cs 416 artificial intelligence lecture 9 logical agents chapter 7 lecture 9 logical agents chapter 7
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CS 416Artificial Intelligence
Lecture 9Lecture 9
Logical AgentsLogical Agents
Chapter 7Chapter 7
Lecture 9Lecture 9
Logical AgentsLogical Agents
Chapter 7Chapter 7
Chess Article
Garry Kasparov reflects on computerized chessGarry Kasparov reflects on computerized chess
• IBM should have released the contents of Deep Blue to chess IBM should have released the contents of Deep Blue to chess community to advance research of computation as it relates to chesscommunity to advance research of computation as it relates to chess
• Kudos to Deep Junior for putting information in public domain so state of Kudos to Deep Junior for putting information in public domain so state of the art can advancethe art can advance
• Deep Blue made one good move that surprised Kasparov (though he Deep Blue made one good move that surprised Kasparov (though he thinks a person was in the loop)thinks a person was in the loop)
• Deep Junior made a fantastic sacrifice that reflects a new Deep Junior made a fantastic sacrifice that reflects a new accomplishment for computerized chessaccomplishment for computerized chess
http://www.opinionjournal.com/extra/?id=110003081http://www.opinionjournal.com/extra/?id=110003081
Garry Kasparov reflects on computerized chessGarry Kasparov reflects on computerized chess
• IBM should have released the contents of Deep Blue to chess IBM should have released the contents of Deep Blue to chess community to advance research of computation as it relates to chesscommunity to advance research of computation as it relates to chess
• Kudos to Deep Junior for putting information in public domain so state of Kudos to Deep Junior for putting information in public domain so state of the art can advancethe art can advance
• Deep Blue made one good move that surprised Kasparov (though he Deep Blue made one good move that surprised Kasparov (though he thinks a person was in the loop)thinks a person was in the loop)
• Deep Junior made a fantastic sacrifice that reflects a new Deep Junior made a fantastic sacrifice that reflects a new accomplishment for computerized chessaccomplishment for computerized chess
http://www.opinionjournal.com/extra/?id=110003081http://www.opinionjournal.com/extra/?id=110003081
Where are we?
We’ve studied classes of search problemsWe’ve studied classes of search problems
• Find the best sequence of actionsFind the best sequence of actions
– Uninformed search (BFS, Iterative Deepening)Uninformed search (BFS, Iterative Deepening)
– Informed search (A*)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 climbingSimulated 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 problemsWe’ve studied classes of search problems
• Find the best sequence of actionsFind the best sequence of actions
– Uninformed search (BFS, Iterative Deepening)Uninformed search (BFS, Iterative Deepening)
– Informed search (A*)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 climbingSimulated 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?”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 wayYes, but knowledge is used in a very specific way
Win the gameWin the game
Not useful for extracting strategies or understanding Not useful for extracting strategies or understanding other aspects of chessother aspects of chess
• We want to develop more general-purpose knowledge We want to develop more general-purpose knowledge systems that support a variety of logical analysessystems that support a variety of logical analyses
What are we talking about, “logical?”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 wayYes, but knowledge is used in a very specific way
Win the gameWin the game
Not useful for extracting strategies or understanding Not useful for extracting strategies or understanding other aspects of chessother aspects of chess
• We want to develop more general-purpose knowledge We want to develop more general-purpose knowledge systems that support a variety of logical analysessystems that support a variety of logical analyses
Why study knowledge-based agents
Partially observable environmentsPartially observable environments
• combine available information (percepts) with general knowledge to combine available information (percepts) with general knowledge to select actionsselect actions
Natural LanguageNatural Language
• Language is too complex and ambiguous. Problem-solving agents are Language is too complex and ambiguous. Problem-solving agents are impeded by high branching factor.impeded by high branching factor.
FlexibilityFlexibility
• Knowledge can be reused for novel tasks. New knowledge can be Knowledge can be reused for novel tasks. New knowledge can be added to improve future performance.added to improve future performance.
Partially observable environmentsPartially observable environments
• combine available information (percepts) with general knowledge to combine available information (percepts) with general knowledge to select actionsselect actions
Natural LanguageNatural Language
• Language is too complex and ambiguous. Problem-solving agents are Language is too complex and ambiguous. Problem-solving agents are impeded by high branching factor.impeded by high branching factor.
FlexibilityFlexibility
• Knowledge can be reused for novel tasks. New knowledge can be Knowledge can be reused for novel tasks. New knowledge can be added to improve future performance.added to improve future performance.
