CS-INFO372-Module3-1-2 - Explorations in Artificial...

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1 Explorations in Artificial Intelligence Prof. Carla P. Gomes gomes@cs.cornell.edu Module 3-1-2 Logic Based Reasoning Proof Methods
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2 Proofs Methods
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Proof methods Proof methods divide into (roughly) two kinds: Application of inference rules Legitimate (sound) generation of new sentences from old Proof = a sequence of inference rule applications Can use inference rules as operators in a standard search algorithm Different types of proofs Model checking truth table enumeration (always exponential in n ) improved backtracking, e.g., Davis--Putnam-Logemann-Loveland (DPLL) heuristic search in model space (sound but incomplete) e.g., min-conflicts-like hill-climbing algorithms we’ve talked about this approach Next modules Current module
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Proof The sequence of wffs (w , w , …, w ) is called a proof (or deduction) of w from a set of wffs Δ iff each w i in the sequence is either in Δ or can be inferred from a wff (or wffs) earlier in the sequence by using a valid rule of inference. If there is a proof of w n from Δ , we say that w n is a theorem of the set Δ. Δ├ w n (read: w n can be proved or inferred from Δ) The concept of proof is relative to a particular set of inference rules used. If we denote the set of inference rules used by R,  we can write the fact that w n can be
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Propositional logic: Rules of Inference or Methods of Proof How to produce additional wffs (sentences) from other ones? What steps can we perform to show that a conclusion follows logically from a set of hypotheses? Example Modus Ponens P P Q ______________ Q The hypotheses are written in a column and the conclusions below the bar; The symbol denotes “therefore”. Given the hypotheses, the conclusion follows. The basis for this rule of inference is the tautology (P (P Q)) Q)
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6 Propositional logic: Rules of Inference or Methods of Proof How to produce additional wffs (sentences) from other ones? What steps can we perform to show that a conclusion follows logically from a set of hypotheses? Example Modus Ponens P P Q ______________ Q The hypotheses (premises) are written in a column and the conclusions below the bar The symbol denotes “therefore”. Given the hypotheses, the conclusion follows. The basis for this rule of inference is the tautology (P (P Q)) Q) [aside: check tautology with truth table to make sure] In words: when P and P Q are True, then Q must be True also. (meaning of second implication)
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7 Propositional logic: Rules of Inference or Methods of Proof Example Modus Ponens If you study the CS 372 material You will pass You study the CS372 material ______________ you will pass Nothing “deep”, but again remember the formal reason is that ((P ^ (P Q)) Q is a tautology.
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Propositional logic: Rules of Inference or Method of Proof Rule of Inference Tautology (Deduction Theorem) Name P P Q P (P Q) Addition P Q P (P Q) P Simplification
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This note was uploaded on 05/22/2008 for the course INFO 3720 taught by Professor Gomes during the Spring '07 term at Cornell University (Engineering School).

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CS-INFO372-Module3-1-2 - Explorations in Artificial...

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