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
1
Tabling is Useful
 Eliminate infinite loops
path(X,Y):-edge(X,Y).
path(X,Y):-edge(X,Z),path(Z,Y).
 Reduce redundant computations
fib(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
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Tabling in OLDT (SLG-WAM)
producer A...
table  suspend/resume
– Complicate implementation
• freeze stacks
• overhead on standard
programs
• garbage collection
...
consumer A’...
A’ is suspended after the
existing answers are exhausted
Linear Tabling
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Linear Tabling
 Advantages
– Easy to implement
pioneer A...
– Space efficient
– Overhead-free
...
 Disadvantage
follower A’...
– Re-computation
A’ fails or becomes a producer  Optimizations
after consuming existing answers – Subgoal optimization
– Semi-naïve evaluation
A needs to be re-evaluated in some cases
table
Linear Tabling
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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
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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
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Definitions
 Loops, pioneers and followers
A derivation Gi…Gj forms a loop if
1.Gi=(A,…) and Gj=(A’,…)
2.A and A’ are variants
3.A is an ancestor of A’
Subgoal A is called a pioneer and A’ is called a
follower of A.
Linear Tabling
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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
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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
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A Linear Tabling Method (cont.)
 memo(A)
– Add A into the table and fail.
Linear Tabling
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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
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Example
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)).
(p1)
(p2)
(p3)
e(a,b).
e(b,c).
(e1)
(e2)
1. p(a,Y0).
p1
2. p(a,Z1),
e(Z1,Y0),
memo(p(a,Y0)).
p2
p3
3. e(a,Y0),
memo(p(a,Y0)).
e1
program
First round
5. check_comp(p(a,Y0)).
4. memo(p(a,b)).
Linear Tabling
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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)).
e(a,b). (e1) e(b,c). (e2)
(p1)
(p2)
(p3)
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)).
use p(a,c)
use p(a,b)
7. e(b,Y0),
memo(p(a,Y0)).
e2
9. e(c,Y0),
memo(p(a,Y0)).
10. check_comp(p(a,Y0)).
p(a,b).
p(a,c).
8. memo(p(a,c)).
Linear Tabling
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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
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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
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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
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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
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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 (i<k) has consumed a new
answer.
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Performance Evaluation
 BP vs. XSB (CPU time)
 BP vs. XSB (Stack space)
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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.
Linear Tabling
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