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yap-6.3/packages/ProbLog/dtproblog.yap
Theofrastos Mantadelis a442d888de Newest ProbLog version
2010-12-16 14:30:50 +01:00

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29 KiB
Prolog

%%% -*- Mode: Prolog; -*-
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% $Date: 2010-08-24 15:14:21 +0200 (Di, 24. Aug 2010) $
% $Revision: 4671 $
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% This file is part of ProbLog
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% ProbLog was developed at Katholieke Universiteit Leuven
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% Guy Van den Broeck
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% DECISION-THEORETIC PROBLOG
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
:- module(dtproblog, [
problog_delta/5,
problog_threshold/5,
problog_low/4,
problog_kbest/4,
problog_kbest_save/6,
problog_max/3,
problog_exact/3,
problog_montecarlo/3,
problog_dnf_sampling/3,
problog_answers/2,
problog_kbest_answers/3,
problog_table/1,
clear_retained_tables/0,
problog_neg/1,
get_fact_probability/2,
set_fact_probability/2,
get_continuous_fact_parameters/2,
set_continuous_fact_parameters/2,
get_fact/2,
tunable_fact/2,
continuous_fact/1,
non_ground_fact/1,
export_facts/1,
problog_help/0,
show_inference/0,
problog_dir/1,
set_problog_flag/2,
problog_flag/2,
problog_flags/0,
reset_problog_flags/0,
problog_assert/1,
problog_assert/2,
problog_retractall/1,
problog_statistics/2,
problog_statistics/0,
grow_atom_table/1,
problog_exact_nested/3,
problog_tabling_negated_synonym/2,
build_trie/2,
build_trie/3,
problog_infer/2,
problog_infer/3,
problog_infer_forest/2,
write_bdd_struct_script/3,
problog_bdd_forest/1,
require/1,
unrequire/1,
'::'/2,
probabilistic_fact/3,
problog_real_kbest/4,
in_interval/3,
below/2,
above/2,
op( 550, yfx, :: ),
op( 550, fx, ?:: ),
op( 1150, fx, problog_table ),
% DTProbLog
set_strategy/1,
unset_strategy/1,
dtproblog_utility_facts/1,
dtproblog_utility_attributes/1,
dtproblog_ev/2,
dtproblog_ev/3,
dtproblog_ev/4,
dtproblog_decisions/1,
dtproblog_decision_ids/1,
dtproblog_decision_ids/2,
dtproblog_solve/2,
dtproblog_solve_specialized/2,
dtproblog_solve_general/2,
dtproblog_solve_local/4,
dtproblog_solve_naive/2,
op( 550, yfx, => )
]).
:- style_check(all).
:- yap_flag(unknown,error).
% problog-related modules
:- use_module('problog').
:- use_module('problog/flags',[
problog_define_flag/4,
problog_define_flag/5,
problog_define_flag/6,
set_problog_flag/2,
reset_problog_flags/0,
problog_flag/2
]).
:- use_module('problog/os', [convert_filename_to_working_path/2,
convert_filename_to_problog_path/2]).
:- use_module('problog/ptree', [delete_ptree/1]).
:- use_module('problog/tabling', [clear_tabling/0]).
% general yap modules
:- use_module(library(system), [delete_file/2, shell/2]).
:- initialization((
problog_define_flag(optimization, problog_flag_validate_atom, 'optimization algorithm [local/global]', global, dtproblog),
problog_define_flag(forest_type, problog_flag_validate_atom, 'type of BDD forest [dependent/independent]', dependent, dtproblog)
)).
init_dtproblog :-
problog_control(off,find_decisions),
problog_control(off,internal_strategy).
:- initialization(init_dtproblog).
:- op( 550, yfx, :: ).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Utility Attributes
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% op for utility attributes
:- op( 550, yfx, => ).
% SETOF DOESNT WORK!!! BUT all/3 is A LOT slower because it doesn't sort the solutions?
dtproblog_utility_facts(Facts) :-
all((Attr => Util),user:(Attr => Util),Facts).
dtproblog_utility_attributes(Attrs) :-
dtproblog_utility_facts(Facts),
facts_to_attributes(Facts,Attrs).
facts_to_attributes([],[]).
facts_to_attributes([A => _|FR],[A|AR]) :- facts_to_attributes(FR,AR).
conditioned_utility_facts([],_,[],[]).
conditioned_utility_facts([(Attr => Reward)|Facts],Condition,[(Condition,Attr)|Attrs],[Reward|Rewards]):-
conditioned_utility_facts(Facts,Condition,Attrs,Rewards).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Strategies (getting/setting/transforming)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Internal strategy representation
% for GROUND strategies (one can specify a ground strategy for a non-ground decision clause)
% e.g. 1 :: market(guy) for ? :: market(P).
set_ground_strategy(GID,LogProb) :- bb_put(GID,LogProb).
get_ground_strategy(GID,LogProb) :- bb_get(GID,LogProb),!.
get_ground_strategy(_,never).
