# Synthesising a Game Solver in Imandra¶

In this notebook we introduce a simple game called "Les Bâtonnets Géants", and show how imandra can be exploited to synthesise a strategy which always wins. The game itself consists of 16 pegs, and opponents take turns in taking 1,2 or 3 pegs from the end. The loser is the player with 1 remaining peg a their turn. ## Game specific rules¶

Let us first set up a very simple representation of this game consisting of a state which is either in play with n pieces or ended with a winner:

In :
type choice = int
;;

type player =
| Imandra
| Opponent
;;

type state =
| Inplay of int
| Terminal of player
;;

let create_initial_state n =
Inplay n;;

let final_state n =
n=1
;;

let remove_pins num_pins state =
match state with
| Terminal p -> Terminal p
| Inplay n -> if n-num_pins >= 1 then Inplay (n - num_pins) else Inplay n
;;

Out:
type choice = Z.t
type player = Imandra | Opponent
type state = Inplay of choice | Terminal of player
val create_initial_state : choice -> state = <fun>
val final_state : choice -> bool = <fun>
val remove_pins : choice -> state -> state = <fun>


Now let us also set up simple functions which describe a "move" in the game, given by the function step, and two functions which find all the valid choices for a given state, and determine if a move is valid for a given state.

In :
let step choice state =
if choice <=0 || choice >3 then state else remove_pins choice state
;;

let find_all_available_choices state =
match state with
| Inplay n ->
if n>3 then [1;2;3] else if n>2 then [2;1] else if n>1 then 
else []
| Terminal _ -> [];;

let is_valid_choice choice state =
choice <=3 && choice >=1 && match state with
| Inplay s -> s > choice
| Terminal _ -> false;;

Out:
val step : choice -> state -> state = <fun>
val find_all_available_choices : state -> choice list = <fun>
val is_valid_choice : choice -> state -> bool = <fun>


Above are the specific functions we need for this game. In what follows a generalised architecture for solving adversarial games is given - this could be viewed as similar to a Functor structure in OCaml where the specific functions given for game of Batonnets Géants are those given above.

## General solver synthesis functions¶

We introduce first a function called one_step which assumes the player of the game is Imandra. The function takes a list of possible states as well as a map between states and choices. For each list of states the choice is played, resulting in a new list of states. If any of these states are in a final state they become "annealed" to the Terminal variant of state declaring Imandra as the winner. For any non-terminal states, every possible opponent play is calcuated using the function find_all_available_choices to calculate all the next possible states.

In :
let one_step (choice_map: (state*choice) list) (states:state list): state list =
let new_states =
List.fold_left (fun acc el ->
match List.find (fun (x,_) -> x=el) choice_map with
| None -> el::acc
| Some (_,choice) ->
let a = step choice el in if List.mem a acc then acc else
(step choice el)::acc) [] states in
let annealed_states = List.map (fun x ->
match x with
| Inplay l ->
if final_state l then Terminal Imandra else Inplay l
| Terminal p -> Terminal p
) new_states in
List.fold_left (fun acc el ->
match el with
| Terminal p -> (Terminal p)::acc
| Inplay l ->
let next_states = List.fold_left (fun acc el ->
(step el (Inplay l))::acc
) [] (find_all_available_choices (Inplay l)) in
List.fold_left (fun acc el -> if List.mem el acc then acc else el::acc) acc next_states
) [] annealed_states
;;

Out:
val one_step : (state * choice) list -> state list -> state list = <fun>


## Using Imandra to synthesise a solver¶

We now introduce a function which takes an initial state and a set of steps and returns true if every resulting list of states is a winning state for Imandra.

In :
let init_state = create_initial_state 16;;

let instance_function init_state steps =
let states,validity_cond = List.fold_left (fun acc el ->
match acc with
| first,second ->
(one_step el first,
second && (
let fsts = List.map fst el in
fsts=first &&
List.for_all (
fun (s,c) ->
match List.find (fun (x,_) -> x = s) el with
| None -> false
| Some (s,c) -> is_valid_choice c s
) el
)
)) ([init_state],true) steps in
validity_cond &&
List.for_all (fun x ->
match x with
| Terminal Imandra -> true
| _ -> false) states;;

Out:
val init_state : state = Inplay 16
val instance_function : state -> (state * choice) list list -> bool = <fun>


Now we can exploit Imandra's technology to find a solution for the game - in this case using [@@blast] to find the solution:

In :
instance (fun steps ->
instance_function init_state steps) [@@blast];;

Out:
- : (state * choice) list list -> bool = <fun>
module CX : sig val steps : (state * choice) list list end

Instance (after 466 steps, 8.956s):
let steps : (state * choice) list list =
[[(Inplay 16, 3)]; [(Inplay 12, 3); (Inplay 11, 2); (Inplay 10, 1)];
[(Inplay 8, 3); (Inplay 7, 2); (Inplay 6, 1)];
[(Inplay 4, 3); (Inplay 3, 2); (Inplay 2, 1)]]

Instance

## Playing against Imandra¶

Now this is a strategy for the game, we can write a simple game player to play against.

In :
[@@@program]
let rec gather_inputs max =
let user_input = read_line () in
if user_input = "" then gather_inputs max else
let n = String.to_nat user_input in
match n with
| None -> gather_inputs max
| Some n ->
if n <=0 || n >max then gather_inputs max else
n;;

let winner_message p =
match p with Imandra -> "imandra wins " | Opponent -> "you win";;

let print_state state =
let rec print_state_aux l =
if l <=0 then "\n"
else "|"^(print_state_aux (l-1)) in
match state with
| Terminal p ->
winner_message p
| Inplay l ->
print_state_aux l;;

let print_choice choice =
String.of_int choice;;

let rec play_against_imandra state solver  =
match state with
| Terminal p ->
print_endline (winner_message p)
| Inplay l ->
if final_state l then
print_endline (winner_message Imandra)
else
begin
match solver with
| [] -> ()
| h::t ->
begin
match List.find (fun (x,_) -> x = state) h with
| None -> print_endline "Solver error"; ()
| Some (_,choice) ->
begin
let next_state = step choice state in
print_endline ("Imandra plays: "^ (print_choice choice)^"\n");
print_endline (print_state next_state);
if (match next_state with |Inplay l -> final_state l | _ -> false)then
print_endline "Imandra wins\n"else
let user_choice = gather_inputs (match next_state with Terminal _ -> 0 | Inplay n -> n-1 ) in
if is_valid_choice user_choice next_state
then
let next_state = step user_choice next_state in
print_endline ("You played: "^ (print_choice user_choice)^"\n");
print_endline (print_state next_state);
if (match next_state with |Inplay l -> final_state l | _ -> false)then
print_endline "You win\n"else
play_against_imandra next_state t
else
print_endline "invalid choices - replaying...";
play_against_imandra state solver
end
end
end
;;

Out:
val gather_inputs : choice -> choice = <fun>
val winner_message : player -> string = <fun>
val print_state : state -> string = <fun>
val print_choice : choice -> string = <fun>
val play_against_imandra : state -> (state * choice) list list -> unit =
<fun>


By invoking the following code in program mode it is possible to play against imandra, but never win:

let play () =
print_endline (print_state init_state);
play_against_imandra init_state CX.steps
;;

play ();;

An example trace is:

  play ();;
||||||||||||||||

Imandra plays: 

|||||||||||||

1
You played: 

||||||||||||

Imandra plays: 

|||||||||

2
You played: 

|||||||

Imandra plays: 

|||||

Imandra wins