Logic and Program modes

Imandra has two modes: logic mode and program mode. When we launch Imandra Terminal (or an Imandra Jupyter Notebook session), we start off in logic mode.

In the terminal, we can identify that Imandra is in logic mode by the pound sign prompt (#).

In a notebook, we can inspect Imandra's current mode using the #config directive.

In [1]:
#config 1
Out[1]:
----------------------------------------------------------------------------
Configuration component: Mode
----------------------------------------------------------------------------
Value: Logic
Help topics: program, logic
----------------------------------------------------------------------------

While in logic mode, we have access to Imandra's reasoning tools, such as verify and theorem.

In [2]:
let succ n = n + 1
Out[2]:
val succ : Z.t -> Z.t = <fun>
In [3]:
verify (fun n -> succ n > n)
Out[3]:
- : int -> bool = <fun>
Proved
proof
ground_instances0
definitions0
inductions0
search_time
0.019s
details
Expand
smt_stats
rlimit count12
mk bool var5
num allocs2094817
memory5.700000
max memory6.010000
Expand
  • start[0.019s] (:var_0: + 1) > :var_0:
  • simplify

    into
    true
    expansions
    []
    rewrite_steps
      forward_chaining
      • unsat

        (mp (asserted (not true)) (rewrite (= (not true) false)) false)

      A future notebook will summarize the various reasoning tools, and explain when to use which one.

      In logic mode, all definitions -- types, values and functions -- are entered into the logic. We can see all previous events in logic mode by inspecting the #history (aliased to #h).

      In [4]:
      #h;;
      
      Out[4]:
      
      

      All events in session

      • 0. Fun: succ
      • 1. Verify: <expr>

      dependency graph
      In [5]:
      #h succ
      
      Out[5]:
      
      

      Fun: succ

      iml
      definition
      fun (n : int) -> n + 1
      def
      namesucc
      typeint -> int
      recursivefalse
      call signaturesucc (n : int)
      validatedin 0.000s
      locationjupyter cell 2:1,0--18
      hashes
      succde952cc1b41b6d630d1aa1e2782313094ac9167e588e9d2ecee3e0c1bbfa4667

      While in logic mode, we are restricted to a purely functional subset of OCaml, and our recursive functions must terminate.

      If we try to define a non-termating function, for example, Imandra will reject it.

      In [6]:
      let rec bad_repeat x = x :: bad_repeat x
      
      Out[6]:
      val bad_repeat : 'a -> 'a list = <fun>
      File "jupyter cell 6", line 1, characters 0-40:
      Error: Validate: no measure provided, and Imandra cannot guess any.
      Are you sure this function is actually terminating?
      See https://docs.imandra.ai/imandra-docs/notebooks/proving-program-termination/
      
      In [7]:
      #show bad_repeat
      
      Out[7]:
      Unknown element.
      

      For more complex recursive functions, we may need to convince Imandra that the function terminates, for example by defining a "measure". See the notebook Proving Program Termination with Imandra for more details.

      Our logic-mode definitions are allowed to call other definitions only if those other definitions have been admitted into the logic.

      In [8]:
      let say_hi () = print_endline "Hello!"
      
      Out[8]:
      val say_hi : unit -> unit = <fun>
      Error: No function Pervasives/0.print_endline was defined in logic-mode
      

      In order to define such a side-effecting function, we switch to program mode. We do this using the #program directive.

      In [9]:
      #program;;
      
      #config 1;;
      
      Out[9]:
      ----------------------------------------------------------------------------
      Configuration component: Mode
      ----------------------------------------------------------------------------
      Value: Program
      Help topics: program, logic
      ----------------------------------------------------------------------------
      

      In the terminal, we can identify that Imandra is in program mode by the angle bracket prompt (>).

      Now that we are in program mode, we have the full power of OCaml at our fingertips!

      In [10]:
      let say_hi () = print_endline "Hello!"
      
      Out[10]:
      val say_hi : unit -> unit = <fun>
      
      In [11]:
      say_hi ()
      
      Out[11]:
      Hello!
      - : unit = ()
      

      When we switch back to logic mode (using the #logic directive), we can still refer to our program-mode definitions at the top level.

      In [12]:
      #logic;;
      
      say_hi ()
      
      Out[12]:
      Hello!
      - : unit = ()
      

      But we are forbidden from using them in our logic-mode definitions.

      In [13]:
      let say_hi_from_logic_mode () = say_hi ()
      
      Out[13]:
      val say_hi_from_logic_mode : unit -> unit = <fun>
      Error: No function say_hi/6913 was defined in logic-mode
      

      Often, we want to define a type in logic mode and then a related function in program mode, for example a pretty-printer. For this case we can use the [@@program] annotation to define a one-off program-mode function while in logic mode.

      In [14]:
      type person = { name : string; favorite_color : string };;
      
      let print_person (person : person) : string =
        Printf.sprintf "%s prefers %s things" person.name person.favorite_color
      [@@program];;
      
      print_person { name = "Matt"; favorite_color = "green" };;
      
      Out[14]:
      type person = { name : string; favorite_color : string; }
      val print_person : person -> string = <fun>
      - : string = "Matt prefers green things"
      

      Sometimes we also need to use program-mode functions in order to generate logic-mode values, this can be done using the [@@reflect] annotation:

      In [15]:
      let one = Z.of_string "1" [@@reflect]
      
      Out[15]:
      - : unit = ()
      val one : Z.t = 1
      

      This can be useful for several reasons, one of the most common ones is reading logic-mode values from files. Imandra also offers the lower-level facility Imandra.port to port program-mode values into logic-mode locals:

      In [16]:
      let x = print_endline "debug"; 1 [@@program];;
      Imandra.port "y" "x";;
      y;;
      
      Out[16]:
      debug
      val x : Z.t = 1
      val y : Z.t = 1
      - : unit = ()
      - : unit = ()
      - : Z.t = 1
      

      That concludes our overview of Imandra's logic and program modes!