# Throwing a coin We throw a coin with uncertainty on its fairness. The coin is fair with 90% probability and biased with 10% probability. If the coin is fair, it lands heads with 50% probability and tails with 50% probability. If the coin is biased, it lands heads with 60% probability and tails with 40% probability. What is the probability that coin lands heads?

prob(heads(coin),Prob).

What is the probability that coin lands heads represented graphically?

prob(heads(coin),Prob),bar(Prob,C).

What is the probability that coin lands tails?

prob(tails(coin),Prob).

What is the probability that coin lands tails represented graphically?

prob(tails(coin),Prob),bar(Prob,C).

What is the probability that coin lands heads given the coin is biased?

prob(heads(coin),biased(coin),Prob).

What is the BDD for query heads(coin)?

bdd_dot_string(heads(coin),BDD,Var).

A solid edge indicates a 1-child, a dashed edge indicates a 0-child and a dotted edge indicates a negated 0-child. Each level of the BDD is associated to a variable of the form XI_J indicated on the left: I indicates the multivalued variable index and J the index of the Boolean variable of I. The table =Var= contains the associations between the rule groundings and the multivalued variables: the first column contains the multivalued variable index, the second column contains the rule index, corresponding to its position in the program, and the last column contains the list of constants grounding the rule, each replacing a variable in the order of appearance in the rule. ## Code Preamble:

:- use_module(library(pita)). :- if(current_predicate(use_rendering/1)). :- use_rendering(c3). :- use_rendering(graphviz). :- use_rendering(table,[header(['Multivalued variable index','Rule index','Grounding substitution'])]). :- endif. :- pita. :- begin_lpad.

If we toss a =Coin= that is not biased then it lands heads with probability 1/2 and tails with probability 1/2:

heads(Coin): 1/2; tails(Coin) : 1/2:-toss(Coin),\+biased(Coin).

If we toss a =Coin= that is biased then it lands heads with probability 0.6 and tails with probability 0.4:

heads(Coin): 0.6 ; tails(Coin) : 0.4:-toss(Coin),biased(Coin).

A =Coin= is fair with probability 0.9 and biased with probability 0.1:

fair(Coin):0.9 ; biased(Coin):0.1.

=coin= is certainly tossed:

toss(coin).

Epilogue:

:- end_lpad.

From J. Vennekens, S. Verbaeten, and M. Bruynooghe. Logic programs with annotated disjunctions. In International Conference on Logic Programming, volume 3131 of LNCS, pages 195-209. Springer, 2004.