% Internal strategy representation
% for NON-GROUND strategies
% e.g. 1 :: market(guy) for ? :: market(P)
:- dynamic(non_ground_strategy/2).
% Get Strategy
strategy(_,_,_) :-
\+ problog_control(check,internal_strategy),
throw(error('Trying to get a strategy that is not set.')).
strategy(ID,Decision,Prob) :-
grounding_id(ID,Decision,GID),
bb_get(GID,LogProb), % because we don't want the default, maybe there is a non-ground strategy
!,
logprob_prob(LogProb, Prob).
strategy(_,Decision,Prob) :-
non_ground_strategy(Decision,LogProb),
!,
logprob_prob(LogProb, Prob).
strategy(_,_,never).
strategy_log(ID,Decision,LogProb) :-
strategy(ID,Decision,Prob),
logprob_prob(LogProb, Prob).
% convert from/to probabilities and their logarithms
logprob_prob(always,always) :- !.
logprob_prob(never,never) :- !.
logprob_prob(0,always) :- !.
logprob_prob(-inf,never) :- !.
logprob_prob(always,1) :- !.
logprob_prob(never,0) :- !.
logprob_prob(LogP,P) :-
number(LogP),
!,
P is exp(LogP).
logprob_prob(LogP,P) :-
number(P),
!,
LogP is log(p).
% Set Strategy
% expects a list of (p :: decision) terms
% - (decision) is interpreted as (1 :: decision)
% - decisions that are not set will evaluate to (0 :: decision)
set_strategy(_) :-
problog_control(check,internal_strategy),
throw(error('A strategy is already set, unset first.')).
set_strategy([]) :- problog_control(on,internal_strategy).
set_strategy([Term|R]) :-
strategy_entry(Term,LogProb,Decision),
(user:problog_user_ground(Decision)->
decision_fact(ID,Decision),
grounding_id(ID,Decision,ID2),
%format("Setting ~q/~q to ~q~n",[Decision,ID2,Prob]),
set_ground_strategy(ID2,LogProb)
;
copy_term(Decision, Decision2),
assertz(non_ground_strategy(Decision2,LogProb))
),
set_strategy(R).
unset_strategy(_) :-
\+ problog_control(check,internal_strategy),
throw(error('Cannot unset a strategy when no strategy is set.')).
unset_strategy([]) :-
retractall(non_ground_strategy(_,_)),
problog_control(off,internal_strategy).
unset_strategy([Term|R]) :-
strategy_entry(Term,LogProb,Decision),
(user:problog_user_ground(Decision)->
decision_fact(ID,Decision),
grounding_id(ID,Decision,ID2),
%format("Unsetting ~q/~q to ~q~n",[Decision,ID2,Prob]),
bb_delete(ID2,LogProb)
;
true
),
unset_strategy(R).
strategy_entry('::'(Prob,Decision),LogProb,Decision) :-
!,logprob_prob(LogProb, Prob).
strategy_entry(Decision,always,Decision).
% Get strategy for a list of decision IDs
% only use when grounding ids are known and strategy is stored internally!
strategy_as_term_list(IDs,List) :- strategy_as_term_list(IDs,[],List).
strategy_as_term_list([],In,In).
strategy_as_term_list([ID|R],In,Out) :-
strategy_as_term(ID,In,In2),
strategy_as_term_list(R,In2,Out).
% Get strategy for a decision ID
strategy_as_term(ID,In,Out) :-
%findall(grounding_is_known(D,I),grounding_is_known(D,I),LGround),
%findall(decision_fact(D,I),decision_fact(D,I),LBasic),
%format("Known IDs: ~q~n",[LGround]),
%format("Known IDs: ~q~n",[LBasic]),
((recover_grounding_id(ID,GID),grounding_is_known(Decision,GID)) ->
%original fact was non-ground
true
;
% original fact was ground
decision_fact(ID,Decision)
),
strategy(ID,Decision,Prob),
strategy_as_term_entry(Decision,Prob,In,Out).
