Crossing the River Safely

For the sake of brain scrambling, we're going to solve this ancient puzzle using Imandra (again!). As most polyvalent farmers will tell you, going to the market with your pet wolf, tastiest goat, and freshest cabbage is sometimes difficult as they tend to have appetite for one another. The good news is that there is a way to cross this river safely anyway.

First we should define the problem by tallying our goods and looking around.

In [1]:
type location =
  | Boat
  | LeftCoast
  | RightCoast
  | Eaten

type boat =
  | Left
  | Right

type good = Cabbage | Goat | Wolf
Out[1]:
type location = Boat | LeftCoast | RightCoast | Eaten
type boat = Left | Right
type good = Cabbage | Goat | Wolf

This problem is delicate and will require multiple steps to be solved. Each step should take us from a state to another state (where, hopefully, no cabbage nor goat was hurt).

In [2]:
type state = {
  cabbage : location;
  goat : location;
  wolf : location;
  boat : boat;
}
Out[2]:
type state = {
  cabbage : location;
  goat : location;
  wolf : location;
  boat : boat;
}

We can define a few helpers:

In [3]:
let get_location (s:state) (g:good) = match g with
  | Cabbage -> s.cabbage
  | Goat -> s.goat
  | Wolf -> s.wolf

let set_location (s:state) (g:good) (l:location) = match g with
  | Cabbage -> { s with cabbage = l}
  | Goat    -> { s with goat    = l}
  | Wolf    -> { s with wolf    = l}

let boat_empty (s:state) =
  (s.cabbage <> Boat) &&
  (s.goat    <> Boat) &&
  (s.wolf    <> Boat)
Out[3]:
val get_location : state -> good -> location = <fun>
val set_location : state -> good -> location -> state = <fun>
val boat_empty : state -> bool = <fun>

Now, transition from a state to the next one is done via actions:

In [4]:
type action =
  | Pick of good
  | Drop of good
  | CrossRiver

let process_action (s:state) (m:action) : state =
  match m with
  | CrossRiver -> { s with boat = match s.boat with Left -> Right | Right -> Left }
  | Pick x -> begin
    if not @@ boat_empty s then s else
    match get_location s x, s.boat with
    |  LeftCoast ,  Left -> set_location s x Boat
    | RightCoast , Right -> set_location s x Boat
    | _ -> s
  end
  | Drop x -> begin
    match get_location s x, s.boat with
    | Boat ,  Left -> set_location s x LeftCoast
    | Boat , Right -> set_location s x RightCoast
    | _ -> s
  end
;;

let process_eating s =
  match s.boat, s.cabbage, s.goat, s.wolf with
  | Right, LeftCoast, LeftCoast, _    -> Some { s with cabbage = Eaten }
  | Right, _ , LeftCoast, LeftCoast   -> Some { s with goat = Eaten }
  |  Left, RightCoast, RightCoast, _  -> Some { s with cabbage = Eaten }
  |  Left, _ , RightCoast, RightCoast -> Some { s with goat = Eaten }
  | _  -> None

(* is… it a bad state? *)
let anything_eaten s =
  s.cabbage = Eaten || s.goat = Eaten

let one_step s a =
  if anything_eaten s then s
  else
    match process_eating s with
    | Some s -> s
    | None ->
      process_action s a

(* process many actions. Note that we have to specify that [acts] is
   the argument that proves termination *)
let rec many_steps s acts =
  match acts with
  | [] -> s
  | a :: acts ->
    let s' = one_step s a in
    many_steps s' acts
[@@adm 1n]


let solved s =
  s.cabbage = RightCoast
  && s.goat = RightCoast
  && s.wolf = RightCoast
  && s.boat = Right
;;
Out[4]:
type action = Pick of good | Drop of good | CrossRiver
val process_action : state -> action -> state = <fun>
val process_eating : state -> state option = <fun>
val anything_eaten : state -> bool = <fun>
val one_step : state -> action -> state = <fun>
val many_steps : state -> action list -> state = <fun>
val solved : state -> bool = <fun>
termination proof

