This case studies tackles encoding the rush hour board game in which cars are packed on a 6-by-6 grid and can either move horizontally or vertically. The goal is to move the red car to the exit. In this particular instance we try to solve the hardest puzzle Nr 40.
MACHINE RushHour /* not a very elegant model; but it seems to work */ /* ProB finds a solution for the hardest puzzle (no. 40) */ DEFINITIONS SET_PREF_MAXINT == 8; /*"RushHour/Puzzle10.def"; */ "RushHour/Puzzle40.def"; INDEX == (1..dim); GOAL == (pos_hcar(red_hcar) >= dim-size_hcar(red_hcar)+1); HEURISTIC_FUNCTION == dim-size_hcar(red_hcar) - pos_hcar(red_hcar) ; /* not a very interesting heuristic function; as red_car can only be moved at very last step */ ANIMATION_IMG0 == "images/sm_empty_box.gif"; ANIMATION_IMG1 == "images/sm_vcar.gif"; ANIMATION_IMG2 == "images/sm_vcar_front.gif"; ANIMATION_IMG3 == "images/sm_hcar.gif"; ANIMATION_IMG4 == "images/sm_red_hcar.gif"; ANIMATION_FUNCTION == ( {r,c,i|r:1..dim & c:1..dim & i=0} <+ {r,c,i|r:1..dim & c:1..dim & i=1 & #j.(j:dom(col_vcar) & c=col_vcar(j) & r>pos_vcar(j) & r<pos_vcar(j)+size_vcar(j)) } <+ {r,c,i|r:1..dim & c:1..dim & i=2 & #j.(j:dom(col_vcar) & c=col_vcar(j) & r=pos_vcar(j)) } <+ {r,c,i|r:1..dim & c:1..dim & i:3..4 & #j.(j:dom(row_hcar) & r=row_hcar(j) & c>=pos_hcar(j) & c<pos_hcar(j)+size_hcar(j) & ((j=red_hcar & i=4) or (j/=red_hcar & i=3)) ) } ); POSs_VCAR(vc) == {c,r|c=col_vcar(vc) & r>=pos_vcar(vc) & r<pos_vcar(vc)+size_vcar(vc)}; POSs_HCAR(hc) == {c,r|r=row_hcar(hc) & c>=pos_hcar(hc) & c<pos_hcar(hc)+size_hcar(hc)} CONSTANTS vcars,hcars,dim, col_vcar, row_hcar, size_vcar, size_hcar, red_hcar PROPERTIES /* The particular puzzle */ STATIC_PROPS & dim = 6 & vcars : NATURAL1 & hcars: NATURAL1 & col_vcar: 1..vcars --> INDEX & row_hcar: 1..hcars --> INDEX & size_vcar: 1..vcars --> INDEX & size_hcar: 1..hcars --> INDEX & red_hcar : 1..hcars & /* vcars are in ascending in row order */ !r.(r:1..(vcars-1) => col_vcar(r)<=col_vcar(r+1)) & /* hcars are in ascending in col order */ !c.(c:1..(hcars-1) => row_hcar(c)<=row_hcar(c+1)) VARIABLES pos_vcar, pos_hcar INVARIANT pos_vcar: 1..vcars --> INDEX & pos_hcar: 1..hcars --> INDEX ASSERTIONS !(vc,hc).(vc:1..vcars & hc:1..hcars => POSs_VCAR(vc) /\ POSs_HCAR(hc) = {}); !(vc1,vc2).(vc1:1..(vcars-1) & vc2:2..vcars & vc1<vc2 => POSs_VCAR(vc1) /\ POSs_VCAR(vc2) = {}); !(hc1,hc2).(hc1:1..(hcars-1) & hc2:2..hcars & hc1<hc2 => POSs_HCAR(hc1) /\ POSs_HCAR(hc2) = {}) INITIALISATION pos_vcar := INIT_VCAR || pos_hcar := INIT_HCAR OPERATIONS move_hcar_right(car) = PRE car:1..hcars & pos_hcar(car)<=dim - size_hcar(car) & /* car not at extreme right */ (car<hcars => (row_hcar(car) /= row_hcar(car+1) or pos_hcar(car+1) > pos_hcar(car)+size_hcar(car))) & !cv.