Thinking in States


If you are used to imperative programming languages and then learn a functional or logic programming language, you may first work successfully through a few easy exercises and then suddenly find yourself unable to solve apparently simple tasks in the declarative language. You may for example find yourself asking: "How do I even increase the value of a variable?", or "How do I even remove an element from a list?", or more generally "How do I even apply a simple transformation to this data structure?" etc.

Invariably, the solution to such problems is to think in terms of relations between different entities. Examples of such entities are variables, lists, trees, states etc.

For example, in an imperative language, changing the state of a variable is easy:
i = i + 1;
After such a statement is executed, the state of i has changed. The old state of i is no longer accessible. Note that declaratively, the equation makes no sense: There is no number i that equals i+1.

In Prolog, the above snippet could become:
I #= I0 + 1
This means that I and I0 are in a certain relation. In this case, (#=)/2 denotes the equivalence relation of integer expressions.

Importantly, such a relation can be used in all directions. The way corresponding to the imperative way shown above would be to have I0 instantiated to a concrete integer, and let I denote the next integer. Due to the generality of Prolog, the same snippet can also be used if I is instantiated and I0 is not known, and even if both of them are still variables. The mental leap you have to perform to benefit from this generality is to think in terms of two variables instead of one. This is necessary because the same variable cannot reflect two different states, old and new, at the same time.

As another example, when you find yourself asking "How do I even remove an element E from a list?", think declaratively and describe a relation between two lists: One list may contain the element E, and the second list contains all elements of the first list which are not equal to E. Actually, as you already see, we are describing a relation between three entities in this case: Two lists, and an element. You can express this relation in Prolog by stating the conditions that make the relation hold:
list1_element_list2([], _, []).
list1_element_list2([E|Ls1], E, Ls2) :-
        list1_element_list2(Ls1, E, Ls2).
list1_element_list2([L|Ls1], E, [L|Ls2]) :-
        dif(L, E),
        list1_element_list2(Ls1, E, Ls2).
This relation has one argument for each of these entities, and we can read each clause declaratively. For example, the first clause means: The relation holds if the first and third argument are the empty list. The second clause means: If the relation holds for Ls1, E and Ls2, then the relation holds for [E|Ls1], E and Ls2. The third clause is read analogously, with the added constraint that it only holds if L and E are different terms. The predicate is usable in all directions: It can answer much more general questions than just "What does a list look like if we remove all occurrences of the element E?". You can also use it to answer for example: "Which element, if any, has been removed in a given example?", or to answer the most general query: "For which 3 entities does this relation even hold?". This generality is the reason why an imperative name like "remove/3" would not be a good fit in this case.

Prolog provides several language constructs to make predicates general, efficient and concise. For example, using the meta-predicate tfilter/3 from library(reif), we can write list1_element_list2/3 equivalently as:
list1_element_list2(Ls1, E, Ls2) :-
        tfilter(dif(E), Ls1, Ls2).
This version is deterministic if all arguments are sufficiently instantiated. For example:
?- list1_element_list2([a,b,c], b, Ls).
Ls = [a, c].
As yet another example, when you find yourself asking: "How do I even apply a transformation to a tree?", again think declaratively and describe a relation between two trees: one tree before the transformation and one tree after the transformation.

Notice that functions are a special kind of relations, so most of the things below hold for functional as well as logic programming languages. Logic programming languages typically allow for more general solutions with less effort, since predicates can often be used in several directions.

In this text, we will see several examples of relations between states: states in puzzles, states in programs, states in compilers etc., so that you see how various tasks can be expressed in declarative languages. The core idea is the same in all these examples: We think in terms of relations between states. This is in contrast to imperative languages, where we often think in terms of destructive modifications to a state.

Global states

If you are used to thinking in terms of imperative programming languages, you will try to find ways to manipulate global states also in declarative languages. For example, you may try to query and change global variables in Prolog.

Prolog does support changing the global state by various means. For example, we can destructively change the global database in several ways. However, if a Prolog program changes a global state by setting a global variable or modifying the global database, important properties we expect from logic programs may break. Such programs may no longer be usable in all directions, may yield different results for identical queries, and can typically no longer be tested and used in isolation from other program fragments that prepare or clean up these global states. For these reasons, this is not the kind of state we discuss in this text.

