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)

2
Tabling is Useful

• Eliminate infinite loops path(X,Y)-edge(X,Y). p
  ath(X,Y)-edge(X,Z),path(Z,Y).
• Reduce redundant computations fib(0,1). fib(1,1)
  . fib(N,F)-Ngt1, N1 is N-1,fib(N1,F1),
  N2 is N-2,fib(N2,F2), F is F1F2.

3
Tabling in OLDT (SLG-WAM)

• suspend/resume
• Complicate implementation
• freeze stacks
• overhead on standard programs
• garbage collection

table
producer
A...
...
consumer
A...
A is suspended after the existing answers are
exhausted
4
Linear Tabling

• Advantages
• Easy to implement
• Space efficient
• Overhead-free
• Disadvantage
• Re-computation
• Optimizations
• Subgoal optimization
• Semi-naïve evaluation

table
A...
pioneer
...
follower
A...
A fails or becomes a producerafter consuming
existing answers A needs to be re-evaluated in
some cases
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)).
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.

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.

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.

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.

10
A Linear Tabling Method (cont.)

• memo(A)
• Add A into the table and fail.

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 As state to temporary complete and resolve A
  by using the answers

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)).
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)-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)).
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

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 pioneers 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

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.

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.

18
Semi-naïve Evaluation in Linear Tabling

• Let H-A1,,Ak,,An be a rule where Ak is 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 (iltk) has consumed a new answer.

19
Performance Evaluation

• BP vs. XSB (CPU time)

• BP vs. XSB (Stack space)

20
Papers

1. N.F. Zhou, Y.D. Shen, L. Yuan, and J. You A
   Linear Tabling Mechanism, The Journal of
   Functional and Logic Programming, 2001.
2. 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.
3. N.F. Zhou, Y. Shen, and T. Sato Semi-naive
   Evaluation in Linear Tabling, ACM PPDP, pp.90-97,
   2004.