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Find solution to Einstein's riddle with Apalache #113

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Jan 29, 2024
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Find solution to Einstein's riddle with Apalache #113

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00b4d5e
Make Einstein work with Apalache
nano-o Jan 21, 2024
575c1a0
Avoid _OF_TYPE postfixes by using type Str
nano-o Jan 22, 2024
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improve comments
nano-o Jan 27, 2024
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65 changes: 38 additions & 27 deletions specifications/EinsteinRiddle/Einstein.tla
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Original file line number Diff line number Diff line change
Expand Up @@ -35,12 +35,14 @@

EXTENDS Naturals, FiniteSets

House == 1..5

\* Note that TLC!Permutations has a Java module override and, thus,
\* would be evaluated faster. However, TLC!Permutations equals a
\* set of records whereas Permutation below equals a set of tuples/
\* sequences. Also, Permutation expects Cardinality(S) = 5.
Permutation(S) ==
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Just by looking at the set of constraints, I think you could define Permutation for the general case too:

Permutation(S) ==
  { p \in [ House -> S ]:
    \A h1 \in House:
      \A h2 \in House:
        h2 /= h1 => p[h2] /= p[h1]
  }

You could even replace House with S, and it should still work in Apalache, as long as S is a finite set. Of course, this definition would produce really a lot of constraints, something like $O(|S|^{|House|}*|House|^2)$

{ p \in [ 1..5 -> S ] :
{ p \in [ House -> S ] :
/\ p[2] \in S \ {p[1]}
/\ p[3] \in S \ {p[1], p[2]}
/\ p[4] \in S \ {p[1], p[2], p[3]}
Expand All @@ -51,17 +53,22 @@ Permutation(S) ==
\* parameterization, the constants are replaced with constant-level
\* operators. As a side-effect, TLC evaluates them eagerly at startup,
\* and Apalache successfully determines their types.
NATIONALITIES == Permutation({ "brit", "swede", "dane", "norwegian", "german" })
DRINKS == Permutation({ "beer", "coffee", "mylk", "tea", "water" })
COLORS == Permutation({ "red", "white", "blue", "green", "yellow" })
PETS == Permutation({ "bird", "cat", "dog", "fish", "horse" })
CIGARS == Permutation({ "blend", "bm", "dh", "pm", "prince" })
NATIONALITIES == Permutation({ "brit_OF_NATIONALITY", "swede_OF_NATIONALITY", "dane_OF_NATIONALITY", "norwegian_OF_NATIONALITY", "german_OF_NATIONALITY" })
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Suggested change
NATIONALITIES == Permutation({ "brit_OF_NATIONALITY", "swede_OF_NATIONALITY", "dane_OF_NATIONALITY", "norwegian_OF_NATIONALITY", "german_OF_NATIONALITY" })
BRIT == "brit_OF_NATIONALITY"
SWEDE == "swede_OF_NATIONALITY"
DANE == "dane_OF_NATIONALITY"
NORWEGIAN == "norwegian_OF_NATIONALITY"
GERMAN == "german_OF_NATIONALITY"
NATIONALITIES == Permutation({ BRIT, SWEDE, DANE, NORWEGIAN, GERMAN })

Here is my suggestion on how to avoid specifying the type all the time.

DRINKS == Permutation({ "beer_OF_DRINK", "coffee_OF_DRINK", "mylk_OF_DRINK", "tea_OF_DRINK", "water_OF_DRINK" })
COLORS == Permutation({ "red_OF_COLOR", "white_OF_COLOR", "blue_OF_COLOR", "green_OF_COLOR", "yellow_OF_COLOR" })
PETS == Permutation({ "bird_OF_PET", "cat_OF_PET", "dog_OF_PET", "fish_OF_PET", "horse_OF_PET" })
CIGARS == Permutation({ "blend_OF_CIGAR", "bm_OF_CIGAR", "dh_OF_CIGAR", "pm_OF_CIGAR", "prince_OF_CIGAR" })

VARIABLES
\* @type: Int -> NATIONALITY;
nationality, \* tuple of nationalities
\* @type: Int -> COLOR;
colors, \* tuple of house colors
\* @type: Int -> PET;
pets, \* tuple of pets
\* @type: Int -> CIGAR;
cigars, \* tuple of cigars
\* @type: Int -> DRINK;
drinks \* tuple of drinks

------------------------------------------------------------
Expand All @@ -70,44 +77,44 @@ VARIABLES
(* Rules *)
(*********)

BritLivesInTheRedHouse == \E i \in 2..5 : nationality[i] = "brit" /\ colors[i] = "red"
BritLivesInTheRedHouse == \E i \in 2..5 : nationality[i] = "brit_OF_NATIONALITY" /\ colors[i] = "red_OF_COLOR"

SwedeKeepsDogs == \E i \in 2..5 : nationality[i] = "swede" /\ pets[i] = "dog"
SwedeKeepsDogs == \E i \in 2..5 : nationality[i] = "swede_OF_NATIONALITY" /\ pets[i] = "dog_OF_PET"

DaneDrinksTea == \E i \in 2..5 : nationality[i] = "dane" /\ drinks[i] = "tea"
DaneDrinksTea == \E i \in 2..5 : nationality[i] = "dane_OF_NATIONALITY" /\ drinks[i] = "tea_OF_DRINK"