Termination proof

call `many_steps (one_step s (List.hd acts)) (List.tl acts)` from `many_steps s acts`
originalmany_steps s acts
submany_steps (one_step s (List.hd acts)) (List.tl acts)
original ordinalOrdinal.Int (_cnt acts)
sub ordinalOrdinal.Int (_cnt (List.tl acts))
path[not Is_a([], acts)]
proof
detailed proof
ground_instances3
definitions0
inductions0
search_time
0.028s
details
Expand
smt_stats
num checks8
arith-make-feasible22
arith-max-columns14
arith-conflicts1
rlimit count2933
mk clause39
datatype occurs check23
mk bool var140
arith-lower16
datatype splits27
decisions49
propagations50
arith-max-rows3
conflicts14
datatype accessor ax23
datatype constructor ax50
num allocs1260782
final checks6
added eqs207
del clause20
arith eq adapter16
arith-upper19
memory7.200000
max memory7.200000
Expand
  • start[0.028s]
      let (_x_0 : int) = count.list count.action acts in
      let (_x_1 : action list) = List.tl acts in
      let (_x_2 : int) = count.list count.action _x_1 in
      not Is_a([], acts) && _x_0 >= 0 && _x_2 >= 0
      ==> Is_a([], _x_1) || Ordinal.( << ) (Ordinal.Int _x_2) (Ordinal.Int _x_0)
  • simplify
    into
    let (_x_0 : action list) = List.tl acts in
    let (_x_1 : int) = count.list count.action _x_0 in
    let (_x_2 : int) = count.list count.action acts in
    (Is_a([], _x_0) || Ordinal.( << ) (Ordinal.Int _x_1) (Ordinal.Int _x_2))
    || not ((not Is_a([], acts) && _x_2 >= 0) && _x_1 >= 0)
    expansions
    []
    rewrite_steps
      forward_chaining
      • unroll
        expr
        (|`count.list count.action[0]`_1733| acts_1721)
        expansions
        • unroll
          expr
          (|`count.list count.action[0]`_1733| (|get.::.1_1717| acts_1721))
          expansions
          • unroll
            expr
            (|Ordinal.<<_126| (|Ordinal.Int_111|
                                (|`count.list count.action[0]`_1733|
                  …
            expansions
            • Unsat

            In [5]:
            (* initial state, on the west bank of Anduin with empty pockets and fuzzy side-kicks *)
            let init_state = {
              cabbage = LeftCoast;
              goat = LeftCoast;
              wolf = LeftCoast;
              boat = Left;
            }
            
            Out[5]:
            val init_state : state =
              {cabbage = LeftCoast; goat = LeftCoast; wolf = LeftCoast; boat = Left}
            

            We are now ready to ask for a solution! Because we're looking for a given solution rather than a universal proof, instance is the most natural.

            In [6]:
            #timeout 10_000;;
            
            instance (fun l -> solved @@ many_steps init_state l) ;;
            