To fully benefit from the advantages of pure Prolog programs, you should always aim to find declarative ways to express changes in states. The declarative way to reason about changes is to describe the relations between states that are induced by such changes.

States in puzzles

The choice of state representation can significantly influence how elegantly you can describe a task. Consider a simple puzzle to see this:
Given water jugs A, B and C of respective capacities 8, 5 and 3 and respective fill states full, empty and empty, measure exactly 4 units into both A and B.

Clearly, an important state in this puzzle is the amount of water in each jug. Using Haskell, let us represent this state as a triple (A,B,C):
type State = (Int,Int,Int)
Now, we are to find a sequence of transfers leading from state (8,0,0) to state (4,4,0). We start with a function that, given a state, yields a list of all proper successor states:
successors :: State -> [State]
successors (a,b,c) =
    let ab = min a (5 - b)
        ac = min a (3 - c)
        ba = min b (8 - a)
        bc = min b (3 - c)
        ca = min c (8 - a)
        cb = min c (5 - b)
        ss = [(ab,a-ab,b+ab,c), (ac,a-ac,b,c+ac), (ba,a+ba,b-ba,c),
              (bc,a,b-bc,c+bc), (ca,a+ca,b,c-ca), (cb,a,b+cb,c-cb)]
      [(a',b',c') | (transfer,a',b',c') <- ss, transfer > 0]
We can test this function interactively:
Main> successors (8,0,0)
Let us now determine whether we can, starting from the initial state, actually reach the target state. We try breadth-first search, a complete and space-inefficient search strategy:
search :: [State] -> Bool
search (s:ss)
    | s == (4,4,0) = True
    | otherwise = search $ ss ++ successors s
In each iteration, we consider the first state in the given list of states. If it's the target state, we're done. Otherwise, its successors are appended to the remaining states (to be considered later), and the search continues. We can now query
search [(8,0,0)]
and know that the puzzle actually has a solution. To find a sequence of actions leading to the target state, we reconsider the state representation. Instead of merely keeping track of the amount of water in the jugs, we also store how we obtained each configuration. We represent this new state as a pair (J,P): J is the jug configuration (A,B,C) like before, and P is a "path" that leads from the starting state to configuration J. A path is a list of FromTo Jug1 Jug2 moves, meaning that we poured water from Jug1 into Jug2. For each successor state, we record how its configuration was reached by appending the corresponding path element to (a copy of) its predecessor's path. The new program (jugs.hs) is:
data Jug = A | B | C deriving Show

data Move = FromTo Jug Jug deriving Show
type Path = [Move]

type State = ((Int,Int,Int), Path)

start :: State
start = ((8,0,0), [])

successors :: State -> [State]

successors ((a,b,c),path) =
    let ab = min a (5 - b)
        ac = min a (3 - c)
        ba = min b (8 - a)
        bc = min b (3 - c)
        ca = min c (8 - a)
        cb = min c (5 - b)
        ss = [(ab, a-ab, b+ab,    c, path ++ [FromTo A B]),
              (ac, a-ac,    b, c+ac, path ++ [FromTo A C]),
              (ba, a+ba, b-ba,    c, path ++ [FromTo B A]),
              (bc,    a, b-bc, c+bc, path ++ [FromTo B C]),
              (ca, a+ca,    b, c-ca, path ++ [FromTo C A]),
              (cb,    a, b+cb, c-cb, path ++ [FromTo C B])]
      [((a',b',c'), path') | (amount,a',b',c',path') <- ss, amount > 0]

search :: [State] -> Path

search (s:ss)
    | fst s == (4,4,0) = snd s
    | otherwise = search $ ss ++ successors s
A path is now readily found:
search [start]
[FromTo A B,FromTo B C,FromTo C A,FromTo B C,FromTo A B,FromTo B C,FromTo C A]
There are various ways to make this more efficient. We could, for example, prepend the new path elements and reverse the path once at the end of the search.