GreenLeftOfWhite == \E i \in 1..4 : colors[i] = "green" /\ colors[i + 1] = "white"
GreenLeftOfWhite == \E i \in 1..4 : colors[i] = "green_OF_COLOR" /\ colors[i + 1] = "white_OF_COLOR"

GreenOwnerDrinksCoffee == \E i \in 1..5 \ {3} : colors[i] = "green" /\ drinks[i] = "coffee"
GreenOwnerDrinksCoffee == \E i \in 1..5 \ {3} : colors[i] = "green_OF_COLOR" /\ drinks[i] = "coffee_OF_DRINK"

SmokesPallmallRearsBirds == \E i \in 1..5 : cigars[i] = "pm" /\ pets[i] = "bird"
SmokesPallmallRearsBirds == \E i \in 1..5 : cigars[i] = "pm_OF_CIGAR" /\ pets[i] = "bird_OF_PET"

YellowOwnerSmokesDunhill == \E i \in 1..5 : colors[i] = "yellow" /\ cigars[i] = "dh"
YellowOwnerSmokesDunhill == \E i \in 1..5 : colors[i] = "yellow_OF_COLOR" /\ cigars[i] = "dh_OF_CIGAR"

CenterDrinksMylk == drinks[3] = "mylk"
CenterDrinksMylk == drinks[3] = "mylk_OF_DRINK"

NorwegianFirstHouse == nationality[1] = "norwegian"
NorwegianFirstHouse == nationality[1] = "norwegian_OF_NATIONALITY"

BlendSmokerLivesNextToCatOwner ==
\E i \in 1..4 :
\/ cigars[i] = "blend" /\ pets[i + 1] = "cat"
\/ pets[i] = "cat" /\ cigars[i + 1] = "blend"
\/ cigars[i] = "blend_OF_CIGAR" /\ pets[i + 1] = "cat_OF_PET"
\/ pets[i] = "cat_OF_PET" /\ cigars[i + 1] = "blend_OF_CIGAR"

HorseKeeperLivesNextToDunhillSmoker ==
\E i \in 1..4 :
\/ cigars[i] = "dh" /\ pets[i + 1] = "horse"
\/ pets[i] = "horse" /\ cigars[i + 1] = "dh"
\/ cigars[i] = "dh_OF_CIGAR" /\ pets[i + 1] = "horse_OF_PET"
\/ pets[i] = "horse_OF_PET" /\ cigars[i + 1] = "dh_OF_CIGAR"

BluemasterSmokerDrinksBeer == \E i \in 1..5 : cigars[i] = "bm" /\ drinks[i] = "beer"
BluemasterSmokerDrinksBeer == \E i \in 1..5 : cigars[i] = "bm_OF_CIGAR" /\ drinks[i] = "beer_OF_DRINK"

GermanSmokesPrince == \E i \in 2..5 : nationality[i] = "german" /\ cigars[i] = "prince"
GermanSmokesPrince == \E i \in 2..5 : nationality[i] = "german_OF_NATIONALITY" /\ cigars[i] = "prince_OF_CIGAR"

NorwegianLivesNextToBlueHouse == colors[2] = "blue" \* since the norwegian lives in the first house
NorwegianLivesNextToBlueHouse == colors[2] = "blue_OF_COLOR" \* since the norwegian_OF_NATIONALITY lives in the first house

BlendSmokerHasWaterDrinkingNeighbor ==
\E i \in 1..4 :
\/ cigars[i] = "blend" /\ drinks[i + 1] = "water"
\/ drinks[i] = "water" /\ cigars[i + 1] = "blend"
\/ cigars[i] = "blend_OF_CIGAR" /\ drinks[i + 1] = "water_OF_DRINK"
\/ drinks[i] = "water_OF_DRINK" /\ cigars[i + 1] = "blend_OF_CIGAR"

------------------------------------------------------------

Expand All @@ -116,9 +123,9 @@ BlendSmokerHasWaterDrinkingNeighbor ==
(************)

Init ==
/\ drinks \in { p \in DRINKS : p[3] = "mylk" }
/\ nationality \in { p \in NATIONALITIES : p[1] = "norwegian" }
/\ colors \in { p \in COLORS : p[2] = "blue" }
/\ drinks \in { p \in DRINKS : p[3] = "mylk_OF_DRINK" }
/\ nationality \in { p \in NATIONALITIES : p[1] = "norwegian_OF_NATIONALITY" }
/\ colors \in { p \in COLORS : p[2] = "blue_OF_COLOR" }
/\ pets \in PETS
/\ cigars \in CIGARS

Expand Down Expand Up @@ -146,4 +153,8 @@ Solution ==

FindSolution == ~Solution

\* To find the solution with the `^Apalache^' model-checker, run:
\* apalache-mc check --init=Init --inv=FindSolution --length=0 Einstein.tla
\* Then look for the file violation.tla in `^Apalache^' output directory.
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In case it is helpful, you can use the --run-dir flag to put the output (including the violation) in a known location:

 apalache-mc --help | grep 'run-dir'
   --run-dir            : additional directory wherein output files for this run will be written directly, default: none (overrides envvar RUN_DIR)


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