            Out[6]:
            - : action list -> bool = <fun>
            
            Unknown (Verification aborted by the solver (after 16 steps))
            Expand
            expanded
            • let (_x_0 : action list) = List.tl (List.tl (List.tl l)) in let (_x_1 : action list) = List.tl _x_0 in let (_x_2 : action list) = List.tl _x_1 in let (_x_3 : action list) = List.tl _x_2 in let (_x_4 : action list) = List.tl _x_3 in let (_x_5 : action list) = List.tl _x_4 in let (_x_6 : action list) = List.tl _x_5 in let (_x_7 : action list) = List.tl _x_6 in many_steps (one_step (one_step (one_step (one_step (one_step (one_step (one_step (one_step (one_step … …) (List.hd _x_0)) (List.hd _x_1)) (List.hd _x_2)) (List.hd _x_3)) (List.hd _x_4)) (List.hd _x_5)) (List.hd _x_6)) (List.hd _x_7)) (List.tl _x_7)
            • let (_x_0 : action list) = List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl l))))))))))) in let (_x_1 : action list) = List.tl _x_0 in many_steps (one_step (one_step (one_step … …) (List.hd _x_0)) (List.hd _x_1)) (List.tl _x_1)
            • let (_x_0 : action list) = List.tl (List.tl (List.tl l)) in let (_x_1 : action list) = List.tl _x_0 in let (_x_2 : action list) = List.tl _x_1 in let (_x_3 : action list) = List.tl _x_2 in let (_x_4 : action list) = List.tl _x_3 in let (_x_5 : action list) = List.tl _x_4 in many_steps (one_step (one_step (one_step (one_step (one_step (one_step (one_step … …) (List.hd _x_0)) (List.hd _x_1)) (List.hd _x_2)) (List.hd _x_3)) (List.hd _x_4)) (List.hd _x_5)) (List.tl _x_5)
            • let (_x_0 : action list) = List.tl (List.tl (List.tl l)) in let (_x_1 : action list) = List.tl _x_0 in let (_x_2 : action list) = List.tl _x_1 in let (_x_3 : action list) = List.tl _x_2 in let (_x_4 : action list) = List.tl _x_3 in let (_x_5 : action list) = List.tl _x_4 in let (_x_6 : action list) = List.tl _x_5 in many_steps (one_step (one_step (one_step (one_step (one_step (one_step (one_step (one_step … …) (List.hd _x_0)) (List.hd _x_1)) (List.hd _x_2)) (List.hd _x_3)) (List.hd _x_4)) (List.hd _x_5)) (List.hd _x_6)) (List.tl _x_6)
            • let (_x_0 : action list) = List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl l))))))))))) in let (_x_1 : action list) = List.tl _x_0 in let (_x_2 : action list) = List.tl _x_1 in many_steps (one_step (one_step (one_step (one_step … …) (List.hd _x_0)) (List.hd _x_1)) (List.hd _x_2)) (List.tl _x_2)
            • let (_x_0 : action list) = List.tl l in let (_x_1 : action list) = List.tl _x_0 in many_steps (one_step (one_step (one_step {cabbage = LeftCoast; goat = LeftCoast; wolf = LeftCoast; boat = Left} (List.hd l)) (List.hd _x_0)) (List.hd _x_1)) (List.tl _x_1)
            • let (_x_0 : action list) = List.tl (List.tl (List.tl l)) in let (_x_1 : action list) = List.tl _x_0 in let (_x_2 : action list) = List.tl _x_1 in let (_x_3 : action list) = List.tl _x_2 in let (_x_4 : action list) = List.tl _x_3 in let (_x_5 : action list) = List.tl _x_4 in let (_x_6 : action list) = List.tl _x_5 in let (_x_7 : action list) = List.tl _x_6 in let (_x_8 : action list) = List.tl _x_7 in many_steps (one_step (one_step (one_step (one_step (one_step (one_step (one_step (one_step (one_step (one_step … …) (List.hd _x_0)) (List.hd _x_1)) (List.hd _x_2)) (List.hd _x_3)) (List.hd _x_4)) (List.hd _x_5)) (List.hd _x_6)) (List.hd _x_7)) (List.hd _x_8)) (List.tl _x_8)
