Methods General of Acquiring Knowledge
Deductive
New knowledge follows from prior knowledge by reasoning E.g., Math proofs or logical inference
Inductive
New knowledge based on observations of the world E.g., we may learn that apples fall from trees as a child by watching apples fall from trees
Deduction-induction combination
E.g., use a theory (e.g., deductive knowledge base) and examples from world (inductive oriented) to generate new information, predictions
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Logic: Chapter 7
Deductive reasoning
Logical inferences in a formal language Preserve truth with each inference Reasoning assumed to be same as logical proof
Logical formalisms
Termed a type of "Knowledge Representation" Use logical languages
E.g., propositional logic or predicate calculus
Application to AI
A new language
E.g., to specify start state, goals, knowledge, inference rules
Inference rules
A logical way to work from start state to goals using knowledge
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Previously
Inputs Goal (task statement) Processing Agent Reasoning (search) Knowledge Goal, or steps to achieve goal Outputs
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Weng (2004)
Now
Inputs Goal (task statement) Processing Logical Agent Outputs
Reasoning (search) Knowledge Goal, or steps to achieve goal
Rely on logical formalisms & methods
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Propositional Logic
Syntax
What are the sentences or well-formed formulas (wffs)?
Semantics
Correspondence of sentences to truth values (true/false)
Proof theory
Deductive inference mechanisms
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Syntax: What are the wffs?
A constant is a wff
true, false
Propositional symbols are wffs
Symbols
Names standing for true or false (and names of other wffs) Call these "variables", but the value that the names stand for cannot be changed dynamically (not variables in a programming sense)
E.g., P, Q, A, B
A wff with parentheses is a wff
E.g., (true), (P)
If P is a wff, and Q is a wff, then
P Q is a wff P Q is a wff P Q is a wff (also written as: P Q) P Q is a wff P is a wff
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Semantics: What Do the wffs Mean?
Defined by truth tables, see Figure 7.8, p. 207
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Implication
P 0 0 1 1 Q 0 1 0 1 PQ 1 1 0 1 P Q 1 1 0 1
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Worlds and Models
world
Logic symbols often used to refer to aspects of real world situation Symbol, J may mean "Jane" Symbol, K may mean "Jane's child is 8 yrs old" We provide this interpretation of the symbols
model
A mapping from symbols to truth values A label for a row in a truth table, enumerating the possible values for the variables in the wff, and assigning a particular wff its true/false value
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Example: Models
What are the models of the wff
(P Q) R
model
A mapping from symbols to truth values A label for a row in a truth table, enumerating the possible values for the variables in the wff, and assigning a particular wff its true/false value
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KB = Knowledge Base
A KB is a collection of wffs
Typically represent as a set E.g., KB = { (AC), (BC) }
Implicitly interpret commas in set notation as conjunctions (conjunction = ) Each element in the set is known as a clause
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When is a KB true?
A KB is true when
Given a row in a truth table, systematically assigning values to the KB's variables, the conjunction of all KB wffs are true
Can consider the KB to be a wff itself by inserting conjunctions in place of the commas (parenthesizing as needed), and dropping the "{" and "}" notation Example
When is KB = {A, B} true?
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Notation
M(KB) = the set of labels for all rows in a truth table, where systematically assigning values to the KB's variables results in the KB being true
Use #<row> to designate the row numbers E.g., {#1, #2, #8}
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Example
KB = {A, B} Truth table systematically assigning values to KB's variables is:
Row #1 #2 #3 #4 A 0 0 1 1 B 0 1 0 1
M(KB) = {#4}
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Method
Form a truth table with columns that are unique variables from wffs of KB
I.e., A, B, C
Write down KB wffs in additional columns of truth table
break down wffs as needed to make calculation of values easier
Determine when conjunction of wffs in KB are true Determine M(KB)
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Example
Definition:
M(KB) = the set of labels for all rows in a truth table, where systematically assigning values to the KB's variables results in the KB being true
Let KB = { (AC), (BC) } What is M(KB)?
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Entailment
KB |=
|= is symbol for entailment; is a wff Definition: KB entails iff is true in all models where KB is true
If KB, then KB |=
Why?
Entailment means
Add to the KB and "preserve truth" is "new knowledge"
Entailment is our first requirement for an inference procedure
Methods to infer new statements that are true
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Inference Procedures
So, we know something about what we want from an inference procedure But, how do we actually compute KB |= ?
