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 (Ordinal.count acts)
sub ordinalOrdinal.Int (Ordinal.count (List.tl acts))
path[not (acts = [])]
proof
detailed proof
ground_instances3
definitions0
inductions0
search_time
0.016s
details
Expand
smt_stats
num checks8
arith-make-feasible21
arith-max-columns16
arith-conflicts1
rlimit count2848
mk clause24
datatype occurs check36
mk bool var120
arith-lower13
datatype splits22
decisions40
propagations27
arith-max-rows6
conflicts13
datatype accessor ax15
datatype constructor ax36
num allocs834569186
final checks9
added eqs144
del clause9
arith eq adapter13
arith-upper16
memory19.660000
max memory19.660000
Expand
  • start[0.016s]
      let (_x_0 : int) = Ordinal.count acts in
      let (_x_1 : action list) = List.tl acts in
      let (_x_2 : int) = Ordinal.count _x_1 in
      not (acts = []) && _x_0 >= 0 && _x_2 >= 0
      ==> _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) = Ordinal.count _x_0 in
    let (_x_2 : int) = Ordinal.count acts in
    (_x_0 = [] || Ordinal.( << ) (Ordinal.Int _x_1) (Ordinal.Int _x_2))
    || not ((not (acts = []) && _x_2 >= 0) && _x_1 >= 0)
    expansions
    []
    rewrite_steps
      forward_chaining
      • unroll
        expr
        (|count_`action list`_2492| acts_2482)
        expansions
        • unroll
          expr
          (|count_`action list`_2492| (|get.::.1_2474| acts_2482))
          expansions
          • unroll
            expr
            (|Ordinal.<<_102| (|Ordinal.Int_93| (|count_`action list`_2492|
                                                …
            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>
            module CX : sig val l : action list end
            
            Instance (after 18 steps, 29.591s):
             let (l : action list) =
              [(Pick (Goat)); CrossRiver; (Drop (Goat)); CrossRiver; (Pick (Cabbage));
               CrossRiver; (Drop (Cabbage)); (Pick (Goat)); CrossRiver; (Drop (Goat));
               (Pick (Wolf)); CrossRiver; (Drop (Wolf)); CrossRiver; (Pick (Goat));
               CrossRiver; (Drop (Goat))]
            
            Instance
            proof attempt
            ground_instances18
            definitions0
            inductions0
            search_time
            29.591s
            details
            Expand
            smt_stats
            num checks37
            arith-make-feasible1
            arith-max-columns4
            rlimit count20071781
            mk clause47589
            datatype occurs check2916
            restarts348
            mk bool var155906
            datatype splits817412
            decisions185066
            propagations3460756
            conflicts46056
            datatype accessor ax8177
            minimized lits492018
            datatype constructor ax138478
            final checks70
            added eqs8888914
            del clause28820
            dyn ack31
            memory192.690000
            max memory193.020000
            num allocs27131488010.000000
            Expand
            • start[29.591s]
                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_60 (|rec_mk.state_2526|
                                   LeftCoast_16
                                   LeftCoast_16
                      …
                  expansions
                  • unroll
                    expr
                    (many_steps_60 (one_step_55 (|rec_mk.state_2526|
                                                  LeftCoast_16
                            …
                    expansions
                    • unroll
                      expr
                      (many_steps_60 (one_step_55 (one_step_55 (|rec_mk.state_2526|
                                                            …
                      expansions
                      • unroll
                        expr
                        (let ((a!1 (one_step_55 (one_step_55 (one_step_55 (|rec_mk.state_2526|
                                                     …
                        expansions
                        • unroll
                          expr
                          (let ((a!1 (one_step_55 (one_step_55 (one_step_55 (|rec_mk.state_2526|
                                                       …
                          expansions
                          • unroll
                            expr
                            (let ((a!1 (one_step_55 (one_step_55 (one_step_55 (|rec_mk.state_2526|
                                                         …
                            expansions
                            • unroll
                              expr
                              (let ((a!1 (one_step_55 (one_step_55 (one_step_55 (|rec_mk.state_2526|
                                                           …
                              expansions
                              • unroll
                                expr
                                (let ((a!1 (one_step_55 (one_step_55 (one_step_55 (|rec_mk.state_2526|
                                                             …
                                expansions
                                • unroll
                                  expr
                                  (let ((a!1 (one_step_55 (one_step_55 (one_step_55 (|rec_mk.state_2526|
                                                               …
                                  expansions
                                  • unroll
                                    expr
                                    (let ((a!1 (one_step_55 (one_step_55 (one_step_55 (|rec_mk.state_2526|
                                                                 …
                                    expansions
                                    • unroll
                                      expr
                                      (let ((a!1 (one_step_55 (one_step_55 (one_step_55 (|rec_mk.state_2526|
                                                                   …
                                      expansions
                                      • unroll
                                        expr
                                        (let ((a!1 (one_step_55 (one_step_55 (one_step_55 (|rec_mk.state_2526|
                                                                     …
                                        expansions
                                        • unroll
                                          expr
                                          (let ((a!1 (one_step_55 (one_step_55 (one_step_55 (|rec_mk.state_2526|
                                                                       …
                                          expansions
                                          • unroll
                                            expr
                                            (let ((a!1 (one_step_55 (one_step_55 (one_step_55 (|rec_mk.state_2526|
                                                                         …
                                            expansions
                                            • unroll
                                              expr
                                              (let ((a!1 (one_step_55 (one_step_55 (one_step_55 (|rec_mk.state_2526|
                                                                           …
                                              expansions
                                              • unroll
                                                expr
                                                (let ((a!1 (one_step_55 (one_step_55 (one_step_55 (|rec_mk.state_2526|
                                                                             …
                                                expansions
                                                • unroll
                                                  expr
                                                  (let ((a!1 (one_step_55 (one_step_55 (one_step_55 (|rec_mk.state_2526|
                                                                               …
                                                  expansions
                                                  • unroll
                                                    expr
                                                    (let ((a!1 (one_step_55 (one_step_55 (one_step_55 (|rec_mk.state_2526|
                                                                                 …
                                                    expansions
                                                    • Sat (Some let (l : action list) = [(Pick (Goat)); CrossRiver; (Drop (Goat)); CrossRiver; (Pick (Cabbage)); CrossRiver; (Drop (Cabbage)); (Pick (Goat)); CrossRiver; (Drop (Goat)); (Pick (Wolf)); CrossRiver; (Drop (Wolf)); CrossRiver; (Pick (Goat)); CrossRiver; (Drop (Goat))] )

                                                    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, 0.171s):
                                                      let l =
                                                        let (_x_0 : action) = Pick Goat in
                                                        let (_x_1 : action) = Drop Goat in
                                                        [_x_0; CrossRiver; _x_1; CrossRiver; Pick Cabbage; CrossRiver;
                                                         Drop Cabbage; _x_0; CrossRiver; _x_1; Pick Wolf; CrossRiver; Drop Wolf;
                                                         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.