Components of knowledge-based agent
Knowledge BaseKnowledge Base• Store informationStore information
– knowledge knowledge representationrepresentation language language
• Add information (Add information (TellTell))
• Retrieve information (Retrieve information (AskAsk))
• Perform Perform inferenceinference
– derive new derive new sentencessentences (knowledge) from existing (knowledge) from existing sentencessentences
Knowledge BaseKnowledge Base• Store informationStore information
– knowledge knowledge representationrepresentation language language
• Add information (Add information (TellTell))
• Retrieve information (Retrieve information (AskAsk))
• Perform Perform inferenceinference
– derive new derive new sentencessentences (knowledge) from existing (knowledge) from existing sentencessentences
The wumpus world
A scary world, indeedA 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 Pits that can entrap you (but not the wumpus for it is too large to fall in)wumpus for it is too large to fall in)
• A heap of gold somewhereA heap of gold somewhere
A scary world, indeedA 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 Pits that can entrap you (but not the wumpus for it is too large to fall in)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)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)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
ActionAction
• 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
ActionAction
• 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
An example
Our agent played well
• Used inference to relate two different percepts observed from Used inference to relate two different percepts observed from different locationsdifferent locations
• Agent is guaranteed to draw correct conclusions if percepts Agent is guaranteed to draw correct conclusions if percepts are correctare correct
• Used inference to relate two different percepts observed from Used inference to relate two different percepts observed from different locationsdifferent locations
• Agent is guaranteed to draw correct conclusions if percepts Agent is guaranteed to draw correct conclusions if percepts are correctare correct
Knowledge Representation
Must be syntactically and semantically correctMust be syntactically and semantically correct
SyntaxSyntax• the formal specification of how information is storedthe formal specification of how information is stored
– a + 2 = c (typical mathematical syntax)a + 2 = c (typical mathematical syntax)
– a2y += (not legal syntax for infix (regular math) notation)a2y += (not legal syntax for infix (regular math) notation)
SemanticsSemantics• 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 (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’)a + 2 = c (the symbol that comes two after ‘a’ in the alphabet is ‘c’)
Must be syntactically and semantically correctMust be syntactically and semantically correct
SyntaxSyntax• the formal specification of how information is storedthe formal specification of how information is stored
– a + 2 = c (typical mathematical syntax)a + 2 = c (typical mathematical syntax)
– a2y += (not legal syntax for infix (regular math) notation)a2y += (not legal syntax for infix (regular math) notation)
SemanticsSemantics• 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 (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’)a + 2 = c (the symbol that comes two after ‘a’ in the alphabet is ‘c’)
Logical Reasoning
EntailmentEntailment• one sentence follows logically from anotherone sentence follows logically from another
–
– the sentence the sentence entails the sentence entails the sentence
• entails entails ( ) if and only if for every ( ) if and only if for every modelmodel in which in which is true, is true, is also true is also true
Model: Model: a description of the world where every a description of the world where every relevant sentence has been assigned truth or relevant sentence has been assigned truth or falsehoodfalsehood
EntailmentEntailment• one sentence follows logically from anotherone sentence follows logically from another
–
– the sentence the sentence entails the sentence entails the sentence
• entails entails ( ) if and only if for every ( ) if and only if for every modelmodel in which in which is true, is true, is also true is also true
Model: Model: a description of the world where every a description of the world where every relevant sentence has been assigned truth or relevant sentence has been assigned truth or falsehoodfalsehood
An example
After one step in wumpus After one step in wumpus worldworld• Knowledge base (KB) isKnowledge base (KB) is
– A set of all game states that are A set of all game states that are possible given:possible given:
rules of gamerules of game
perceptspercepts
• How does the KB represent the How does the KB represent the game?game?
After one step in wumpus After one step in wumpus worldworld• Knowledge base (KB) isKnowledge base (KB) is
– A set of all game states that are A set of all game states that are possible given:possible given:
rules of gamerules of game
perceptspercepts
• How does the KB represent the How does the KB represent the game?game?
