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Course: CS 625, Fall 2009
School: Texas A&M
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Word Count: 918

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Resolving Announcement Two Clauses: Revisited Midterm exam: Friday 10/18 in class! Resolving two clauses and with the most general unier ! : Overview This is basically: Resolvents 1. Find the most general unier . Resolution in rst order logic: example 2. Apply to both and Theorem proving strategies 3. Remove the complimentary literal from and ....

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Resolving Announcement Two Clauses: Revisited Midterm exam: Friday 10/18 in class! Resolving two clauses and with the most general unier ! : Overview This is basically: Resolvents 1. Find the most general unier . Resolution in rst order logic: example 2. Apply to both and Theorem proving strategies 3. Remove the complimentary literal from and . Uncertainty 1 2 Resolving Two Clauses: Example Revisited Resolvent: A Full Example Example: resolve the two clauses ! ! ! ! 7 ' 2 ' 1 8 2 ! Example: Resolve the following (hint: ! ! ' 1 ! ) 0 2 ) ! ' ( 1. 2. and ! ! ! 7 ( 8 ) 0 ' 9 ! and " # \$ % & ) " # \$ % . & ! ! ! ( ! ' ' 1 0 2 ) 1. Get the factor of : ! ! ! ! 7 7 2 2 ' 1 8 2 ! 3 3 2. Resolve factor of 7 4 2 ! ! and ! ! ' ) : ! ! ! ! 7 ! ( 1 8 2 8 4 vs. ! 2 8 ! ( ! ' ' 1 ) 0 2 3. 3 ! 5 : ! ! ! ! ! ! ! ! ! 7 7 ( 1 ' 8 8 8 8 3 0 ' 9 vs. " % " # \$ @A BCD A 0 ! # 5 ) 0 @A BCD \$ remove 4 remove 4 4. Result: ! 5 1 ! ! ! ! ( 2 1 ) 6 ) 8 8 ' 9 ! ! 5 ( 6 ! 3 4 A % ' 9 ! Example Proof Using Resolution a Example: Predicates E Given: (1) The customs ofcials searched everyone who entered the 1. E F E country who was not a VIP. (2) Some of the drug dealers entered the 2. E G E : : : E P E P is a customs ofcial entered the country is a VIP country, and they were only searched by drug dealers. (3) No drug dealer was a VIP. 3. H E I Prove: (4) Some of the customs ofcials were drug dealers. 4. Q E : E was searched by is a drug dealer 5. : a Chang & Lee, Example 5.22 5 6 Example: English to First Order Logic (1) a b V Example: Standard Form (I) ! ! ! ! ! ! R W X S T U V 2 2 T 0 Y 2 ! ! ! ! ! ! ! (1) The customs ofcials searched everyone who entered the country who was R W X S T U ' 2 2 T 0 not a VIP. (2) Some of the drug dealers entered the country, and they were only 0 = c ad d e ! ! Y ! ! ! 2 R W X S ' U ' searched by drug dealers. (3) No drug dealer was a VIP. (4) Some of the 2 2 T 0 customs ofcials were drug dealers. g i f h ! p p = = = p q r f ! ! Y ! ! 2 ! ! ! ! ! R X 7 S ' U ' T 7 0 ! ! ! Y ! ! ! ! ! ! R X 7 S ' U ' T 7 ! ! ! ! ! ! R W X S U ! 0 T V 2 2 T 0 1. Y 2 ! ! ! X ! 2 V ` 2 ! ! Y ! R X S ' U ' 7 0 = " Y ! ! W ! R # S T ` T 2 2. Y \$ % ! ! ! ! S T ' U ' 7 " 0 ! # ! ! R ` V U 3. 0 ! Clauses: ! ! ! X S ' U ' 7 0 ! ! W ` T ! 4. (1a) (1b) ! ! Y ! S ' U ' 7 7 0 ! 8 ! \$ % ! ! Example: Standard Form (II) ! ! ! ! ! ! ! ! R X ` ` R ` T T V 2 2 2 V U Y Example: Standard Form (III) (3) ! ! ! ! 0 ! R ` 2 V ' U a b ! W S (2) ! ! ! R ` X T T 2 2 ' ` 0 Y W V S a b = ! ! ! ! ! ` X T T 2 ' ` 0 Y 2 ! = 0 W R c ad d e = Clause: ! ! ! ! ! ` ` ' ' U 2 ! ! T ` T X 2 0 Y 2 S g i R S f h 2 = % " % " # \$ # \$ % 0 ! ! W ` T (3) (4) ! ! ! W r ` d e 8 d s T 0 ! ! # " Clauses: ! R ` T \$ ! ! S (2a) c ad d e ! ! ! ` (2b) = 0 ! R ` ' 0 t u ! X 2 ' ` 2 ! ! ! 0 Y (2c) = Clause: ! ` ' 0 (4) 0 9 ! 10 Example: Clauses ! ! Basic Theorem Proving Algorithm Level saturation resolution method (or two-pointer method) Generate all possible resolvents: w H H I I I H x x ! ! X S ' U ' 0 (1a) ! ! ! ! Y S ' U ' 7 (1b) 0 S (2a) ! ` ! 7 0 ! ! (2b) Generate sequences of clauses H w x , where = y H H ! X 2 ' ` 0 (2c) Y ! ` ' U ! 2 ! (original set of clauses) = (3) 0 ! ` ' 0 0 ! (4) all possible resolvents of clauses w H Hy Hy x x x Note: The input to your theorem prover will be in a standard form like the above. and Exercise 1: rewrite the above in LISP representation. v This is basically a breadth rst search method, and it can be extremely inefcient except for small problems. Exercise 2: use resolution to derive . The problem is that irrelevant derivations are made: in generating an n-step proof, we also generate all possible derivations of n-1 steps. 11 12 Deletion Strategy Example: E Subsumed Clause: Example (I) To reduce the huge number of generated clauses, we would like to delete clauses whenever possible. We can delete: . E Q E 1. Any tautology, e.g. 2. Any clause which duplicates an existing clause. 3. Any clause which is subsumed by an existing clause. If Q , then Q A clause I subsumes a clause iff there is a substitution such Q that (recall that a clause can be represented as a set of Since . Q Q literals). , is called a subsumed clause. subsumes , and Q can be deleted. Deletion strategy will be complete if it is used with certain resolution algorithms (such as level saturation). 13 14 Strategies to Improve Resolution Advantages and Disadvantages of Resolution 1. Deletion strategy: remove tautology, dupl...

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