 ^ (P 2,2 V P 3,1 ) What are rules of ~, ^, VWhat are rules of ~, ^, V use a truth table to compute the value of any sentence with respect to a model by recursive evaluationuse a truth table to compute the value of any sentence with respect to a model by recursive evaluationSemantics given a particular model (situation), what are the rules that determine the truth of a sentence?given a particular model (situation), what are the rules that determine the truth of a sentence? –m 1 = {P 1,2 =false, P 2,2 =false, P 3,1 =true} –What is ~P 1,2 ^ (P 2,2 V P 3,1 ) What are rules of ~, ^, VWhat are rules of ~, ^, V use a truth table to compute the value of any sentence with respect to a model by recursive evaluationuse a truth table to compute the value of any sentence with respect to a model by recursive evaluation

Truth table (our semantics)

Example from wumpus A square is breezy if and only if a neighboring square has a pit B 1,1  (P 1,2 V P 2,1 )B 1,1  (P 1,2 V P 2,1 ) A square is breezy if a neighboring square has a pit (P 1,2 V P 2,1 ) => B 1,1 (P 1,2 V P 2,1 ) => B 1,1 Former is more powerful and true to Wumpus rules A square is breezy if and only if a neighboring square has a pit B 1,1  (P 1,2 V P 2,1 )B 1,1  (P 1,2 V P 2,1 ) A square is breezy if a neighboring square has a pit (P 1,2 V P 2,1 ) => B 1,1 (P 1,2 V P 2,1 ) => B 1,1 Former is more powerful and true to Wumpus rules

A wumpus knowledge base Initial conditionsInitial conditions –R 1 : ~P 1,1 no pit in [1,1] Rules of Breezes (for a few example squares)Rules of Breezes (for a few example squares) –R 2 : B 1,1  (P 1,2 V P 2,1 ) –R 3 : B 2,1  (P 1,1 V P 2,1 V P 3,1 ) PerceptsPercepts –R 4 : ~B 1,1 –R 5 : B 2,1 We know: R 1 ^ R 2 ^ R 3 ^ R 4 ^ R 5 Initial conditionsInitial conditions –R 1 : ~P 1,1 no pit in [1,1] Rules of Breezes (for a few example squares)Rules of Breezes (for a few example squares) –R 2 : B 1,1  (P 1,2 V P 2,1 ) –R 3 : B 2,1  (P 1,1 V P 2,1 V P 3,1 ) PerceptsPercepts –R 4 : ~B 1,1 –R 5 : B 2,1 We know: R 1 ^ R 2 ^ R 3 ^ R 4 ^ R 5

Inference Does KB entail  (KB ->  ?) Is there a pit in [1,2]: P 1,2 ?Is there a pit in [1,2]: P 1,2 ? Consider only what we needConsider only what we need –B 1,1 B 2,1 P 1,1 P 1,2 P 2,1 P 2,2 P 3,1 –2 7 permutations of models to check For each model, see if KB is trueFor each model, see if KB is true For all KB = True, see if  is trueFor all KB = True, see if  is true Does KB entail  (KB ->  ?) Is there a pit in [1,2]: P 1,2 ?Is there a pit in [1,2]: P 1,2 ? Consider only what we needConsider only what we need –B 1,1 B 2,1 P 1,1 P 1,2 P 2,1 P 2,2 P 3,1 –2 7 permutations of models to check For each model, see if KB is trueFor each model, see if KB is true For all KB = True, see if  is trueFor all KB = True, see if  is true This algorithm is called Model Checking

Inference Truth table There is no pit in P 1,2

Concepts related to entailment logical equivalence Two sentences a and b are logically equivalent if they are true in the same set of models… a  bTwo sentences a and b are logically equivalent if they are true in the same set of models… a  b validity (or tautology) a sentence that is valid in all modelsa sentence that is valid in all models –P V ~P –deduction theorem: a entails b if and only if a implies b satisfiability a sentence that is true in some modela sentence that is true in some modelunsatisfiable a entails b  (a ^ ~b)a entails b  (a ^ ~b) logical equivalence Two sentences a and b are logically equivalent if they are true in the same set of models… a  bTwo sentences a and b are logically equivalent if they are true in the same set of models… a  b validity (or tautology) a sentence that is valid in all modelsa sentence that is valid in all models –P V ~P –deduction theorem: a entails b if and only if a implies b satisfiability a sentence that is true in some modela sentence that is true in some modelunsatisfiable a entails b  (a ^ ~b)a entails b  (a ^ ~b)

Logical Equivalences Know these equivalences (no need to memorize)

Reasoning w/ propositional logic Inference Rules Modus Ponens:Modus Ponens: –Whenever sentences of form  =>  and  are given the sentence  can be inferred  R 1 : Green => Martian  R 2 : Green  Inferred: Martian Inference Rules Modus Ponens:Modus Ponens: –Whenever sentences of form  =>  and  are given the sentence  can be inferred  R 1 : Green => Martian  R 2 : Green  Inferred: Martian

Reasoning w/ propositional logic Inference Rules And-EliminationAnd-Elimination –Any of conjuncts can be inferred  R 1 : Martian ^ Green  Inferred: Martian  Inferred: Green Use truth tables if you want to confirm inference rules Inference Rules And-EliminationAnd-Elimination –Any of conjuncts can be inferred  R 1 : Martian ^ Green  Inferred: Martian  Inferred: Green Use truth tables if you want to confirm inference rules

Example of a proof of ~P 1,2 ~P ~B B P?

Example of a proof of ~P 1,2 ~P ~B B ~P P?

Constructing a proof Proving is like searching Find sequence of logical inference rules that lead to desired resultFind sequence of logical inference rules that lead to desired result Note the explosion of propositionsNote the explosion of propositions –Good proof methods ignore the countless irrelevant propositions Proving is like searching Find sequence of logical inference rules that lead to desired resultFind sequence of logical inference rules that lead to desired result Note the explosion of propositionsNote the explosion of propositions –Good proof methods ignore the countless irrelevant propositions

Monotonicity of knowledge base Knowledge base can only get larger Adding new sentences to knowledge base can only make it get largerAdding new sentences to knowledge base can only make it get larger –If (KB entails  )  ((KB ^  ) entails  ) This is important when constructing proofsThis is important when constructing proofs –A logical conclusion drawn at one point cannot be invalidated by a subsequent entailment Knowledge base can only get larger Adding new sentences to knowledge base can only make it get largerAdding new sentences to knowledge base can only make it get larger –If (KB entails  )  ((KB ^  ) entails  ) This is important when constructing proofsThis is important when constructing proofs –A logical conclusion drawn at one point cannot be invalidated by a subsequent entailment