Building the KB
Consider a KB intended to represent the presence Consider a KB intended to represent the presence of pits in a Wumpus world where [1,1] is clear and of pits in a Wumpus world where [1,1] is clear and [2,1] has a breeze[2,1] has a breeze
• There are three cells withThere are three cells withtwo conditions each two conditions each
• 2233 = Eight possible models = Eight possible models
• According to percepts andAccording to percepts andrules, KB is well definedrules, KB is well defined
Consider a KB intended to represent the presence Consider a KB intended to represent the presence of pits in a Wumpus world where [1,1] is clear and of pits in a Wumpus world where [1,1] is clear and [2,1] has a breeze[2,1] has a breeze
• There are three cells withThere are three cells withtwo conditions each two conditions each
• 2233 = Eight possible models = Eight possible models
• According to percepts andAccording to percepts andrules, KB is well definedrules, KB is well defined
Model Checking
The agent wishes to check all The agent wishes to check all modelsmodels of the game of the game in which a pit is in the three candidate spotsin which a pit is in the three candidate spots
• Enumerate all modelsEnumerate all modelswhere three candidatewhere three candidatespots may have pitsspots may have pits
• 3 pits, two conditions each3 pits, two conditions each
• 2233 = Eight models = Eight models
The agent wishes to check all The agent wishes to check all modelsmodels of the game of the game in which a pit is in the three candidate spotsin which a pit is in the three candidate spots
• Enumerate all modelsEnumerate all modelswhere three candidatewhere three candidatespots may have pitsspots may have pits
• 3 pits, two conditions each3 pits, two conditions each
• 2233 = Eight models = Eight models
Checking entailment
Can “Can “11:There is no pit in [1, 2]” be true?:There is no pit in [1, 2]” be true?
• Enumerate all states whereEnumerate all states where11 is true is true
• For all models where KBFor all models where KBis true, is true, 11 is true also is true also
• The KB entails The KB entails 11
Can “Can “11:There is no pit in [1, 2]” be true?:There is no pit in [1, 2]” be true?
• Enumerate all states whereEnumerate all states where11 is true is true
• For all models where KBFor all models where KBis true, is true, 11 is true also is true also
• The KB entails The KB entails 11
Checking entailment
Can “Can “22: There is no pit in [2, 2]” be true?: There is no pit in [2, 2]” be true?
• Enumerate all statesEnumerate all stateswhere where 22 is true is true
• For all models where KBFor all models where KBis true, is true, 22 is not always is not always
true alsotrue also
• KB does not entail KB does not entail 22
Can “Can “22: There is no pit in [2, 2]” be true?: There is no pit in [2, 2]” be true?
• Enumerate all statesEnumerate all stateswhere where 22 is true is true
• For all models where KBFor all models where KBis true, is true, 22 is not always is not always
true alsotrue also
• KB does not entail KB does not entail 22
Logical inference
Entailment permitted logicEntailment permitted logic
• we inferred new knowledge from entailmentswe inferred new knowledge from entailments
Inference algorithmsInference algorithms
• The method of logical inference we demonstrated is called The method of logical inference we demonstrated is called model checkingmodel checking because we enumerated all possibilities to because we enumerated all possibilities to find the inferencefind the inference
Entailment permitted logicEntailment permitted logic
• we inferred new knowledge from entailmentswe inferred new knowledge from entailments
Inference algorithmsInference algorithms
• The method of logical inference we demonstrated is called The method of logical inference we demonstrated is called model checkingmodel checking because we enumerated all possibilities to because we enumerated all possibilities to find the inferencefind the inference
Inference algorithms
There are many inference algorithmsThere are many inference algorithms
• If an inference algorithm, If an inference algorithm, ii, can derive , can derive from KB, we write from KB, we write
There are many inference algorithmsThere are many inference algorithms
• If an inference algorithm, If an inference algorithm, ii, can derive , can derive from KB, we write from KB, we write
Inference Algorithms
SoundSound
• Algorithm derives only entailed sentencesAlgorithm derives only entailed sentences
• An unsound algorithm derives falsehoodsAn unsound algorithm derives falsehoods
CompleteComplete
• inference algorithm can derive any sentence that is entailedinference algorithm can derive any sentence that is entailed
– Means inference algorithm cannot become caught in Means inference algorithm cannot become caught in infinite loop?infinite loop?
SoundSound
• Algorithm derives only entailed sentencesAlgorithm derives only entailed sentences
• An unsound algorithm derives falsehoodsAn unsound algorithm derives falsehoods
CompleteComplete
• inference algorithm can derive any sentence that is entailedinference algorithm can derive any sentence that is entailed
– Means inference algorithm cannot become caught in Means inference algorithm cannot become caught in infinite loop?infinite loop?