% Convert strategy for a decision to term representation
strategy_as_term_entry(_,0,In,In) :- !.
strategy_as_term_entry(Decision,1,In,[Decision|In]) :- !.
strategy_as_term_entry(_,never,In,In) :- !.
strategy_as_term_entry(Decision,always,In,[Decision|In]) :- !.
strategy_as_term_entry(Decision,P,In,[P'::'Decision|In]).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Utility inference
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Unconditional expected value for all utilities
dtproblog_ev(Strategy,Ev) :- dtproblog_ev(Strategy,true,Ev).
% Conditional expected value for a given strategy and all utility attributes
dtproblog_ev(Strategy,Condition,Ev) :-
(dtproblog_utility_facts(UtilFacts) ->
dtproblog_ev(Strategy,Condition,UtilFacts,Ev)
;
format('There are no utility facts in the program.~n',[]),
Ev = 0
).
% Conditional expected value for a given strategy and utility attributes
dtproblog_ev(Strategy,Condition,UtilFacts,Ev) :-
require(keep_ground_ids),
set_strategy(Strategy),
ev_for_internal_strategy(Condition,UtilFacts,Ev),
unset_strategy(Strategy),
unrequire(keep_ground_ids),
reset_non_ground_facts.
% Conditional expected value for internal strategy and given utility attributes
% assumes that problog_control(on,internal_strategy)
ev_for_internal_strategy(Condition,UtilFacts,Ev) :-
require(keep_ground_ids),
(problog_infer_forest_supported ->
% specialized version for inference using forests
conditioned_utility_facts(UtilFacts,Condition,Goals,Utilities),
problog_infer_forest([Condition|Goals],[CondProb|GoalProbs]),
!, % forest inference was supported, don't try general purpose
summed_utils(Utilities,GoalProbs,0,EvUncond),
Ev is EvUncond/CondProb
;
% general-purpose version
ev_loop(Condition, UtilFacts, 0, EvUnnormalized),
problog_infer(Condition, Prob),
%format("Dividing the utilities by the conditional probability ~q~n",[Prob]),
(Prob > 0.000001 ->
Ev is EvUnnormalized/Prob
;
format('Impossible condition: ~q has probability ~q.~n', [Condition,Prob]),
%throw(error(improbable_condition(Condition)))
Ev = -inf
)
),
unrequire(keep_ground_ids),
reset_non_ground_facts.
summed_utils([],[],Ev,Ev).
summed_utils([Util|Utils],[Prob|Probs],Acc,Ev) :-
Acc2 is Acc + (Util * Prob),
summed_utils(Utils,Probs,Acc2,Ev).
ev_loop(_, [],Acc,Acc).
ev_loop(Condition,[(Attr => Util)|R],Acc,Ev) :-
problog_infer((Condition,Attr), Prob),
Acc2 is Acc + (Prob * Util),
%format('The probability of ~q is ~q, yielding a utility of ~q.~n', [(Condition,Attr),Prob,Prob * Util]),
ev_loop(Condition,R,Acc2,Ev).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Finding all decisions used in proofs
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Finding all decisions used in proofs
dtproblog_decisions(Decisions) :-
require(keep_ground_ids),
dtproblog_decision_ids(IDs),
ids_as_decisions(IDs,Decisions),
unrequire(keep_ground_ids),
reset_non_ground_facts.
% Get decisions for a list of IDs
ids_as_decisions(IDs,List) :- ids_as_decisions(IDs,[],List).
ids_as_decisions([],In,In).
ids_as_decisions([ID|R],In,Out) :-
id_as_decision(ID,In,In2),
ids_as_decisions(R,In2,Out).
id_as_decision(ID,In,[Decision|In]) :-
%findall(grounding_is_known(D,I),grounding_is_known(D,I),LGround),
%findall(decision_fact(D,I),decision_fact(D,I),LBasic),
%format("Known IDs: ~q~n",[LGround]),
%format("Known IDs: ~q~n",[LBasic]),
((recover_grounding_id(ID,GID),grounding_is_known(Decision,GID)) ->
% original fact was non-ground
true
;
% original fact was ground
decision_fact(ID,Decision)
).
% Finding all decision IDs used in proofs
dtproblog_decision_ids(Decisions) :-
(dtproblog_utility_attributes(UtilityAttrs) ->
dtproblog_decision_ids(UtilityAttrs,Decisions)
;
Decisions = []
).
dtproblog_decision_ids(UtilityAttrs,Decisions) :-
require(keep_ground_ids),
problog_control(on,find_decisions),
reset_decisions,
add_decisions_all(UtilityAttrs),
unrequire(keep_ground_ids),
reset_non_ground_facts,
get_decisions(Decisions),
problog_control(off,find_decisions),
reset_decisions.