            • let (_x_0 : action list) = List.tl (List.tl (List.tl l)) in let (_x_1 : action list) = List.tl _x_0 in let (_x_2 : action list) = List.tl _x_1 in let (_x_3 : action list) = List.tl _x_2 in let (_x_4 : action list) = List.tl _x_3 in let (_x_5 : action list) = List.tl _x_4 in let (_x_6 : action list) = List.tl _x_5 in let (_x_7 : action list) = List.tl _x_6 in let (_x_8 : action list) = List.tl _x_7 in let (_x_9 : action list) = List.tl _x_8 in many_steps (one_step (one_step (one_step (one_step (one_step (one_step (one_step (one_step (one_step (one_step (one_step … …) (List.hd _x_0)) (List.hd _x_1)) (List.hd _x_2)) (List.hd _x_3)) (List.hd _x_4)) (List.hd _x_5)) (List.hd _x_6)) (List.hd _x_7)) (List.hd _x_8)) (List.hd _x_9)) (List.tl _x_9)
            • let (_x_0 : action list) = List.tl (List.tl (List.tl l)) in let (_x_1 : action list) = List.tl _x_0 in let (_x_2 : action list) = List.tl _x_1 in many_steps (one_step (one_step (one_step (one_step … …) (List.hd _x_0)) (List.hd _x_1)) (List.hd _x_2)) (List.tl _x_2)
            • many_steps (one_step {cabbage = LeftCoast; goat = LeftCoast; wolf = LeftCoast; boat = Left} (List.hd l)) (List.tl l)
            • let (_x_0 : action list) = List.tl (List.tl (List.tl l)) in let (_x_1 : action list) = List.tl _x_0 in let (_x_2 : action list) = List.tl _x_1 in let (_x_3 : action list) = List.tl _x_2 in let (_x_4 : action list) = List.tl _x_3 in many_steps (one_step (one_step (one_step (one_step (one_step (one_step … …) (List.hd _x_0)) (List.hd _x_1)) (List.hd _x_2)) (List.hd _x_3)) (List.hd _x_4)) (List.tl _x_4)
            • let (_x_0 : action list) = List.tl l in many_steps (one_step (one_step {cabbage = LeftCoast; goat = LeftCoast; wolf = LeftCoast; boat = Left} (List.hd l)) (List.hd _x_0)) (List.tl _x_0)
            • let (_x_0 : action list) = List.tl (List.tl (List.tl l)) in let (_x_1 : action list) = List.tl _x_0 in let (_x_2 : action list) = List.tl _x_1 in let (_x_3 : action list) = List.tl _x_2 in many_steps (one_step (one_step (one_step (one_step (one_step … …) (List.hd _x_0)) (List.hd _x_1)) (List.hd _x_2)) (List.hd _x_3)) (List.tl _x_3)
            • let (_x_0 : action list) = List.tl l in let (_x_1 : action list) = List.tl _x_0 in let (_x_2 : action list) = List.tl _x_1 in many_steps (one_step (one_step (one_step (one_step {cabbage = LeftCoast; goat = LeftCoast; wolf = LeftCoast; boat = Left} (List.hd l)) (List.hd _x_0)) (List.hd _x_1)) (List.hd _x_2)) (List.tl _x_2)
            • let (_x_0 : action list) = List.tl (List.tl (List.tl l)) in let (_x_1 : action list) = List.tl _x_0 in many_steps (one_step (one_step (one_step … …) (List.hd _x_0)) (List.hd _x_1)) (List.tl _x_1)
            • many_steps {cabbage = LeftCoast; goat = LeftCoast; wolf = LeftCoast; boat = Left} l
            blocked
            • let (_x_0 : action list) = List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl l))))))))))) in let (_x_1 : action list) = List.tl _x_0 in let (_x_2 : action list) = List.tl _x_1 in let (_x_3 : action list) = List.tl _x_2 in many_steps (one_step (one_step (one_step (one_step (one_step … …) (List.hd _x_0)) (List.hd _x_1)) (List.hd _x_2)) (List.hd _x_3)) (List.tl _x_3)
            proof attempt
            ground_instances16
            definitions0
            inductions0
            search_time
            45.490s
            Expand
            • start[45.490s]
                let (_x_0 : state)
                    = many_steps
                      {cabbage = LeftCoast; goat = LeftCoast; wolf = LeftCoast; boat = Left}
                      ( :var_0: )
                in
                _x_0.cabbage = RightCoast
                && _x_0.goat = RightCoast && _x_0.wolf = RightCoast && _x_0.boat = Right
            • simplify