Propositional (Boolean) Logic
Syntax of allowable sentencesSyntax of allowable sentences
• atomic sentencesatomic sentences
– indivisible syntactic elementsindivisible syntactic elements
– Use uppercase letters to represent a proposition that can Use uppercase letters to represent a proposition that can be true or falsebe true or false
– True and False are predefined propositions where True True and False are predefined propositions where True means always true and False means always falsemeans always true and False means always false
Syntax of allowable sentencesSyntax of allowable sentences
• atomic sentencesatomic sentences
– indivisible syntactic elementsindivisible syntactic elements
– Use uppercase letters to represent a proposition that can Use uppercase letters to represent a proposition that can be true or falsebe true or false
– True and False are predefined propositions where True True and False are predefined propositions where True means always true and False means always falsemeans always true and False means always false
Atomic sentences
Syntax of atomic sentencesSyntax of atomic sentences
• indivisible syntactic elementsindivisible syntactic elements
• Use uppercase letters to represent a proposition that can be Use uppercase letters to represent a proposition that can be true or falsetrue or false
• True and False are predefined propositions where True True and False are predefined propositions where True means always true and False means always falsemeans always true and False means always false
Syntax of atomic sentencesSyntax of atomic sentences
• indivisible syntactic elementsindivisible syntactic elements
• Use uppercase letters to represent a proposition that can be Use uppercase letters to represent a proposition that can be true or falsetrue or false
• True and False are predefined propositions where True True and False are predefined propositions where True means always true and False means always falsemeans always true and False means always false
Complex sentences
Formed from atomic sentences using connectivesFormed 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 connectivesFormed 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
Backus-Naur Form (BNF)
Propositional (Boolean) Logic
SemanticsSemantics
• given a particular model (situation), what are the rules that given a particular model (situation), what are the rules that determine the truth of a sentence?determine the truth of a sentence?
• use a truth table to compute the value of any sentence with use a truth table to compute the value of any sentence with respect to a model by recursive evaluationrespect to a model by recursive evaluation
SemanticsSemantics
• given a particular model (situation), what are the rules that given a particular model (situation), what are the rules that determine the truth of a sentence?determine the truth of a sentence?
• use a truth table to compute the value of any sentence with use a truth table to compute the value of any sentence with respect to a model by recursive evaluationrespect to a model by recursive evaluation
Truth table
Example from wumpus
A square is breezy A square is breezy only if only if a neighboring square has a pit a neighboring square has a pit
• BB1,11,1 (P (P1,21,2 V P V P2,12,1))
A square is breezy A square is breezy if if a neighboring square has a pita neighboring square has a pit
• (P(P1,21,2 V P V P2,12,1) => B) => B1,11,1
Former is more powerful and true to Wumpus rulesFormer is more powerful and true to Wumpus rules
A square is breezy A square is breezy only if only if a neighboring square has a pit a neighboring square has a pit
• BB1,11,1 (P (P1,21,2 V P V P2,12,1))
A square is breezy A square is breezy if if a neighboring square has a pita neighboring square has a pit
• (P(P1,21,2 V P V P2,12,1) => B) => B1,11,1
Former is more powerful and true to Wumpus rulesFormer is more powerful and true to Wumpus rules
A wumpus knowledge base• Initial conditionsInitial conditions
– RR11: ~P: ~P1,11,1 no pit in [1,1] no pit in [1,1]
• Rules of Breezes (for a few Rules of Breezes (for a few example squares)example squares)
– RR22: B: B1,11,1 (P (P1,21,2 V P V P2,12,1) )
– RR33: B: B2,12,1 (P (P1,11,1 V P V P2,12,1 V P V P3,13,1))
• PerceptsPercepts
– RR44: ~B: ~B1,11,1
– RR55: B: B2,12,1
We know: RWe know: R11 ^ R ^ R22 ^ R ^ R33 ^ R ^ R44 ^ R ^ R55
• Initial conditionsInitial conditions
– RR11: ~P: ~P1,11,1 no pit in [1,1] no pit in [1,1]
• Rules of Breezes (for a few Rules of Breezes (for a few example squares)example squares)
– RR22: B: B1,11,1 (P (P1,21,2 V P V P2,12,1) )
– RR33: B: B2,12,1 (P (P1,11,1 V P V P2,12,1 V P V P3,13,1))
• PerceptsPercepts
– RR44: ~B: ~B1,11,1
– RR55: B: B2,12,1
We know: RWe know: R11 ^ R ^ R22 ^ R ^ R33 ^ R ^ R44 ^ R ^ R55
Inference
Does KB entail Does KB entail (KB -> (KB -> ?)?)
• Is there a pit in [1,2]: PIs there a pit in [1,2]: P1,21,2??
• Consider only what we needConsider only what we need
– BB1,1 1,1 BB2,1 2,1 PP1,1 1,1 PP1,2 1,2 PP2,1 2,1 PP2,2 2,2 PP3,13,1
– 2277 permutations of models to check 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 For all KB = True, see if is true is true
Does KB entail Does KB entail (KB -> (KB -> ?)?)
• Is there a pit in [1,2]: PIs there a pit in [1,2]: P1,21,2??