% TODO generalize so that it works with every inference method, not just exact.
add_decisions_all([]) :-
clear_tabling.
add_decisions_all([Goal|R]) :-
add_decisions(Goal),
add_decisions_all(R).
% UGLY - needs to actually build tries for tabling to work.
% TODO change tabling.yap to do nothing when problog_control(on,find_decisions)
% then, simplify this predicate so that it doesn't build tries
% setting problog_control(on,mc) might work, but will maybe prune some decisions away?
add_decisions(Goal) :-
problog_control(on, exact),
build_trie(exact, Goal, Trie),
delete_ptree(Trie),
problog_control(off, exact).
reset_decisions :- bb_put(problog:decisions,[]).
get_decisions(D) :- bb_get(problog:decisions,D).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Strategy optimization
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
dtproblog_solve(Strategy,EV) :-
(dtproblog_solve_specialized_supported -> % try to go specialized
dtproblog_solve_specialized(Strategy,EV)
;
format('Flag settings not supported by specialized solution algorithm.~nTrying general purpose version.~n',[]),
(dtproblog_solve_general_supported -> % try to go general
dtproblog_solve_general(Strategy,EV)
;
throw(error('Flag settings not supported by dtproblog_solve/2.'))
)
).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Strategy optimization (specialized in BDD)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
dtproblog_solve_specialized(Strategy,EV) :-
(dtproblog_solve_specialized_supported ->
require(keep_ground_ids),
dtproblog_utility_facts(UtilFacts),
conditioned_utility_facts(UtilFacts,true,Goals,Utilities),
write_util_file(Utilities),
problog_bdd_forest(Goals),
length(Goals,N),
bdd_optimization(N,EV,DecisionIDs,ok),
(problog_flag(save_bdd,true) -> true ; delete_util_file ),
strategy_as_term_list(DecisionIDs,Strategy),
unset_strategy(Strategy), % was set by bdd_optimization/4
require(keep_ground_ids),
reset_non_ground_facts
;
throw(error('Flag settings not supported by dtproblog_solve_specialized/2.'))
).
dtproblog_solve_specialized_supported :-
problog_bdd_forest_supported,
(
problog_flag(forest_type, dependent)
;
problog_flag(optimization, local)
).
% Write a utility file, with for every utility attribute BDD, its reward value on a new line
write_util_file(Utils) :-
bdd_util_file(UtilFile),
open(UtilFile,'write',UtilFileStream),
tell(UtilFileStream),
length(Utils,N),
format("~w~n",[N]),
write_util_file_line(Utils),
flush_output,
told.
write_util_file_line([]).
write_util_file_line([U|R]) :-
format("~w~n",[U]),
write_util_file_line(R).
bdd_util_file(UtilFile) :-
problog_flag(bdd_file,BDDFileFlag),
atomic_concat([BDDFileFlag,'_',utils],UtilFileName),
convert_filename_to_working_path(UtilFileName, UtilFile).
delete_util_file :-
bdd_util_file(UtilFile),
delete_file(UtilFile,[]).
bdd_optimization(N,EV,Decisions,Status) :-
bdd_files(BDDFile,BDDParFile),
problog_flag(bdd_time,BDDTime),
(problog_flag(dynamic_reorder, true) -> ParamD = '' ; ParamD = ' -dreorder'),
(problog_flag(bdd_static_order, true) ->
problog_flag(static_order_file, FileName),
convert_filename_to_working_path(FileName, SOFileName),
atomic_concat([ParamD, ' -sord ', SOFileName], Param)
;
Param = ParamD
),
convert_filename_to_problog_path('problogbdd', ProblogBDD),
problog_flag(bdd_result,ResultFileFlag),
convert_filename_to_working_path(ResultFileFlag, ResultFile),
bdd_util_file(UtilFile),
(problog_flag(optimization,local) -> LocalPar = ' -lo';LocalPar = ''),
(problog_flag(forest_type,independent) -> Forest = ' -if';Forest = ''),
%(problog_flag(verbose,true) -> Debug = ' -d';Debug = ''), % messes up result parsing
atomic_concat([ProblogBDD, Param, ' -l ',BDDFile,' -i ',BDDParFile,' -u ',UtilFile,' -m s',LocalPar,Forest,' -t ', BDDTime,' > ', ResultFile],Command),
statistics(walltime,_),
% format(user,'$ ~w~n',[Command]),
shell(Command,Return),
(Return =\= 0 ->
Status = timeout
;
statistics(walltime,[_,E3]),
(problog_flag(verbose,true) -> format(user,'~w ms BDD processing~n',[E3]);true),
see(ResultFile),
read(expected_value(EV)),
read_strategy(Decisions),
seen,
Status = ok,
% cleanup
(problog_flag(save_bdd,true) ->
true
;
delete_file(BDDFile,[]),
delete_file(BDDParFile,[]),
delete_file(ResultFile,[]),
delete_bdd_forest_files(N)
)
).
