1. Mode-Directed Tabling for Dynamic Programming, Machine Learning, and Constraint Solving
Neng-Fa Zhou
(City Univ. of New York) Yoshitaka Kameya and Taisuke Sato
(Tokyo Inst. of Technology)
ICTAI'10, Zhou&Kameya&Sato

2. ICTAI'10, Zhou&Kameya&Sato
What is Tabling?
The idea of tabling is to memorize answers to tabled subgoals and use the answers to resolve subsequent variant or subsumed subgoals.
Eliminate infinite loops
Reduce redundancy
:-table path/2. path(X,Y):-edge(X,Y). path(X,Y):-
edge(X,Z),
path(Z,Y).
:-table fib/2. fib(N,F):-N=<1,!,F=1. fib(N,F):- N1 is N-1, fib(N1,F1), N2 is N-2, fib(N2,F2), F is F1+F2.
ICTAI'10, Zhou&Kameya&Sato

3. The Table-All Approach
Characteristics
All the arguments of a tabled subgoal are used in variant or subsumption checking
All answers are tabled
Problems
The number of answers may be too large or even infinite for DP and ML problems
When computing aggregates, tabling noncontributing answers is a waste
ICTAI'10, Zhou&Kameya&Sato

4. Problems of Table-All (Ex-1)
:-table path/3. path(X,Y,[(X,Y)]):-edge(X,Y). path(X,Y,[(X,Z)|P]):-
edge(X,Z),
path(Z,Y,P).
It’s infeasible to table all the answers because the number of paths is infinite.
ICTAI'10, Zhou&Kameya&Sato

5. Problems of Table-All (Ex-2) (From the PRISM User’s Manual)
pcfg(L):- pcfg(s,L-[]).
pcfg(LHS,L0-L1):-
( nonterminal(LHS)->
msw(LHS,RHS),
proj(RHS,L0-L1) ;
L0 = [LHS|L1] ).
proj([],L-L).
proj([X|Xs],L0-L1):-
pcfg(X,L0-L2),
proj(Xs,L2-L1).
It is infeasible to table all the answers of pcfg/2 if the number of parse trees is huge.
ICTAI'10, Zhou&Kameya&Sato

6. Mode-Directed Tabling in B-Prolog
Table mode declaration
C: Cardinality limit
Modes
+: input, used in variant checking
-: output, not used in variant checking
nt : not tabled, not used in variant checking or answer tabling
min: output, table answers with minimum values
max: output, table answers with maximum values
:-table p(M1,...,Mn):C.
ICTAI'10, Zhou&Kameya&Sato

7. Ex-1: Shortest Path Problem
:-table sp(+,+,-,min).
sp(X,Y,[(X,Y)],W) :-
edge(X,Y,W).
sp(X,Y,[(X,Z)|Path],W) :-
edge(X,Z,W1),
sp(Z,Y,Path,W2),
W is W1+W2.
sp(X,Y,P,W)
P is a shortest path between X and Y with weight W.
ICTAI'10, Zhou&Kameya&Sato

8. ICTAI'10, Zhou&Kameya&Sato
Ex-2: Knapsack Problem
:- table knapsack(+,+,-,max).
knapsack(_,0,[],0).
knapsack([_|L],K,Selected,V):-
knapsack(L,K,Selected,V).
knapsack([F|L],K,[F|Selected],V):-
K1 is K - F, K1 >= 0,
knapsack(L,K1,Selected,V1),
V is V1 + 1.
knapsack(L,K,Selected,V)
L: the list of items
K: the total capacity
Selected: the list of selected items
V: the length of Selected
ICTAI'10, Zhou&Kameya&Sato

9. Mode-Directed Tabling for Dynamic Programming
General approach
Use recursion to generate tuples of a relation
Use mode-directed tabling to guide indexing subgoals and tabling answers
More examples
Second ASP solver competition (
How to Solve it With B-Prolog (17th Prolog Programming Contest)
ICTAI'10, Zhou&Kameya&Sato

10. PRISM Prolog + Probabilistic reasoning and learning
Probability distributions and switches
Sample execution: sample(Goal)
Probability calculation: prob(Goal,P)
Learning: learn(Facts)
values(coin,[head:0.5,tail:0.5]).
direction(D):-
msw(coin,Face),
(Face==head->D=left;D=right).
ICTAI'10, Zhou&Kameya&Sato

11. ICTAI'10, Zhou&Kameya&Sato
An Example in PRISM
hmm(L) :- msw(init,Si), hmm(Si,L). hmm(S,[]). hmm(S,[C|L]) :- msw(out(S),C), msw(tr(S),NextS), hmm(NextS,L).
values(init,[s0,s1]). values(out(_),[a,b]). values(tr(_),[s0,s1]).
hmm([a,b,a])
hmm(s0,[a,b,a])
hmm(s1,[a,b,a])
hmm(s0,[b,a])
hmm(s1,[b,a])
hmm(s0,[a])
hmm(s1,[a])
hmm(s0,[])
hmm(s1,[])
ICTAI'10, Zhou&Kameya&Sato

12. ICTAI'10, Zhou&Kameya&Sato
Use of Tabling in PRISM
:-table expl_hmm/2. expl_hmm(S,[]). expl_hmm(S,[CL]) :- expl_msw(out(S),C), expl_msw(tr(S),NextS), expl_hmm(NextS,L), add_explanation(path(hmm(S,[CL]), [hmm(NextS,L)], [msw(out(S),C), msw(tr(S),NextS)])).
ICTAI'10, Zhou&Kameya&Sato

13. Use of Mode-Directed Tabling in PRISM
:- table viterbi_hmm(+,-,-,max):3.
viterbi_hmm(Os,E0,E1,W) :-
str_length(L),
viterbi_msw(init,S,W1),
viterbi_hmm(1,L,S,Os,E2,E1,W2),
W is W1+W2,
E0=[node(hmm(Os),
[path([hmm(1,L,S,Os)],
[msw(init,S)])])|E2].
Perform partial Viterbi computation by declaring a cardinality limit
ICTAI'10, Zhou&Kameya&Sato

14. Performance Improvement
Fig.~\ref{performance} (left) shows the execution time in Viterbi
inference for the HMM program with various sequence lengths when
the number of states is fixed at $50$.
Fig.~\ref{performance} (right) shows the execution time with various
numbers of states when the sequence length is fixed at $500$.
The curve ``plain'' indicates the case {\em without}
mode-directed tabling, and the curve ``mode-directed'' indicates
the case {\em with} mode-directed tabling. In all runs, the cardinality limit was
set to be one, and the probabilities over {\tt msw(I,V)} were set randomly. We can see from
these results that the improvement by mode-directed tabling gets
more significant as the model/data gets larger, and thus
mode-directed tabling is effective, especially for large data and complex models. }
ICTAI'10, Zhou&Kameya&Sato

15. Mode-Directed Tabling for Evaluating ASP Programs
Use tabling for programs with stratified negation
Use propagation and labeling to handle non-stratified negation
Use mode-directed tabling in seeking supports
ICTAI'10, Zhou&Kameya&Sato

16. ICTAI'10, Zhou&Kameya&Sato
Conclusion
Mode-directed tabling is useful
For dynamic programming problems
For machine learning applications
For evaluating ASP programs
More information
B-Prolog version 7.4 & up (bprolog.com)
ICTAI'10, Zhou&Kameya&Sato