1. Discussion #20 1/11 Discussion #20 Resolution in Datalog

2. Discussion #20 2/11 Topics Datalog –Prolog –Notation –Goals  success and failure –Examples –Recursive Rules Termination Infinite recursion We are not trying to do all of Prolog (or even all of Datalog).

3. Discussion #20 3/11 236 Datalog  Datalog  Prolog Database oriented (i.e. query answering) Does not include some features. (Prolog does have the power of procedural languages, but we won't do general I/O, arithmetic, and negation.) The exact distinction is fuzzy (and doesn't matter). For the project, we'll do exactly the language we've defined, 236 Datalog. –Syntactically taken from a real Prolog –Is an actual subset of Datalog

4. Discussion #20 4/11 236 Datalog  Prolog (Differences…) Constants –Prolog: numbers or alphanumeric beginning with lower case letter –236 Datalog: only strings Variables –Prolog: begin with a capital letter (anonymous variables, i.e. _ ) –236 Datalog: begin with a letter (no anonymous variables) Predicates –Prolog: begin with lower case letter –236 Datalog: begin with a letter Comments –Prolog: % … –236 Datalog: #-comments and #|…|#-comments Queries –Prolog: | ?- is the interactive prompt, so | ?- p(Y). is the query –236 Datalog: not interactive, no prompt, just P(Y)?

5. Discussion #20 5/11 236 Datalog  Prolog (Similarities…) Same syntax for Facts and Rules –Comma means  –No semicolon for (  )  use 2 rules. parent(x,y) :- mother(x,y); father(x,y).  changes to: parent(x,y) :- mother(x,y). parent(x,y) :- father(x,y). Recursive rules the same Derivations the same –Instantiation –Unification –Resolution (derivations of empty clause) –Backtracking Results same: “can be derived from database”

6. Discussion #20 6/11 Prolog Derivations Although derivations are the same, there is a different way of looking at the derivation in terms of –Goals  query and atomic formulas in the body of the rule. –Success  can be derived –Failure  cannot be derived We can use this same view in 236 Datalog. We’ll use the following database to explain these ideas. Facts: sister('ann','bob'). parent('bob','jay'). parent('bob','kay'). Rules: aunt(x,y) :- sister(x,z), parent(z,y). Queries: aunt('ann','jay')? aunt('ann',x)?

7. Discussion #20 7/11 Resolution 1.  aunt('ann','jay') 2. aunt('ann','jay') :- sister('ann',z), parent(z,'jay'). 2a.  sister('ann','ann') 2b. Fail! Backtrack 2a.  sister('ann','bob') 2b. res. with fact: sister('ann','bob') 2c.  parent('bob','jay') 2c. res. with fact: parent('bob','jay') Yes! Prolog 1. aunt('ann','jay') goal 2. aunt('ann','jay') :- sister('ann',z), parent(z,'jay'). 2a. sister('ann','ann') subgoal 2b. Fail! Backtrack 2a. sister('ann','bob') subgoal 2b. Matches a Fact (directly) 2c. parent('bob','jay') subgoal 2c. Matches a Fact (directly) Success! Yes! Prolog Derivations (continued…)

8. Discussion #20 8/11 236 Datalog aunt('ann',x)? 1. aunt('ann','ann') goal 2. aunt('ann','ann') :- sister('ann',z),parent(z,'ann'). 2a. sister('ann','ann') subgoalFail! Backtrack 2a. sister('ann','bob') subgoalSucceed! 2b. parent('bob','ann') subgoalFail! Backtrack 2a. sister('ann','jay') subgoalFail! Backtrack 2a. sister('ann','kay') subgoalFail! Backtrack 1. aunt('ann','bob') goal 2. aunt('ann','bob') :- sister('ann',z),parent(z,'bob'). 2a. sister('ann','ann') subgoalFail! Backtrack 2a. sister('ann','bob') subgoalSucceed! 2b. parent('bob','bob') subgoalFail! Backtrack 2a. sister('ann','jay') subgoalFail! Backtrack 2a. sister('ann',‘kay') subgoalFail! Backtrack Facts: sister('ann','bob'). parent('bob','jay'). parent('bob','kay'). Rules: aunt(x,y) :- sister(x,z), parent(z,y). Queries: aunt('ann',x)?

9. Discussion #20 9/11 1. aunt('ann','jay') goal 2. aunt('ann','jay') :- sister('ann',z),parent(z,'jay'). 2a. sister('ann','ann') subgoalFail! Backtrack 2a. sister('ann','bob') subgoalSucceed! 2b. parent('bob','jay') subgoalSucceed! Output “x='jay'” 2a. sister('ann','jay') subgoalFail! Backtrack 2a. sister('ann','kay') subgoalFail! Backtrack 1. aunt('ann','kay') goal 2. aunt('ann','kay') :- sister('ann',z),parent(z,'kay'). 2a. sister('ann','ann') subgoalFail! Backtrack 2a. sister('ann','bob') subgoalSucceed! 2b. parent('bob','kay') subgoalSucceed! Output “x='kay'” 2a. sister('ann','jay') subgoalFail! Backtrack 2a. sister('ann',‘kay') subgoalFail! Backtrack Facts: sister('ann','bob'). parent('bob','jay'). parent('bob','kay'). Rules: aunt(x,y) :- sister(x,z), parent(z,y). Queries: aunt('ann',x)?

10. Discussion #20 10/11 Tree View Facts: sister('ann','bob'). parent('bob','jay'). parent('bob','kay'). Rules: aunt(x,y) :- sister(x,z), parent(z,y). Queries: aunt('ann',x)? aunt('ann',x) sister('ann',z),parent(z,'ann'). sister('ann',z),parent(z,'jay'). sister('ann','ann') sister('ann','bob'),parent('bob‘,'jay'). x = 'ann' z = 'ann' x = 'bob'x = 'jay'x = 'kay' z = 'bob' z = 'ann' fail succeed … … … … Note that we only need to keep track of one path from root to leaf at a time.

11. Discussion #20 11/11 Potential Infinite Recursion 1. f(1,1). 2. f(1,2). 3. f(2,3). 4. b(x,y):-f(x,y). 5. b(x,y):-f(x,z),b(z,y). b(1,3)? b(1,3) f(1,3) fail rule 4 f(1,z),b(z,3) rule 5 f(1,1),b(1,3) succeed z=1 f(1,3),b(3,3) fail z=3 f(1,2),b(2,3) succeed z=2 f(2,3) succeed rule 4 Infinite Recursion! fail Infinite recursion! Keep current path stack  if recursive call already in path, fail! Domain = {1,2,3}