(cv:1..vcars & col_vcar(cv)=pos_hcar(car)+size_hcar(car) => row_hcar(car) /: pos_vcar(cv)..pos_vcar(cv)+size_vcar(cv)-1) THEN pos_hcar(car) := pos_hcar(car)+1 END; move_hcar_left(car) = PRE car:1..hcars & pos_hcar(car)> 1 & /* car not at extreme left */ (car>1 => (row_hcar(car) /= row_hcar(car-1) or pos_hcar(car-1)+size_hcar(car-1) <= pos_hcar(car)-1))& !cv.(cv:1..vcars & col_vcar(cv)=pos_hcar(car)-1 => row_hcar(car) /: pos_vcar(cv)..pos_vcar(cv)+size_vcar(cv)-1) THEN pos_hcar(car) := pos_hcar(car)-1 END; move_vcar_down(car) = PRE car:1..vcars & pos_vcar(car)<=dim - size_vcar(car) & /* car not at extreme bottom */ (car<vcars => (col_vcar(car) /= col_vcar(car+1) or pos_vcar(car+1) > pos_vcar(car)+size_vcar(car))) & !cv.(cv:1..hcars & row_hcar(cv)=pos_vcar(car)+size_vcar(car) => col_vcar(car) /: pos_hcar(cv)..pos_hcar(cv)+size_hcar(cv)-1) THEN pos_vcar(car) := pos_vcar(car)+1 END; move_vcar_up(car) = PRE car:1..vcars & pos_vcar(car)> 1 & /* car not at extreme top */ (car>1 => (col_vcar(car) /= col_vcar(car-1) or pos_vcar(car-1)+size_vcar(car-1) <= pos_vcar(car)-1)) & !cv.(cv:1..hcars & row_hcar(cv)=pos_vcar(car)-1 => col_vcar(car) /: pos_hcar(cv)..pos_hcar(cv)+size_hcar(cv)-1) THEN pos_vcar(car) := pos_vcar(car)-1 END END
The encoding of hardest puzzle Nr 40 in the file RushHour/Puzzle40.def is as follows:
DEFINITIONS /* The particular puzzle (nr. 40) */ STATIC_PROPS == (vcars=7 & hcars = 6 & col_vcar = {1|->1, 2|->2, 3|->3, 4|->3, 5|->4, 6|->5, 7|->6} & size_vcar = {1|->3, 2|->2, 3|->2, 4|->2, 5|->2, 6|->2, 7|->3} & row_hcar = {1|->1, 2|->3, 3|->4, 4|->5, 5|->6, 6|->6} & size_hcar = {1|->2, 2|->2, 3|->3, 4|->2, 5|->2, 6|->2} & red_hcar = 2); /* red hcar */ INIT_VCAR == {1|->1, 2|->2, 3|->2, 4|->5, 5|->4, 6|->1, 7|->2 }; INIT_HCAR == {1|->2, 2|->4, 3|->1, 4|->5, 5|->1, 6|->4}
ProB 1.3.7 took about 26 seconds to solve this puzzle (on my Mac Book Air 1.8 GHz i7; runtimes may vary as ProB uses a randomized depth-first/breadth-first search):
The solution found has 125 steps:
SETUP_CONSTANTS(6,7,6,[1,2,3,3,4,5,6],[1,3,4,5,6,6],[3,2,2,2,2,2,3],[2,2,3,2,2,2],2) INITIALISATION([1,2,2,5,4,1,2],[2,4,1,5,1,4]) move_hcar_right(6) move_vcar_down(5) move_hcar_right(3) move_vcar_down(1) move_vcar_up(7) move_hcar_left(1) move_vcar_down(1) move_hcar_right(3) move_vcar_down(2) move_hcar_right(3) move_vcar_down(2) move_vcar_up(4) move_vcar_up(3) move_hcar_right(5) move_vcar_down(1) move_hcar_left(2) move_vcar_down(6) move_hcar_left(2) move_hcar_left(2) move_vcar_down(3) move_hcar_right(1) move_hcar_right(1) move_hcar_right(1) move_vcar_up(3) move_hcar_right(2) move_vcar_up(1) move_vcar_up(1) move_vcar_up(1) move_hcar_right(2) move_vcar_up(2) move_vcar_up(2) move_hcar_left(5) move_vcar_down(4) move_vcar_up(2) move_hcar_left(3) move_hcar_left(3) move_hcar_left(3) move_hcar_left(2) move_vcar_up(5) move_vcar_up(5) move_vcar_up(5) move_vcar_down(7) move_hcar_right(1) move_hcar_right(3) move_vcar_up(5) move_hcar_right(2) move_vcar_down(1) move_vcar_down(2) move_hcar_right(3) move_vcar_down(2) move_vcar_down(2) move_hcar_left(2) move_vcar_down(5) move_vcar_down