More importantly, we can also make it more elegant: Clearly, the code above contains some redundancy, which we can avoid with a different state representation. The following Prolog version illustrates this:
jug_capacity(a, 8).
jug_capacity(b, 5).
jug_capacity(c, 3).

moves(Jugs) -->
        { memberchk(jug(a,4), Jugs),
          memberchk(jug(b,4), Jugs) }.
moves(Jugs0) -->
        { select(jug(AID,AF), Jugs0, Jugs1), AF #> 0,
          select(jug(BID,BF), Jugs1, Jugs2), jug_capacity(BID, BC), BF #< BC,
          Move #= min(AF,BC-BF),
          AF1 #= AF - Move,
          BF1 #= BF + Move },
With this state representation, moves can be described uniformly and need not be enumerated explicitly. We use iterative deepening to find a shortest solution:
?- length(Ms, _), phrase(moves([jug(a,8),jug(b,0),jug(c,0)]), Ms).
Ms = [from_to(a,b),from_to(b,c),from_to(c,a),from_to(b,c),from_to(a,b),from_to(b,c),from_to(c,a)] .

Exercise: Use this state representation in the Haskell version.

In a similar manner, you can solve other puzzles involving search like "wolf and goat", 8-puzzles, Escape from Zurg and "missionary and cannibal".

States in programs

Let us now build an interpreter for a simple programming language over integers in Prolog. Using Prolog terms, we represent programs as abstract syntax trees (ASTs) like:
function(Name, Parameter, Body)
call(Name, Expression)
assign(Variable, Expression)
if(Condition, Then, Else)
while(Condition, Body)
sequence(First, Second)
To symbolically distinguish variables from numerals in arithmetic expressions, we use the unary functors "v" and "n", respectively. Look up the definition of is_program/2 in the source code ( for a complete declarative specification of the chosen representation. Also included, you find a parser to automatically generate this term representation from more readable syntax. For instance, the following program (recursively) computing and printing the fourth Catalan number
catalan (n) {
        if (n == 0) {
                return 1;
        } else {
                c = catalan(n-1);
                r = 2*(2*n + 1)*c / (n + 2);
                return r;

print catalan(4);
is converted to a syntax tree like this:
?- string_ast("catalan (n) { if (n == 0) { return 1; } else { c = catalan(n-1);
          r = 2*(2*n + 1)*c / (n + 2); return r; } } print catalan(4);", AST).

AST = sequence(function(catalan, n,
                         if(bin(=, v(n), n(0)),
                            sequence(assign(c, call(catalan, bin(-, v(n), n(1)))),
                                     sequence(assign(r, bin(/,
                                                            bin(+, v(n), n(2)))),
               print(call(catalan, n(4))))

To interpret such programs, we have to keep track of the state of computation. It consists of: These two, collectively referred to as the environment, are represented as a pair of association lists, associating variable names with values, and function heads with function bodies. This makes defining and referring to functions as well as accessing variables O(log(N)) operations in the number of encountered functions and variables, respectively.

Clearly, the predicates responsible for interpreting syntax trees define relations between such environments and thus between states. This is how we interpret imperative programs in a purely declarative way.

To evaluate expressions with respect to the current environment, we use the predicate eval/3:
eval(bin(Op,A,B), Env, Value) :-
        eval(A, Env, VA),
        eval(B, Env, VB),
        eval_(Op, VA, VB, Value).
eval(v(V), Env, Value) :-
        env_get_var(Env, V, Value).
eval(n(N), _, N).
eval(call(Name, Arg), Env0, Value) :-
        eval(Arg, Env0, ArgVal),
        env_func_body(Env0, Name, ArgName, Body),
        env_clear_variables(Env0, Env1),
        env_put_var(ArgName, ArgVal, Env1, Env2),
        interpret(Body, Env2, Value).

eval_(+, A, B, V) :- V #= A + B.
eval_(-, A, B, V) :- V #= A - B.
eval_(*, A, B, V) :- V #= A * B.
eval_(/, A, B, V) :- V #= A // B.
eval_(=, A, B, V) :- goal_truth(A #= B, V).
eval_(>, A, B, V) :- goal_truth(A #> B, V).
eval_(<, A, B, V) :- goal_truth(A #< B, V).

