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:
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 nnum_pins >= 1 then Inplay (n  num_pins) else Inplay n
;;
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.
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 [1]
else []
 Terminal _ > [];;
let is_valid_choice choice state =
choice <=3 && choice >=1 && match state with
 Inplay s > s > choice
 Terminal _ > false;;
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 nonterminal states, every possible opponent play is calcuated using the function find_all_available_choices
to calculate all the next possible states.
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
;;
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.
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;;
Now we can exploit Imandra's technology to find a solution for the game  in this case using [@@blast]
to find the solution:
Playing against Imandra¶
Now this is a strategy for the game, we can write a simple game player to play against.
[@@@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 (l1)) 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
print_endline "Enter your choices";
let user_choice = gather_inputs (match next_state with Terminal _ > 0  Inplay n > n1 ) 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
;;
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: [3]

Enter your choices
1
You played: [1]

Imandra plays: [3]

Enter your choices
2
You played: [2]

Imandra plays: [2]

Enter your choices
1
You played: [1]

Imandra plays: [3]

Imandra wins