• Consider only what we needConsider only what we need
– BB1,1 1,1 BB2,1 2,1 PP1,1 1,1 PP1,2 1,2 PP2,1 2,1 PP2,2 2,2 PP3,13,1
– 2277 permutations of models to check 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 For all KB = True, see if is true is true
ModelChecking
Inference
Truth tableTruth tableTruth tableTruth table
Concepts related to entailmentlogical equivalencelogical equivalence
• Two sentences a and b are logically equivalent if they are true in the same set of Two sentences a and b are logically equivalent if they are true in the same set of models… amodels… a b b
validity (or tautology)validity (or tautology)
• a sentence that is valid in all modelsa sentence that is valid in all models
– P V ~PP V ~P
– deduction theoremdeduction theorem: a entails b if and only if a implies b: a entails b if and only if a implies b
satisfiabilitysatisfiability
• a sentence that is true in a sentence that is true in somesome model model
• a entails b a entails b (a ^ ~b) is unsatisfiable (a ^ ~b) is unsatisfiable
logical equivalencelogical equivalence
• Two sentences a and b are logically equivalent if they are true in the same set of Two sentences a and b are logically equivalent if they are true in the same set of models… amodels… a b b
validity (or tautology)validity (or tautology)
• a sentence that is valid in all modelsa sentence that is valid in all models
– P V ~PP V ~P
– deduction theoremdeduction theorem: a entails b if and only if a implies b: a entails b if and only if a implies b
satisfiabilitysatisfiability
• a sentence that is true in a sentence that is true in somesome model model
• a entails b a entails b (a ^ ~b) is unsatisfiable (a ^ ~b) is unsatisfiable
Logical Equivalences
Know these equivalencesKnow these equivalencesKnow these equivalencesKnow these equivalences
Reasoning w/ propositional logic
Inference RulesInference Rules
• Modus Ponens: Modus Ponens:
– Whenever sentences of form Whenever sentences of form => => and and are given are giventhe sentence the sentence can be inferred can be inferred
RR11: Green => Martian: Green => Martian
RR22: Green: Green
Inferred: MartianInferred: Martian
Inference RulesInference Rules
• Modus Ponens: Modus Ponens:
– Whenever sentences of form Whenever sentences of form => => and and are given are giventhe sentence the sentence can be inferred can be inferred
RR11: Green => Martian: Green => Martian
RR22: Green: Green
Inferred: MartianInferred: Martian
Reasoning w/ propositional logic
Inference RulesInference Rules• And-EliminationAnd-Elimination
– Any of conjuncts can be inferredAny of conjuncts can be inferred
RR11: Martian ^ Green: Martian ^ Green
Inferred: MartianInferred: Martian
Inferrred: GreenInferrred: Green
Use truth tables if you want to confirm inference Use truth tables if you want to confirm inference rulesrules
Inference RulesInference Rules• And-EliminationAnd-Elimination
– Any of conjuncts can be inferredAny of conjuncts can be inferred
RR11: Martian ^ Green: Martian ^ Green
Inferred: MartianInferred: Martian
Inferrred: GreenInferrred: Green
Use truth tables if you want to confirm inference Use truth tables if you want to confirm inference rulesrules
Example of a proof
~P ~B
BP?
P?
P?
P?
Example of a proof
~P ~B
B~P
~P
P?
P?
Constructing a proof
Proving Proving is like is like searchingsearching
• Find sequence of logical inference rules that lead to desired Find sequence of logical inference rules that lead to desired resultresult
• Note the explosion of propositionsNote the explosion of propositions
– Good proof methods ignore the countless irrelevant Good proof methods ignore the countless irrelevant propositionspropositions
Proving Proving is like is like searchingsearching
• Find sequence of logical inference rules that lead to desired Find sequence of logical inference rules that lead to desired resultresult
• Note the explosion of propositionsNote the explosion of propositions
– Good proof methods ignore the countless irrelevant Good proof methods ignore the countless irrelevant propositionspropositions
Monotonicity of knowledge base
Knowledge base can only get largerKnowledge 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 If (KB entails ))
((KB ^ ((KB ^ ) entails ) entails ))
• This is important when constructing proofsThis is important when constructing proofs
– A logical conclusion drawn at one point cannot be invalidated by a A logical conclusion drawn at one point cannot be invalidated by a subsequent entailmentsubsequent entailment
Knowledge base can only get largerKnowledge 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 If (KB entails ))
((KB ^ ((KB ^ ) entails ) entails ))
• This is important when constructing proofsThis is important when constructing proofs
– A logical conclusion drawn at one point cannot be invalidated by a A logical conclusion drawn at one point cannot be invalidated by a subsequent entailmentsubsequent entailment
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