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Enumeration Method
Given
KB and a wff
Question: Is KB |= Method
Compute M(KB) Compute M()
KB |= iff M() M(KB)
(This fits our first definition of entailment: KB entails iff is true in all models where KB is true)
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An Interpretation
M() gives the conditions under which is true
I.e., the possible combinations of values of the logical variables in for which is true
If M() M(KB), then M() is true under at least all the conditions that M(KB) is true
I.e., If M() M(KB), then M() will not weaken the conditions under which the KB is true
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Example
Let KB = { (AC), (BC) } Let = AB Use the enumeration method to determine if KB |=
KB |= iff M() M(KB)
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Inference Procedures
An inference procedure lets us mechanically preserve truth
I.e., it provides an algorithm that lets us ensure a formula is entailed
Example Inference Procedure
"conjunction introduction"
A If KB, and B KB, then KB |= A B
I.e., If A and B in KB, then we can insert new wff: A B into KB
Proof:
A B is true in all models where KB is true, so KB |= A B
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Justification
B, } is the conjunction of all other clauses that are in the KB (or just "true" if no other clauses)
KB = {A,
Need to show that M(A B) M(KB)
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More Notation
KB |-i
can be derived from KB with a specific inference procedure e.g., KB |-and A B, if A KB, and B KB
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Inference Procedure Properties
Generation
Create new sentence, E.g., KB |-and lets us create new sentences
Verification
Check if some is entailed
Soundness
Inference procedures are sound when they derive only sentences that are entailed E.g., |-and is sound
Completeness
An inference procedure is complete if it can derive any wff that is entailed Is |-and complete? 25
Is |-and Complete?
Say that KB = {A, B}
1) Is KB |= (A B)? 2) If it is, then |-and is not complete because |-and cannot derive (A B) and (A B) is entailed
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Inference Problem & Approaches
Inference problem
Given a KB How do we determine if a new wff, , logically follows from the KB? i.e., how do we determine if: KB |= ?
E.g., KB = {P, P Q},
Approaches
Enumeration method Conjunction introduction Other inference rules & proof construction
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Inference Procedure: Modus Ponens, KB |-mp
Given that: KB And KB Then: KB |=
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Justification
KB = {
, , } is the conjunction of all other clauses that are in the KB (or just "true" if no other clauses)
Need to show that M() M(KB)
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Example: Modus Ponens
Given: KB = { P Q => R, P Q } Prove: R using KB |-mp
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Solution
KB = { P Q => R, P Q } Let = P Q, and Let = R KB has wffs of the form: , By modus ponens, KB |= Since R = , KB |= R
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Inference Procedure: Unit Resolution
Given that: KB And KB Then: KB |=
Exercise: Prove this using our enumeration method
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Justification
KB = {
, , } is the conjunction of all other clauses that are in the KB (or just "true" if no other clauses)
Need to show that M() M(KB)
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Example
Given: KB = { (P R) Q, Q } Prove: P R
Using unit resolution
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Solution
KB = { (P R) Q, Q } Let = (P R), Let = Q KB has wffs of form: , By unit resolution, KB |= (P R)
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Inference Procedure #4: Resolution
Given that: KB And KB Then: KB |=
Example: Prove this using our enumeration method
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Justification
KB = {
, , } is the conjunction of all other clauses that are in the KB (or just "true" if no other clauses)
Need to show that M( ) M(KB)
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Resolution Example
Given: KB = { (P R) Q, Q (R S) } Prove that: KB |= (P R) (R S)
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Solution
KB = { (P R) Q, Q (R S) } Let = (P R) Let = (R S) Let = Q KB has wffs of form: , By resolution, KB |= (P R) (R S)
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Monotonic vs. Non-monotonic
Monotonic logic
The set of entailed sentences can only increase as clauses are added to KB
Non-Monotonic logic
When the set of entailed sentences can decrease when a clause is added to the KB
http://en.wikipedia.org/wiki/Non-monotonic_logic
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Completeness
Resolution in propositional logic is "refutation complete" (p. 214, text) Resolution can always be used to prove or disprove that a sentence is true, but it cannot be used to generate all entailed sentences E.g., Given that A is true, A B is true...