PRISM¶

% model part from the direction.psm example in Sato's PRISM distribution http://rjida.meijo-u.ac.jp/prism/ % adapted to ProbLog, with arbitrary probabilities added 0.6::msw(coin,head); 0.4::msw(coin,tail). % ProbLog does not support if-then-else with ->, so we untangle direction(left) :- msw(coin,head). direction(right) :- msw(coin,tail). query(direction(_)).

% model part from the dcoin.psm example in Sato's PRISM distribution http://rjida.meijo-u.ac.jp/prism/ % adapted to ProbLog % as coins are used repeatedly, ProbLog needs extra identifier 0.5::msw(coin(1),ID,head); 0.5::msw(coin(1),ID,tail). 0.7::msw(coin(2),ID,head); 0.3::msw(coin(2),ID,tail). dcoin(N,Rs) :- % Rs is a list with length N of outcomes dcoin(N,coin(1),Rs). % from two Bernoulli trials processes. dcoin(N,Coin,[R|Rs]) :- N > 0, msw(Coin,N,R), % replace N by some constant here to see the difference ( R == head, NextCoin = coin(2) ; R == tail, NextCoin = coin(1) ), N1 is N-1, dcoin(N1,NextCoin,Rs). dcoin(0,_,[]). query(dcoin(4,_)).

% model part from the agree.psm example in Sato's PRISM distribution http://rjida.meijo-u.ac.jp/prism/ % adapted to ProbLog 0.8::msw(coin(a),heads); 0.2::msw(coin(a),tails). 0.1::msw(coin(b),heads); 0.9::msw(coin(b),tails). failure :- not(success). % failure is defined as not(success). success :- agree(_). % success <=> exist([X],agree(X)) agree(A):- % flip two coins and msw(coin(a),A), % output the result only when msw(coin(b),B), % both outcomes agree. A=B. query(msw(_,_)). evidence(success).

% model part from the bloodABO.psm example in Sato's PRISM distribution http://rjida.meijo-u.ac.jp/prism/ % adapted to ProbLog % in contrast to PRISM, ProbLog memoizes random variables, so we need to add an ID to this switch 0.5::msw(gene,ID,a); 0.2::msw(gene,ID,b); 0.3::msw(gene,ID,o). % ProbLog does not support if-then-else with ->, so we untangle % clauses 2 and 3 overlap with the first, but unlike PRISM ProbLog does not require them to be mutex bloodtype(X) :- genotype(X,X). bloodtype(Y) :- genotype(o,Y). bloodtype(X) :- genotype(X,o). bloodtype(ab) :- genotype(X,Y), X \= Y, X \= o, Y \= o. % tell ProbLog not to reuse X as Y by adding different extra arguments genotype(X,Y) :- msw(gene,1,X),msw(gene,2,Y). query(bloodtype(_)).

% model part from the bloodAaBb.psm example in Sato's PRISM distribution http://rjida.meijo-u.ac.jp/prism/ % adapted to ProbLog, with arbitrary probabilities added 0.2::msw(locus1,'A'); 0.8::msw(locus1,a). 0.7::msw(locus2,'B'); 0.3::msw(locus2,b). % ProbLog does not support if-then-else with ->, so we untangle bloodtype(o) :- genotype(locus1,a,a), genotype(locus2,b,b). bloodtype(a) :- genotype(locus1,'A',_Y1), genotype(locus2,b,b). bloodtype(a) :- genotype(locus1,_,'A'), genotype(locus2,b,b). bloodtype(b) :- genotype(locus1,a,a), genotype(locus2,'B',_). bloodtype(b) :- genotype(locus1,a,a), genotype(locus2,_,'B'). bloodtype(ab) :- \+ bloodtype(o), \+ bloodtype(a), \+ bloodtype(b). genotype(L,X,Y) :- msw(L,X),msw(L,Y). query(bloodtype(_)).