(1) move_hcar_left(1) move_vcar_up(7) move_hcar_right(3) move_vcar_up(4) move_hcar_right(5) move_vcar_down(1) move_hcar_left(2) move_vcar_down(3) move_hcar_left(1) move_hcar_left(1) move_hcar_left(1) move_vcar_up(3) move_hcar_right(2) move_vcar_up(1) move_hcar_left(5) move_hcar_left(4) move_vcar_up(5) move_vcar_down(4) move_hcar_right(2) move_vcar_up(2) move_vcar_up(6) move_hcar_right(2) move_vcar_up(1) move_vcar_up(4) move_vcar_up(4) move_vcar_down(2) move_vcar_down(1) move_vcar_down(4) move_vcar_down(3) move_vcar_up(2) move_vcar_up(2) move_hcar_left(6) move_hcar_right(1) move_vcar_up(1) move_vcar_up(1) move_vcar_down(4) move_vcar_down(3) move_vcar_down(2) move_vcar_down(2) move_vcar_up(3) move_vcar_up(4) move_vcar_down(1) move_vcar_down(1) move_vcar_down(4) move_vcar_up(2) move_vcar_up(1) move_hcar_left(3) move_vcar_up(2) move_vcar_down(7) move_vcar_up(1) move_vcar_down(7) move_vcar_down(7) move_vcar_down(2) move_vcar_down(2) move_vcar_up(7) move_vcar_up(7) move_hcar_right(6) move_vcar_up(7) move_vcar_up(2) move_vcar_up(2) move_hcar_left(3) move_vcar_down(7) move_vcar_down(7) move_vcar_down(1) move_vcar_down(1) move_hcar_left(6) move_vcar_down(7) move_vcar_up(1) move_hcar_right(2)
Using our translator to TLA+ we can solve an adapted version of the above model in 11 seconds plus the time for the translation and starting up the translator and TLC (a few seconds). (The model has to be rewritten slightly for TLC, as it's enumeration and constraint solving capabilities are more limited.)
Here is the slightly rewritten version which works with TLC (in addition to ProB):
MACHINE RushHour_TLC /* not a very elegant model; but it seems to work */ /* ProB finds a solution for the hardest puzzle (no. 40) */ DEFINITIONS SET_PREF_MAXINT == 8; /*"RushHour/Puzzle40.def"; */ STATIC_PROPS == (vcars=7 & hcars = 6 & col_vcar = {1|->1, 2|->2, 3|->3, 4|->3, 5|->4, 6|->5, 7|->6} & size_vcar = {1|->3, 2|->2, 3|->2, 4|->2, 5|->2, 6|->2, 7|->3} & row_hcar = {1|->1, 2|->3, 3|->4, 4|->5, 5|->6, 6|->6} & size_hcar = {1|->2, 2|->2, 3|->3, 4|->2, 5|->2, 6|->2} & red_hcar = 2); /* red hcar */ INIT_VCAR == {1|->1, 2|->2, 3|->2, 4|->5, 5|->4, 6|->1, 7|->2 }; INIT_HCAR == {1|->2, 2|->4, 3|->1, 4|->5, 5|->1, 6|->4}; INDEX == (1..dim); GOAL == (pos_hcar(red_hcar) >= dim-size_hcar(red_hcar)+1); HEURISTIC_FUNCTION == dim-size_hcar(red_hcar) - pos_hcar(red_hcar) ; /* not a very interesting heuristic function; as red_car can only be moved at very last step */ ANIMATION_IMG0 == "images/sm_empty_box.gif"; ANIMATION_IMG1 == "images/sm_vcar.gif"; ANIMATION_IMG2 == "images/sm_vcar_front.gif"; ANIMATION_IMG3 == "images/sm_hcar.gif"; ANIMATION_IMG4 == "images/sm_red_hcar.gif"; ANIMATION_FUNCTION == ( {r,c,i|r:1..dim & c:1..dim & i=0} <+ {r,c,i|r:1..dim & c:1..dim & i=1 & #j.(j:dom(col_vcar) & c=col_vcar(j) & r>pos_vcar(j) & r<pos_vcar(j)+size_vcar(j)) } <+ {r,c,i|r:1..dim & c:1..dim & i=2 & #j.