<div class="notebook"> <div class="nb-cell markdown"> # Dice In this section we illustrate an example that models a game with a six-sided dice. The dice is repeatedly thrown until the outcome is six. When the outcome is six, the game stops. We will show how to perform a simple query, how to perform a conditional query and how to execute a query whose results are graphically represented by a histogram. ### Writing the program step by step First of all we want to write a fact which states that at time 0 the die land on one of its faces with a uniform probability distribution (1/6 for each face). We use the predicate =|on(T,F)|= which means that the die landed on face F at time T. == on(0,1):1/6;on(0,2):1/6;on(0,3):1/6; on(0,4):1/6;on(0,5):1/6;on(0,6):1/6. == The following rule states that at time T the die lands on one of its faces with equal probability if at the previous time point it was thrown and it did not land on face 6. == on(X,1):1/6;on(X,2):1/6;on(X,3):1/6; on(X,4):1/6;on(X,5):1/6;on(X,6):1/6:- X1 is X-1,X1>=0,on(X1,_), \+ on(X1,6). == </div> <div class="nb-cell markdown"> ### Full program with Prolog editor Below we can see the full LPAD of the example. </div> <div class="nb-cell program prolog"> % Load 'pita' library to perform inference :- use_module(library(pita)). :- pita. % to be written before the program :- begin_lpad. % T = 0 on(0,1):1/6;on(0,2):1/6;on(0,3):1/6; on(0,4):1/6;on(0,5):1/6;on(0,6):1/6. % T > 0 on(X,1):1/6;on(X,2):1/6;on(X,3):1/6; on(X,4):1/6;on(X,5):1/6;on(X,6):1/6:- X1 is X-1,X1>=0,on(X1,_), \+ on(X1,6). % to be written after the program evidence:- on(0,1), on(1,1). :- end_lpad. </div> <div class="nb-cell markdown"> Now we ask for the probability that the die will land on face 1 at time 0. </div> <div class="nb-cell query"> prob(on(0,1),P). </div> <div class="nb-cell markdown"> At this point we ask for the probability that the die will land on face 1 at time 2. If we submit this query, we can note that the probability that the die will land on one of its faces at time T (with T > 0) decreases. This is because it is the probability that the dice will land on one of its faces at time T and that at time T-1 it did not land on face 6. </div> <div class="nb-cell query"> prob(on(2,1),P). </div> <div class="nb-cell markdown"> We can ask conditional queries with the predicate == prob(:Query:atom,:Evidence:atom,-Probability:float). == For example, we can ask for the probability that the die will land on face 1 at time 2 given that it landed on face 1 at time 0. </div> <div class="nb-cell query"> prob(on(2,1),on(0,1),P). </div> <div class="nb-cell markdown"> If the evidence is composed of more than one atom, add a clause of the form == evidence:- e1,...,en. == to the program, where `e1,...,en` are the evidence atoms, and use the query == ?- prob(goal,evidence,P). == as for example in </div> <div class="nb-cell query"> prob(on(2,1),evidence,P). </div> <div class="nb-cell markdown"> ### How to execute a query with graphical results *cplint on SWISH* can show the probabilistic results of a query as histograms. What we have to do is to use the predicate prob_bar/2 instead of prob/2. This feature, however, it is only supported if we are using the "Prolog" editor, it is *NOT* supported with the "LPAD" editor. The syntax is the same as prob/2. == prob_bar(:Atom,-P). == Where =Atom= is the query that we want to ask, while =P= is the variable that will contain a bar chart with two bars, one for the probability of the atom of being true and one for the probability of the atom of being false (1- the first). It provides a graphical representation of the difference between the two values. However, before submitting this kind of query, we need to specify that we want to use the renderer =c3= by adding the following line before the =|:- begin_lpad.|= goal == :- use_rendering(c3). == Therefore our program becomes </div> <div class="nb-cell program prolog"> % load the 'pita' library to perform inference :- use_module(library(pita)). :- pita. % load the graphical renderer :- use_rendering(c3). % to be written before the program :- begin_lpad. % Program on(0,1):1/6;on(0,2):1/6;on(0,3):1/6; on(0,4):1/6;on(0,5):1/6;on(0,6):1/6. on(X,1):1/6;on(X,2):1/6;on(X,3):1/6; on(X,4):1/6;on(X,5):1/6;on(X,6):1/6:- X1 is X-1,X1>=0,on(X1,_), \+ on(X1,6). % to be written after the program :- end_lpad. </div> <div class="nb-cell markdown"> For istance let us consider again the previous query, this time with a graphical result. </div> <div class="nb-cell query"> prob_bar(on(2,1), P). </div> <div class="nb-cell markdown"> -- Complete example: [dice.pl](example/inference/dice.pl) -- Complete example with the LPAD editor: [dice.cpl](example/inference/dice.cpl) -- - Reference: J. Vennekens, S. Verbaeten, and M. Bruynooghe. _Logic programs with annotated disjunctionsi_. In International Conference on Logic Programming, volume 3131 of LNCS, pages 195-209. Springer, 2004. </div> <div class="nb-cell markdown"> -- [Back to Tutorial](tutorial/tutorial.swinb) </div> </div>