% set the strategy in the internal format and returns a list of all decisions
read_strategy(_) :-
problog_control(check,internal_strategy),
throw(error('A strategy is already set, unset first.')).
read_strategy(DecisionIDs) :-
problog_control(on,internal_strategy),
read_strategy_intern(DecisionIDs).
read_strategy_intern(DecisionIDs) :-
read(T),
(T = end_of_file ->
DecisionIDs = []
;
T = strategy(ID,Prob),
logprob_prob(LProb,Prob),
set_ground_strategy(ID,LProb),
DecisionIDs = [ID|Rest],
read_strategy_intern(Rest)
).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Strategy optimization (general purpose)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
dtproblog_solve_general(Strategy,EV) :-
(dtproblog_solve_general_supported ->
dtproblog_utility_facts(UtilFacts),
dtproblog_solve_local(true,UtilFacts,Strategy,EV)
;
throw(error('Flag settings not supported by dtproblog_solve_specialized/2.'))
).
dtproblog_solve_general_supported :- problog_flag(optimization, local).
dtproblog_solve_local(Condition,Utils,Strategy,EV) :-
require(keep_ground_ids),
facts_to_attributes(Utils,Attrs),
dtproblog_decision_ids(Attrs,DecisionIDs),
problog_control(on,internal_strategy), % consider strategy set, even though everything is default
ev_for_internal_strategy(Condition,Utils,EvStart),
optimization_iteration_loop(DecisionIDs,Condition,Utils,EvStart,EV),
strategy_as_term_list(DecisionIDs,Strategy),
unset_strategy(Strategy),
unrequire(keep_ground_ids),
reset_non_ground_facts.
optimization_iteration_loop(Decisions,Condition,Utils,EVIn,EVOut) :-
optimization_decision_loop(Decisions,Condition,Utils,EVIn,EVTemp),
strategy_as_term_list(Decisions,Strategy),
(problog_flag(verbose,true) -> format("Found strategy ~q with EV=~q~n",[Strategy,EVTemp]);true),
(EVIn == EVTemp ->
EVOut = EVTemp
;
optimization_iteration_loop(Decisions,Condition,Utils,EVTemp,EVOut)
).
optimization_decision_loop([],_,_,Ev,Ev).
optimization_decision_loop([ID|Rest],Condition,Utils,EvIn,EvOut) :-
get_ground_strategy(ID,ProbBefore),
flip(ProbBefore, ProbAfter),
set_ground_strategy(ID,ProbAfter),
ev_for_internal_strategy(Condition,Utils,EvTest),
(EvTest>EvIn ->
EvTemp = EvTest,
(problog_flag(verbose,true) -> format("Changing strategy for #~q to ~q for EV of ~q~n",[ID,ProbAfter,EvTest]);true)
;
EvTemp = EvIn,
set_ground_strategy(ID,ProbBefore),
(problog_flag(verbose,true) -> format("Keeping strategy for #~q at ~q because EV is ~q~n",[ID,ProbBefore,EvTest]);true)
),
optimization_decision_loop(Rest,Condition,Utils,EvTemp,EvOut).
flip(always,never) :- !.
flip(never,always) :- !.
flip(P,always) :- P<1, !.
flip(_,never).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Naive Search
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
dtproblog_solve_naive(Strategy,EV) :-
require(keep_ground_ids),
dtproblog_decisions(Decisions),
all_subsets(Decisions,Strategies),
max_strategy(Strategies,[],-inf,Strategy,EV),
unrequire(keep_ground_ids),
reset_non_ground_facts.
max_strategy([],Strategy,EV,Strategy,EV).
max_strategy([S1|Rest],S2,U2,S3,U3) :-
dtproblog_ev(S1,true,U1),
%format("EV of ~q is ~q~n",[S1,U1]),
(U1>U2 ->
max_strategy(Rest,S1,U1,S3,U3)
;
max_strategy(Rest,S2,U2,S3,U3)
).
% List of all sublists
all_subsets([], [[]]).
all_subsets([X|Xs], Subsets) :-
all_subsets(Xs, Subsets1),
attach_first_element(Subsets1, X, Subsets, Subsets1).
attach_first_element([], _, S, S).
attach_first_element([Sub|Subs], X, [[X|Sub]|XSubs], S) :-
attach_first_element(Subs, X, XSubs, S).