(j:dom(col_vcar) & c=col_vcar(j) & r=pos_vcar(j)) } <+ {r,c,i|r:1..dim & c:1..dim & i:3..4 & #j.(j:dom(row_hcar) & r=row_hcar(j) & c>=pos_hcar(j) & c<pos_hcar(j)+size_hcar(j) & ((j=red_hcar & i=4) or (j/=red_hcar & i=3)) ) } ); POSs_VCAR(vc) == {c,r|c=col_vcar(vc) & r : pos_vcar(vc)..(pos_vcar(vc)+size_vcar(vc))-1}; /* r>=pos_vcar(vc) & r<pos_vcar(vc)+size_vcar(vc) */ POSs_HCAR(hc) == {c,r|r=row_hcar(hc) & c : pos_hcar(hc) ..(pos_hcar(hc)+size_hcar(hc))-1} /* c>=pos_hcar(hc) & c<pos_hcar(hc)+size_hcar(hc) */ CONSTANTS vcars,hcars,dim, col_vcar, row_hcar, size_vcar, size_hcar, red_hcar PROPERTIES /* The particular puzzle */ STATIC_PROPS & dim = 6 & vcars : NATURAL1 & hcars: NATURAL1 & col_vcar: 1..vcars --> INDEX & row_hcar: 1..hcars --> INDEX & size_vcar: 1..vcars --> INDEX & size_hcar: 1..hcars --> INDEX & red_hcar : 1..hcars & /* vcars are in ascending in row order */ !r.(r:1..(vcars-1) => col_vcar(r)<=col_vcar(r+1)) & /* hcars are in ascending in col order */ !c.(c:1..(hcars-1) => row_hcar(c)<=row_hcar(c+1)) VARIABLES pos_vcar, pos_hcar INVARIANT pos_vcar: 1..vcars --> INDEX & pos_hcar: 1..hcars --> INDEX ASSERTIONS !(vc,hc).(vc:1..vcars & hc:1..hcars => POSs_VCAR(vc) /\ POSs_HCAR(hc) = {}); !(vc1,vc2).(vc1:1..(vcars-1) & vc2:2..vcars & vc1<vc2 => POSs_VCAR(vc1) /\ POSs_VCAR(vc2) = {}); !(hc1,hc2).(hc1:1..(hcars-1) & hc2:2..hcars & hc1<hc2 => POSs_HCAR(hc1) /\ POSs_HCAR(hc2) = {}) INITIALISATION pos_vcar := INIT_VCAR || pos_hcar := INIT_HCAR OPERATIONS move_hcar_right(car) = PRE car:1..hcars & pos_hcar(car)<=dim - size_hcar(car) & /* car not at extreme right */ (car<hcars => (row_hcar(car) /= row_hcar(car+1) or pos_hcar(car+1) > pos_hcar(car)+size_hcar(car))) & !cv.(cv:1..vcars & col_vcar(cv)=pos_hcar(car)+size_hcar(car) => row_hcar(car) /: pos_vcar(cv)..pos_vcar(cv)+size_vcar(cv)-1) THEN pos_hcar(car) := pos_hcar(car)+1 END; move_hcar_left(car) = PRE car:1..hcars & pos_hcar(car)> 1 & /* car not at extreme left */ (car>1 => (row_hcar(car) /= row_hcar(car-1) or pos_hcar(car-1)+size_hcar(car-1) <= pos_hcar(car)-1))& !cv.(cv:1..vcars & col_vcar(cv)=pos_hcar(car)-1 => row_hcar(car) /: pos_vcar(cv)..pos_vcar(cv)+size_vcar(cv)-1) THEN pos_hcar(car) := pos_hcar(car)-1 END; move_vcar_down(car) = PRE car:1..vcars & pos_vcar(car)<=dim - size_vcar(car) & /* car not at extreme bottom */ (car<vcars => (col_vcar(car) /= col_vcar(car+1) or pos_vcar(car+1) > pos_vcar(car)+size_vcar(car))) & !cv.(cv:1..hcars & row_hcar(cv)=pos_vcar(car)+size_vcar(car) => col_vcar(car) /: pos_hcar(cv)..pos_hcar(cv)+size_hcar(cv)-1) THEN pos_vcar(car) := pos_vcar(car)+1 END; move_vcar_up(car) = PRE car:1..vcars & pos_vcar(car)> 1 & /* car not at extreme top */ (car>1 => (col_vcar(car) /= col_vcar(car-1) or pos_vcar(car-1)+size_vcar(car-1) <= pos_vcar(car)-1)) & !cv.(cv:1..hcars & row_hcar(cv)=pos_vcar(car)-1 => col_vcar(car) /: pos_hcar(cv)..pos_hcar(cv)+size_hcar(cv)-1) THEN pos_vcar(car) := pos_vcar(car)-1 END END