Tabled Prolog and Linear Tabling

1. Tabled Prolog and Linear Tabling Neng-Fa Zhou (City Univ. of New York)Yi-Dong Shen (Chinese Academy of Sciences) Taisuke Sato (Tokyo Inst. of Technology) Linear Tabling

2. Tabling is Useful • Eliminate infinite loopspath(X,Y):-edge(X,Y). path(X,Y):-edge(X,Z),path(Z,Y). • Reduce redundant computationsfib(0,1). fib(1,1). fib(N,F):-N>1, N1 is N-1,fib(N1,F1), N2 is N-2,fib(N2,F2), F is F1+F2. Linear Tabling

3. suspend/resume Complicate implementation freeze stacks overhead on standard programs garbage collection Tabling in OLDT (SLG-WAM) table producer A... ... consumer A’... A’ is suspended after the existing answers are exhausted Linear Tabling

4. Advantages Easy to implement Space efficient Overhead-free Disadvantage Re-computation Optimizations Subgoal optimization Semi-naïve evaluation Linear Tabling table A... pioneer ... follower A’... A’ fails or becomes a producerafter consuming existing answers A needs to be re-evaluated in some cases Linear Tabling

5. The Linear Tabling Framework • Augmented programs p(X,Y):-p(X,Z),e(Z,Y).p(X,Y):-e(X,Y). p(X,Y):-p(X,Z),e(Z,Y),memo(p(X,Y)).p(X,Y):-e(X,Y),memo(p(X,Y)). p(X,Y):-check_completion(p(X,Y)). Linear Tabling

6. The Linear Tabling Framework(cont.) • table_start(A) • Executed when a tabled subgoal A is encountered • memo(A) • Executed when a clause succeeds • check_completion(A) • Executed after all clauses have been tried. Linear Tabling

7. Definitions • Loops, pioneers and followers A derivation Gi…Gj forms a loop if • Gi=(A,…) and Gj=(A’,…) • A and A’ are variants • A is an ancestor of A’ • Subgoal A is called a pioneer and A’ is called a follower of A. Linear Tabling

8. Definitions (cont.) • Top-most looping nodes and subgoals A node in an SLD-tree is called a top-most looping node if the selected subgoal of the node is the pioneer of a loop that is not contained in any other loops. Linear Tabling

9. A Linear Tabling Method • table_start(A) • If A is complete, resolve A by using answers. • If A is a pioneer, register A and resolve A by using program clauses. • If A is a follower, resolve A by using answers and fail A after all existing answers are exhausted. Linear Tabling

10. A Linear Tabling Method (cont.) • memo(A) • Add A into the table and fail. Linear Tabling

11. A Linear Tabling Method (cont.) • check_completion(A) • If A has never occurred in a loop, complete A and resolve A by using the answers. • If A is a top-most looping subgoal • If no new answer was produced in the last round, then complete A and resolve A by using the answers • Otherwise, start a new round of evaluation of A. • If A is a looping subgoal but not a top-most one • Set A’s state to temporary complete and resolve A by using the answers Linear Tabling

12. Example p(X,Y):-p(X,Z),e(Z,Y),memo(p(X,Y)). (p1)p(X,Y):-e(X,Y),memo(p(X,Y)). (p2) p(X,Y):-check_completion(p(X,Y)). (p3) e(a,b). (e1) e(b,c). (e2) program 1. p(a,Y0). First round p1 p3 p2 2. p(a,Z1), e(Z1,Y0), memo(p(a,Y0)). 3. e(a,Y0), memo(p(a,Y0)). 5. check_comp(p(a,Y0)). e1 4. memo(p(a,b)). Linear Tabling

13. p(X,Y):-p(X,Z),e(Z,Y),memo(p(X,Y)). (p1)p(X,Y):-e(X,Y),memo(p(X,Y)). (p2)p(X,Y):-p(X,Z),e(Z,Y),memo(p(X,Y)). (p1)p(X,Y):-e(X,Y),memo(p(X,Y)). (p2) p(X,Y):-check_completion(p(X,Y)). (p3) e(a,b). (e1) e(b,c). (e2) p(a,b). table program Second round 1. p(a,Y0). p1 p3 p2 6. p(a,Z1), e(Z1,Y0), memo(p(a,Y0)). 10. check_comp(p(a,Y0)). … use p(a,c) use p(a,b) 9. e(c,Y0), memo(p(a,Y0)). 7. e(b,Y0), memo(p(a,Y0)). p(a,b). p(a,c). e2 8. memo(p(a,c)). Linear Tabling

14. Characteristics of the Method • Fixpoints are computed by iterating the evaluation of top-most looping subgoals • Followers consume answers only • Pioneers consume answers lazily • Top-most looping subgoals consume answers after they are complete • Other looping subgoals consume answers after all clauses have been tried Linear Tabling

15. Adopted and Related Tabling Strategies • Lazy answer consumption • Local scheduling strategy in SLG-WAM [Freire96] • What to do after a follower consumes all available answers? • Steals the pioneer’s choice pointer [Zhou00] • Fails the follower [Guo & Gupta 01] • Where to start re-computation? • At the top-most looping subgoal [Shen98] • At every looping subgoal [Guo01] Linear Tabling

16. Strengths and Weaknesses • Lazy answer consumption is suitable for all-solution search • A basic operation used in PRISM • Not suitable for single-solution search or programs with cuts • For the query, once(p(X)), all solutions are computed even though only one is needed. Linear Tabling

17. Optimization Techniques • Subgoal Optimization • In each round of evaluation of a top-most looping subgoal, each subgoal needs to be evaluated only once. • Semi-naïve Optimization • Mimic the semi-naïve technique in bottom-up evaluation: at least one new answer is involved in the join of answers for each rule. Linear Tabling

18. Semi-naïve Evaluation in Linear Tabling • Let H:-A1,…,Ak,…,Anbe a rulewhere Akis the last dependent subgoal of H. For a subgoal C of H, it is safe for Ak to consume only new answers if: • 1. C has occurred in an early round • 2. No subgoal Ai (i<k) has consumed a new answer. Linear Tabling

19. BP vs. XSB (CPU time) BP vs. XSB (Stack space) Performance Evaluation Linear Tabling

20. Papers • N.F. Zhou, Y.D. Shen, L. Yuan, and J. You: A Linear Tabling Mechanism, The Journal of Functional and Logic Programming, 2001. • N.F. Zhou and T. Sato: Efficient Fixpoint Computation in Linear Tabling, ACM SIGPLAN International Conference on Principles and Practice of Declarative Programming (PPDP), pp.275-283, 2003. • N.F. Zhou, Y. Shen, and T. Sato: Semi-naive Evaluation in Linear Tabling, ACM PPDP, pp.90-97, 2004. Linear Tabling