              into
              let (_x_0 : state)
                  = many_steps
                    {cabbage = LeftCoast; goat = LeftCoast; wolf = LeftCoast; boat = Left}
                    ( :var_0: )
              in
              ((_x_0.cabbage = RightCoast && _x_0.goat = RightCoast)
               && _x_0.wolf = RightCoast)
              && _x_0.boat = Right
              expansions
              []
              rewrite_steps
                forward_chaining
                • unroll
                  expr
                  (many_steps_96 (|rec_mk.state_1729|
                                   LeftCoast_16
                                   LeftCoast_16
                      …
                  expansions
                  • unroll
                    expr
                    (many_steps_96 (one_step_91 (|rec_mk.state_1729|
                                                  LeftCoast_16
                            …
                    expansions
                    • unroll
                      expr
                      (many_steps_96 (one_step_91 (one_step_91 (|rec_mk.state_1729|
                                                            …
                      expansions
                      • unroll
                        expr
                        (let ((a!1 (one_step_91 (one_step_91 (one_step_91 (|rec_mk.state_1729|
                                                     …
                        expansions
                        • unroll
                          expr
                          (let ((a!1 (one_step_91 (one_step_91 (one_step_91 (|rec_mk.state_1729|
                                                       …
                          expansions
                          • unroll
                            expr
                            (let ((a!1 (one_step_91 (one_step_91 (one_step_91 (|rec_mk.state_1729|
                                                         …
                            expansions
                            • unroll
                              expr
                              (let ((a!1 (one_step_91 (one_step_91 (one_step_91 (|rec_mk.state_1729|
                                                           …
                              expansions
                              • unroll
                                expr
                                (let ((a!1 (one_step_91 (one_step_91 (one_step_91 (|rec_mk.state_1729|
                                                             …
                                expansions
                                • unroll
                                  expr
                                  (let ((a!1 (one_step_91 (one_step_91 (one_step_91 (|rec_mk.state_1729|
                                                               …
                                  expansions
                                  • unroll
                                    expr
                                    (let ((a!1 (one_step_91 (one_step_91 (one_step_91 (|rec_mk.state_1729|
                                                                 …
                                    expansions
                                    • unroll
                                      expr
                                      (let ((a!1 (one_step_91 (one_step_91 (one_step_91 (|rec_mk.state_1729|
                                                                   …
                                      expansions
                                      • unroll
                                        expr
                                        (let ((a!1 (one_step_91 (one_step_91 (one_step_91 (|rec_mk.state_1729|
                                                                     …
                                        expansions
                                        • unroll
                                          expr
                                          (let ((a!1 (one_step_91 (one_step_91 (one_step_91 (|rec_mk.state_1729|
                                                                       …
                                          expansions
                                          • unroll
                                            expr
                                            (let ((a!1 (one_step_91 (one_step_91 (one_step_91 (|rec_mk.state_1729|
                                                                         …
                                            expansions
                                            • unroll
                                              expr
                                              (let ((a!1 (one_step_91 (one_step_91 (one_step_91 (|rec_mk.state_1729|
                                                                           …
                                              expansions
                                              • unroll
                                                expr
                                                (let ((a!1 (one_step_91 (one_step_91 (one_step_91 (|rec_mk.state_1729|
                                                                             …
                                                expansions

                                                That seems to take a bit of time, because this problem is not that easy for Imandra's unrolling algorithm. Let's try [@@blast] to see if we can get a result faster:

                                                In [7]:
                                                instance (fun l -> solved @@ many_steps init_state l)
                                                [@@blast] ;;
                                                
                                                Out[7]:
                                                - : action list -> bool = <fun>
                                                module CX : sig val l : action list end
                                                
                                                Instance (after 72 steps, 0.136s):
                                                  let l =
                                                    let (_x_0 : action) = Pick Goat in
                                                    let (_x_1 : action) = Drop Goat in
                                                    [_x_0; CrossRiver; _x_1; CrossRiver; Pick Wolf; CrossRiver; Drop Wolf;
                                                     _x_0; CrossRiver; _x_1; Pick Cabbage; CrossRiver; Drop Cabbage;
                                                     CrossRiver; _x_0; CrossRiver; _x_1]
                                                
                                                Instance

                                                It only took a fraction of second! 🎉

                                                Now we have a clear plan for crossing the river. How to sell the goat and cabbage is left as an exercise to the reader.