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Dusa is answer set programming

Answer set programming is a way of writing Datalog-like programs to compute acceptable models (answer sets) that meet certain constraints. Whereas traditional datalog aims to compute just one solution, answer set programming introduces choices that let multiple possible solutions diverge.

Dusa allows choices by supporting mutually exclusive assignments to a functional predicate. Any assignment which satisfies all integrity constraints is a valid solution. (Answer set programming calls them “answer sets” or “models,” Dusa calls them “solutions,” but they’re the same concept.)

Generating suspects

If we have three suspects, Amy, Harry, and Sally, we can assign each of them guilt or innocence with the following program:

guilt amy is { innocent, guilty }.
guilt harry is { innocent, guilty }.
guilt sally is { innocent, guilty }.

This makes “guilt” a functional predicate: each of the three suspects can have one guilt quality: either guilt amy is innocent can hold, or guilt amy is guilty can hold, but Dusa won’t let both of these hold at the same time (and the rule forces the attribute guilt amy to take one of these two values).

Dusa will, in no particular order, generate eight different solutions, corresponding to the two different guilt assignments for the three suspects.

  • guilt amy is innocent, guilt harry is innocent, guilt sally is innocent
  • guilt amy is innocent, guilt harry is innocent, guilt sally is guilty
  • guilt amy is innocent, guilt harry is guilty, guilt sally is innocent
  • guilt amy is innocent, guilt harry is guilty, guilt sally is guilty
  • guilt amy is guilty, guilt harry is innocent, guilt sally is innocent
  • guilt amy is guilty, guilt harry is innocent, guilt sally is guilty
  • guilt amy is guilty, guilt harry is guilty, guilt sally is innocent
  • guilt amy is guilty, guilt harry is guilty, guilt sally is guilty

The word is is a keyword separating the attribute guilt amy from the value guilty or innocent.

Constraints

It’s not a good murder mystery if everyone is guilty or everyone is innocent! We can reject any solutions that don’t have someone guilty with #demand constraints:

#demand guilt _ is guilty.
#demand guilt _ is innocent.

These declarations demand that someone (the underscore means we don’t care who) is innocent, and that else is guilty. This will cause us to reject the two solutions where everyone is innocent or guilty.

Those two constraints above are equivalent to the single constraint:

#demand guilt Someone is guilty, guilt SomeoneElse is innocent.

When a #demand constraint has multiple parts separated by commas, all must hold for the demand to be met. The uppercase Someone and SomeoneElse are variables that can be replaced by any suspect (Amy, Harry, or Sally).

If we want to ensure that there’s only one innocent, we could write a #forbid constraint to forbid any solution where two or more suspects are innocent.

If every clause of a #forbid constraint can be satisfied, then the solution will be rejected.

#forbid guilt Someone is innocent,
guilt SomeoneElse is innocent,
Someone != SomeoneElse.

It’s very important that we add Someone != SomeoneElse. If we do not, Dusa can assign the same suspect to Someone and SomeoneElse, which means that this rule would forbid any solution where anyone was innocent.

Rules

Let’s say that we want to add a new character, Harold, to our little band of potential criminals. Harold’s innocence is tied to others: if Amy innocent, then Harold is guilty, and if Harry is guilty, then Harold is guilty.

This is a job for rules, which describe if-then conditions… but we write them backwards, as is traditional. The logic above is handled like this:

guilt harold is guilty :- guilt amy is innocent.
guilt harold is guilty :- guilt harry is guilty.

This doesn’t say that Harold is potentially guilty and potentially innocent, it provides specific conditions where Harold must be guilty.

But that’s not everything! To complete our story, we want to say that Harold is innocent unless Amy is innocent or Harry is guilty. We can capture this “unless” reasoning with an open rule. (All the rules we’ve seen so far have been closed, because they insist on only particular values for attributes.)

The open rule we want looks like this:

guilt harold is? innocent.

In the case where nothing forces Harold to be guilty, this rule will conclude that Harold is innocent. However, in the case where one of the other rules forces Harold to be guilty, this indifferent rule won’t contradict that other conclusion.

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Boolean satisfiability

Answer set programming is sometimes compared to boolean satisfiability solving, and is sometimes implemented with general purpose boolean satisfiability solvers. We can use Dusa as a (pretty bad) boolean satisfiability solver by assigning every proposition we come across the value true or false. The Wikipedia article describing a basic SAT-solving algorithm, DPLL describes this SAT instance:

(a' + b + c ) * (a + c + d ) *
(a + c + d') * (a + c' + d ) *
(a + c' + d') * (b' + c' + d ) *
(a' + b + c') * (a' + b' + c )

We can ask Dusa to solve this problem by negating each the OR-ed together clauses and making each one a #forbid constraint. (Needing one of the literals in a clause to hold is the same as needing all of the negations to hold.)

a is { true, false }.
b is { true, false }.
c is { true, false }.
d is { true, false }.
#forbid a is true, b is false, c is false.
#forbid a is false, c is false, c is false.
#forbid a is false, c is false, d is true.
#forbid a is false, c is true, d is false.
#forbid a is false, c is true, d is true.
#forbid b is true, c is true, d is false.
#forbid a is true, b is false, c is true.
#forbid a is true, b is true, c is false.

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Rock-paper-scissors

Rock-paper-scissors is a two-person game. The two players simultaneously choose rock, paper, or scissors, and the pair of choices determines the winner: paper covers rock, rock crushes scissors, and scissors cut paper.

One round

We can use Dusa to simulate a single round of rock-paper-scissors.

player player1.
player player2.
throw Player is { rock, paper, scissors } :- player Player.

There’s no “is” in the player _ proposition: this is a relational proposition, instead of the functional propositions we’ve seen so far.

This Dusa program has nine different solutions:

  • Player 1 throws rock, player 2 throws rock (Tie)
  • Player 1 throws rock, player 2 throws paper (Player 2 wins, paper covers rock)
  • Player 1 throws rock, player 2 throws scissors (Player 1 wins, rock crushes scissors)
  • Player 1 throws paper, player 2 throws rock (Player 1 wins, paper covers rock)
  • …and so on…

We can capture the information about who wins a new relational proposition outcome _ _ _, with three facts, and two more rules:

outcome rock crushes scissors.
outcome scissors cuts paper.
outcome paper covers rock.
tie :- throw player1 is Throw, throw player2 is Throw.
winner Win :- outcome Victor _ Loser, throw Win is Victor, throw _ is Loser.

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Multiple rounds

Often, we want to play rock paper scissors until someone wins. To play multiple rounds, we replace the predicate throw Player is Move with the predicate throw Player Round is Move which records the player’s move in a particular round. There’s always a round one, and there are always two players, player1 and player2.

round 1.
player player1.
player player2.
throw Player Round is { rock, paper, scissors } :-
player Player,
round Round.

As before, as soon as a player casts a winning move, they win.

outcome rock crushes scissors.
outcome scissors cuts paper.
outcome paper covers rock.
winner Win :-
outcome Victor _ Loser,
throw Win Round is Victor,
throw _ Round is Loser.

If (and only if) the players pick the same answer, then we need to play another round. This uses built-in addition to use concise round numbers 1, 2, 3…

#builtin INT_PLUS plus
round (plus Round 1) :-
throw player1 Round is Throw,
throw player2 Round is Throw.

Now we have a working simulator for arbitrary games of rock-paper-scissors! If we want to control the amount of drama we’re exploring, we can use constraints to control the number of rounds. These two constraints require there to be at least three rounds but no more than eight.

#demand round 3.
#forbid round 8.

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