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Reference: propositional WaterWorld

Module by: Ian Barland, John Greiner, Phokion Kolaitis, Moshe Vardi, Matthias Felleisen

Summary: The domain axioms for WaterWorld in propositional logic.

We summarize the details of how we choose to model WaterWorld boards in propositional logic: exactly what propositions we make up, and the formal domain axioms which capture the game's rules.

The board is fixed at 6×4, named AA,…,ZZ (with I and O omitted).

Figure 1: A Sample WaterWorld board
Figure 1 (hwA-1.png)

Propositions

There are a myriad of propositions for WaterWorld, which can be grouped:

  • Whether or not a location contains a pirate: A-unsafeA-unsafe, B-unsafeB-unsafe, …, Z-unsafeZ-unsafe.
  • Whether or not a location contains no pirate: A-safeA-safe, B-safeB-safe, …, Z-safeZ-safe.

    aside:

    Yes, using the intended interpretation, these are redundant with the previous ones. Some domain axioms below will formalize this.
  • Propositions indicating the number of neighboring pirates, to a location: A-has-0A-has-0, A-has-1A-has-1, A-has-2A-has-2, B-has-0B-has-0, B-has-1B-has-1, B-has-2B-has-2, …, H-has-0H-has-0, H-has-1H-has-1, H-has-2H-has-2, H-has-3H-has-3, …, Z-has-0Z-has-0, Z-has-1Z-has-1. These are all true/false propositions; — there are no explicit numbers in the logic. A domain axiom below will assert that whenever (say) B-has-1B-has-1 is true, then B-has-0B-has-0 and B-has-2B-has-2 are both false.

    aside:

    There is no proposition A-has-3A-has-3 — since location AA has only two neighbors. Similarly, there is no proposition B-has-3B-has-3. We could have chosen to include those, but under the intended interpretation they'd always be false.

These propositions describe the state of the underlying board — the model — and not our particular view of it. Our particular view will be reflected in which formulas we'll accept as premises. So we'll accept A-has-2A-has-2 as a premise only when AA has been exposed and shows a 2.

The domain axioms

Axioms asserting that the neighbor counts are correct:

  • Count of 0:
    • "A0": (A-has-0→(B-safe∧G-safe))(A-has-0→(B-safe∧G-safe))
    • …
    • "H0": (H-has-0→(G-safe∧J-safe∧P-safe))(H-has-0→(G-safe∧J-safe∧P-safe))
    • …
    • "Z0": (Z-has-0→(Y-safe))(Z-has-0→(Y-safe))
  • Count of 1:
    • "A1": (A-has-1→((B-safe∧G-unsafe)∨(B-unsafe∧G-safe)))(A-has-1→((B-safe∧G-unsafe)∨(B-unsafe∧G-safe)))
    • …
    • "H1": (H-has-1→((G-safe∧J-safe∧P-unsafe)∨(G-safe∧J-unsafe∧P-safe)∨(G-unsafe∧J-safe∧P-safe)))(H-has-1→((G-safe∧J-safe∧P-unsafe)∨(G-safe∧J-unsafe∧P-safe)∨(G-unsafe∧J-safe∧P-safe)))
    • …
    • "Z1": (Z-has-1→(Y-unsafe))(Z-has-1→(Y-unsafe))
  • Count of 2:
    • "A2": (A-has-2→(B-unsafe∧G-unsafe))(A-has-2→(B-unsafe∧G-unsafe))
    • …
    • "H2": (H-has-2→((G-safe∧J-unsafe∧P-unsafe)∨(G-unsafe∧J-safe∧P-unsafe)∨(G-unsafe∧J-unsafe∧P-safe)))(H-has-2→((G-safe∧J-unsafe∧P-unsafe)∨(G-unsafe∧J-safe∧P-unsafe)∨(G-unsafe∧J-unsafe∧P-safe)))
    • …
    There aren't any such axioms for locations with only one neighbor.
  • Count of 3:
    • "H3": (H-has-3→(G-unsafe∧J-unsafe∧P-unsafe))(H-has-3→(G-unsafe∧J-unsafe∧P-unsafe))
    • …
    There aren't any such axioms for locations with only one or two neighbors.

Axioms asserting that the neighbor count propositions; are consistent:

  • (A-has-0∨A-has-1)(A-has-0∨A-has-1)
  • (A-has-0→¬A-has-1)(A-has-0→¬A-has-1)
  • (A-has-1→¬A-has-0)(A-has-1→¬A-has-0)
  • (B-has-0∨B-has-1∨B-has-2)(B-has-0∨B-has-1∨B-has-2)
  • (B-has-0→(¬B-has-1∧¬B-has-2))(B-has-0→(¬B-has-1∧¬B-has-2))
  • (B-has-1→(¬B-has-0∧¬B-has-2))(B-has-1→(¬B-has-0∧¬B-has-2))
  • (B-has-2→(¬B-has-0∧¬B-has-1))(B-has-2→(¬B-has-0∧¬B-has-1))
  • …
  • (H-has-0∨H-has-1∨H-has-2∨H-has-3)(H-has-0∨H-has-1∨H-has-2∨H-has-3)
  • (H-has-0→(¬H-has-1∧¬H-has-2∧¬H-has-3))(H-has-0→(¬H-has-1∧¬H-has-2∧¬H-has-3))
  • (H-has-1→(¬H-has-0∧¬H-has-2∧¬H-has-3))(H-has-1→(¬H-has-0∧¬H-has-2∧¬H-has-3))
  • (H-has-2→(¬H-has-0∧¬H-has-1∧¬H-has-3))(H-has-2→(¬H-has-0∧¬H-has-1∧¬H-has-3))
  • (H-has-3→(¬H-has-0∧¬H-has-1∧¬H-has-2))(H-has-3→(¬H-has-0∧¬H-has-1∧¬H-has-2))
  • …

Axioms asserting that the safety propositions are consistent:

  • (A-safe→¬A-unsafe)(A-safe→¬A-unsafe),
  • (¬A-safe→A-unsafe)(¬A-safe→A-unsafe),
  • …
  • (Z-safe→¬Z-unsafe)(Z-safe→¬Z-unsafe),
  • (¬Z-safe→Z-unsafe)(¬Z-safe→Z-unsafe).

This set of axioms is not quite complete, as explored in an exercise.

As mentioned, it is a bit redundant to have both A-safeA-safe and A-unsafeA-unsafe as propositions;. Furthermore, having both allows us to express inconsistent states (ones that would contradict the safety axioms). If implementing this in a program, you might use both as variables, but have a safety-check function to make sure that a given board representation is consistent. Even better, you could implement WaterWorld so that these propositions; wouldn't be variables, but instead be calls to a lookup (accessor) functions. These would examine the same internal state, to eliminate the chance of inconsistent data.

Using only true/false propositions; without recourse to numbers makes these domain axioms unwieldy. We'll see later how relations and quantifiers help us model the game of WaterWorld more concisely.

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