% model part from the alarm.psm example in Sato's PRISM distribution http://rjida.meijo-u.ac.jp/prism/ % adapted to ProbLog % switches defining CPTs 0.1::msw(fi,yes); 0.9::msw(fi,no). 0.15::msw(ta,yes); 0.85::msw(ta,no). 0.95::msw(sm(yes),yes); 0.05::msw(sm(yes),no). 0.05::msw(sm(no),yes); 0.95::msw(sm(no),no). 0.5::msw(al(yes,yes),yes); 0.5::msw(al(yes,yes),no). 0.9::msw(al(yes,no),yes); 0.1::msw(al(yes,no),no). 0.85::msw(al(no,yes),yes); 0.15::msw(al(no,yes),no). 0.05::msw(al(no,no),yes); 0.95::msw(al(no,no),no). 0.88::msw(le(yes),yes); 0.12::msw(le(yes),no). 0.01::msw(le(no),yes); 0.99::msw(le(no),no). 0.75::msw(re(yes),yes); 0.25::msw(re(yes),no). 0.1::msw(re(no),yes); 0.9::msw(re(no),no). % unchanged from PRISM world(Fi,Ta,Al,Sm,Le,Re) :- %% Define a distribution for world/5 such that e.g. %% P(Fire=yes,Tapering=yes,Smoke=no,Alarm=no,Leaving=no,Report=no) %% = P(world(yes,yes,no,no,no,no)) msw(fi,Fi), % P(Fire) msw(ta,Ta), % P(Tampering) msw(sm(Fi),Sm), % CPT P(Smoke | Fire) msw(al(Fi,Ta),Al), % CPT P(Alarm | Fire,Tampering) msw(le(Al),Le), % CPT P(Leaving | Alarm) msw(re(Le),Re). % CPT P(Report | Leaving) world(Sm,Re):- %% Define marginal distribution for `Smoke' and `Report' world(_,_,_,Sm,_,Re). query(world(_,_)).

% model part from the hmm.psm example in Sato's PRISM distribution http://rjida.meijo-u.ac.jp/prism/ % adapted to ProbLog, with arbitrary probabilities added 0.5::msw(init,s0); 0.5::msw(init,s1). % state initialization % ProbLog needs extra argument to avoid reusing previous values 0.7::msw(out(X),Y,a); 0.3::msw(out(X),Y,b). % symbol emission 0.4::msw(tr(X),Y,s0); 0.6::msw(tr(X),Y,s1). % state transition hmm(L):- % To observe a string L: str_length(N), % Get the string length as N msw(init,S), % Choose an initial state randomly hmm(1,N,S,L). % Start stochastic transition (loop) % ProbLog does not support cut (!) to stop at basecase, so we add check to recursive clause instead hmm(T,N,_,[]):- T>N. % Stop the loop hmm(T,N,S,[Ob|Y]) :- % Loop: current state is S, current time is T T =< N, msw(out(S),T,Ob), % Output Ob at the state S using T to avoid reuse msw(tr(S),T,Next), % Transit from S to Next using T to avoid reuse T1 is T+1, % Count up time hmm(T1,N,Next,Y). % Go next (recursion) str_length(4). % String length query(hmm(_)).

% model part from the german.psm example in Sato's PRISM distribution http://rjida.meijo-u.ac.jp/prism/ % adapted to ProbLog 0.2::msw(det,[m,nom,der]); 0.8::msw(det,[f,nom,die]). 0.6::msw(n,[m,nom,mann]); 0.4::msw(n,[f,nom,frau]). % in contrast to PRISM, ProbLog does not depend on compiling negation away, % and we therefore can directly use the parsing program to define failure failure :- not(success). % failure :- not(exist([X],np(X))). success :- np(_). np(L) :- np(L,[]). np(L1,L3):- det(Gen,Case,L1,L2), n(Gen,Case,L2,L3). det(Gen,Case,[Wd|L2],L2):- msw(det,R),R = [Gen,Case,Wd]. n(Gen,Case,[Wd|L3],L3):- msw(n,R), R = [Gen,Case,Wd]. query(np(_)). query(failure).

% model part from the chmm.psm example in Sato's PRISM distribution http://rjida.meijo-u.ac.jp/prism/ % adapted to ProbLog % add time argument to avoid reuse 0.7::msw(tr(s0),T,s0); 0.3::msw(tr(s0),T,s1). 0.7::msw(tr(s1),T,s1); 0.3::msw(tr(s1),T,s0). 0.4::msw(lunch(s0),T,p) ; 0.6::msw(lunch(s0),T,s). % pizza:900, sandwich:400 0.5::msw(lunch(s1),T,h) ; 0.5::msw(lunch(s1),T,s). % hanburger:400, sandwich:500 failure:- not(success). success:- success(_). success(L):- chmm(L,s0,0,3). failure(L):- not(success(L)). chmm(L,S,C,N):- N>0, msw(tr(S),N,S2), msw(lunch(S),N,D), ( S == s0, ( D = p, C2 is C+900 ; D = s, C2 is C+400 ) ; S == s1, ( D = h, C2 is C+400 ; D = s, C2 is C+500 ) ), L=[D|L2], N2 is N-1, chmm(L2,S2,C2,N2). chmm([],_,C,0):- C < 2000. % given that the total number of calories in 3 days is below 2000, what was the choice of menu? query(success(_)). evidence(failure,false).