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### Chapter7

Course: CS 5541, Fall 2008
School: Minnesota
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Word Count: 1880

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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...

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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 1 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 2 Previously Inputs Goal (task statement) Processing Agent Reasoning (search) Knowledge Goal, or steps to achieve goal Outputs 3 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 4 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 5 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 6 Semantics: What Do the wffs Mean? Defined by truth tables, see Figure 7.8, p. 207 7 Implication P 0 0 1 1 Q 0 1 0 1 PQ 1 1 0 1 P Q 1 1 0 1 8 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 9 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 10 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 11 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? 12 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} 13 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} 14 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) 15 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)? 16 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 17 Inference Procedures So, we know something about what we want from an inference procedure But, how do we actually compute KB |= ? 18 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) 19 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 20 Example Let KB = { (AC), (BC) } Let = AB Use the enumeration method to determine if KB |= KB |= iff M() M(KB) 21 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 22 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) 23 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 24 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 26 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 27 Inference Procedure: Modus Ponens, KB |-mp Given that: KB And KB Then: KB |= 28 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) 29 Example: Modus Ponens Given: KB = { P Q => R, P Q } Prove: R using KB |-mp 30 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 31 Inference Procedure: Unit Resolution Given that: KB And KB Then: KB |= Exercise: Prove this using our enumeration method 32 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) 33 Example Given: KB = { (P R) Q, Q } Prove: P R Using unit resolution 34 Solution KB = { (P R) Q, Q } Let = (P R), Let = Q KB has wffs of form: , By unit resolution, KB |= (P R) 35 Inference Procedure #4: Resolution Given that: KB And KB Then: KB |= Example: Prove this using our enumeration method 36 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) 37 Resolution Example Given: KB = { (P R) Q, Q (R S) } Prove that: KB |= (P R) (R S) 38 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) 39 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 40 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...

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