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General Methods of Acquiring Knowledge

Logic: Chapter 7. Deductive reasoningLogical inferences in a formal languagePreserve truth with each inferenceReasoning assumed to be same as logical proofLogical formalismsTermed ?Knowledge representation"Use logical languagesE.g., propositional logic or predicate calculusApplication to AI

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General Methods of Acquiring Knowledge

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    1. General Methods 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

    2. 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 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

    3. Overall Approach

    4. Propositional Logic Syntax What are the sentences or well-formed formulas (wffs)? Semantics Correspondence 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: variables standing for true or false & names of other wffs 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 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 world Logic symbols refer to aspects of real world situation Symbol, J may mean Jane Symbol, K may mean Janes 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

    7. When is a KB true? KB = Knowledge Base A KB is a collection of wffs E.g., KB = { (A?C), (B??C) } A KB is true when the conjunction of all KB wffs in a row in the corresponding truth table are true M(KB) = the set of all rows in the truth table that are true for KB

    8. Example Let KB = { (A?C), (B??C) } What is M(KB)? Method Form a truth table with columns A, B, C Write down wffs of KB in additional columns Determine when conjunction of wffs in KB are true Determine M(KB)

    9. Entailment KB |= ? |= is symbol for entailment; ? is another wff KB entails ? iff ? is true in all models where KB is true Entailment means Add ? to the KB and preserve truth ? is new knowledge Entailment is the first requirement for an inference procedure or mechanism We want to infer new statements that are true

    10. Inference Procedures An inference procedure lets us mechanically preserve truth E.g., if A and B are in the KB, then we can insert the new wff: A ? B into the KB A ? B is true in all models where KB is true, so KB |= A ? B Additional requirements for a specific inference procedure Generation: create new sentences ? Verification: check if some ? is entailed KB |-i ? ? can be derived from KB with a specific inference procedure

    11. Inference Problem & Approaches Inference problem Given a KB E.g., KB = {P, P ? Q}, How do we determine if a new wff, ?, logically follows from the KB? i.e., how do we determine if: KB |= ? Approaches Enumeration method Inference rules & proof construction

    12. Enumeration Method Given KB and a wff ? Question: Is KB |= ? Method Compute M(KB) Compute M(?) KB |= ? iff M(KB) ? M(?)

    13. Why? KB |= ? iff M(KB) ? M(?) Consider again KB = { (A?C), (B??C) } Recall that M(KB) = {4, 5, 7, 8} Let ? be (A ? B ? C) M(?) = {2, 3, 4, 5, 6, 7, 8} Would we expect KB |= ? ? Let ? be (A ? B) M(?) = {7, 8} Would we expect KB |= ? ?

    14. Example Let KB = { (A?C), (B??C) } Let ? = A?B Is it true that KB |= ? ?

    15. Inference Rules & Proof Construction: A Simple Inference Procedure Say KB = {A1, A2, A3, } How to derive new true wffs? Assume our logical language has and Usual notation: ^ Then, define the following inference procedure Given two wffs, X and Y from KB Create a new wff by: X ^ Y Call this procedure |-and KB |-and A1 ^ A2 KB |-and A2 ^ A2

    16. Inference Procedure #2: Modus Ponens Justification To see that modus ponens preserves truth, recall: ? ? ? = ?? ? ? We know that: ? ? KB (I.e., ? is true) Therefore ?? ? KB (because, otherwise, M(KB) = {}) Therefore because ?? ? ? ? KB (i.e., ? ? ? ? KB), ? must be true Therefore KB |= ?

    17. Example: Modus Ponens Given: KB = { P ? Q => R, P ? Q } Prove: R using KB |-mp

    18. Solution KB = { P ? Q => R, P ? Q } Let ? = P ? Q, and Let ? = R KB has wffs of form: ? ? ?, ? By modus ponens, KB |= R

    19. Example Using our enumeration procedure, and our definition of entailment: KB |= ? iff M(KB) ? M(?) Prove that: { ? ? ?, ?} |= ?

    20. Inference Procedure #3: Unit Resolution Justification To see that unit resolution preserves truth: Note that ?? is true (i.e., ?? ? KB), Therefore, ? is false Therefore because ? ? ? ? KB, ? must be true Therefore KB |= ?

    21. Example Given: KB = { (P? R) ? Q, ?Q } Prove: P? R Using unit resolution

    22. Solution KB = { (P? R) ? Q, ?Q } Let ? = (P? R), Let ? = Q KB has wffs of form: ? ? ?, ? ? By unit resolution, KB |= (P? R)

    23. Inference Procedure #4: Resolution Justification To see that resolution preserves truth: Note that for any symbol P, either P is true or ?P is true So, either ? or ?? is true Case 1: ? is true ?? is false, therefore ? is true (since ? ? ?? ? KB) Case 2: ?? is true ? is false, therefore ? is true (since ? ? ? ? KB) Therefore since either Case 1 or Case 2 must apply ? ? ? must be true Therefore, KB |= ? ? ?

    24. Resolution Example Given: KB = { (P? R) ? Q, ?Q ? (R ?S) } Prove: (P? R) ? (R ?S)

    25. 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)

    26. Example Application Reasoning about statements in natural language Given Statements in natural language And a goal (theorem) to prove Convert statements in natural language to (propositional) logic Convert theorem into logic Apply inference operators until theorem (goal) is reached

    27. Example (Exercise 7.9, p. 238) If the unicorn is mythical, then it is immortal, but if it is not mythical, then it is a mortal mammal. If the unicorn is either immortal or a mammal, then it is horned. The unicorn is magical if it is horned. Prove that the unicorn is magical. 1) Translate English sentences into logic 2) Use inference to prove

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