goal_truth(Goal, V) :- ( Goal -> V = 1 ; V = 0).
The predicates accessing the environment (env_get_var/3 etc.) are straight-forward, and you can look up their definitions in the source code. Finally, the predicate interpret/3 specifies how, if at all, each construct of our language changes the environment:
interpret(print(P), Env, Env) :-
        eval(P, Env, Value),
        format("~w\n", [Value]).
interpret(sequence(A,B), Env0, Env) :-
        interpret(A, Env0, Env1),
        (   A = return(_) ->
            Env = Env1
        ;   interpret(B, Env1, Env)
interpret(call(Name, Arg), Env0, Env0) :-
        eval(Arg, Env0, ArgVal),
        env_func_body(Env0, Name, ArgName, Body),
        env_clear_variables(Env0, Env1),
        env_put_var(ArgName, ArgVal, Env1, Env2),
        interpret(Body, Env2, _).
interpret(function(Name,Arg,Body), Env0, Env) :-
        env_put_func(Name, Arg, Body, Env0, Env).
interpret(if(Cond,Then,Else), Env0, Env) :-
        eval(Cond, Env0, Value),
        (   Value #\= 0 ->
            interpret(Then, Env0, Env)
        ;   interpret(Else, Env0, Env)
interpret(assign(Var, Expr), Env0, Env) :-
        eval(Expr, Env0, Value),
        env_put_var(Var, Value, Env0, Env).
interpret(while(Cond, Body), Env0, Env) :-
        eval(Cond, Env0, Value),
        (   Value #\= 0 ->
            interpret(Body, Env0, Env1),
            interpret(while(Cond, Body), Env1, Env)
        ;   Env = Env0
interpret(return(Expr), Env0, Value) :-
        eval(Expr, Env0, Value).
interpret(nop, Env, Env).
Two things deserve special attention: For one, the print statement produces a side-effect: It is meant to show output on the terminal, and this cannot be expressed by transforming the binding environment. The interpreter is thus not purely logical. To fix this, we could incorporate a suitable representation of the state of the "world" into our environment and adjust it appropriately whenever a print statement is encountered. Second, return statements are special in that their resulting environment consists of a single value. The eval/3 predicate makes use of this when evaluating function calls.

To interpret a program, we start with a fresh environment and discard the resulting environment:
run(AST) :-
        interpret(AST, Env, _).
We can run our simple example program like this:

?- string_ast("catalan (n) { if (n == 0) { return 1; } else { c = catalan(n-1);
          r = 2*(2*n + 1)*c / (n + 2); return r; } } print catalan(4);", AST), run(AST).


States in compilers

To get rid of the interpreter's overhead incurred by looking up variables and function definitions in the environment, we now compile programs to virtual machine code in which variables and functions are addressed by offsets into specific regions of the virtual machine's "memory". Using a programming language permitting efficient array indexing, you can thus interpret variable access and function calls in O(1).

Our virtual machine (VM) shall be stack-based and have the following instructions:


halt stop execution
alloc n push n zeros on top of stack
pushc c push constant c on top of stack
pushv v push value of variable v on top of stack
pop v remove topmost element from stack and assign its value to variable v
add replace topmost two elements of stack by their sum
sub ... subtract
mul ... multiply
div ... integer division
jmp adr continue execution at instruction adr
jne adr remove topmost two stack elements and jump to adr if they are not equal
jge adr jump if greater or equal
jle adr jump if less or equal
call adr call subroutine starting at adr
print remove topmost stack element and print its value
ret return from subroutine

Variables are now actually integers, denoting an offset into the stack frame of the function being executed. 0 (zero) corresponds to a function's sole argument and is copied on the stack when encountering a call instruction. Also, call saves the current frame pointer and program counter on the stack. A function can allocate additional space for local variables by means of the alloc instruction. The return instruction (ret) removes all stack elements allocated by the current function, restores the frame pointer and program counter, and pushes the function's return value on the stack.

The following example program and corresponding VM code illustrate most of the instructions:

fac(n) {
        f = 1;
        while (n > 1) {
                f = f*n;
                n = n - 1;

print fac(4);
 0:     jmp 33
 2:     alloc 1
 4:     pushc 1
 6:     pop 1
 8:     pushv 0
10:     pushc 1
12:     jle 30
14:     pushv 1
16:     pushv 0
18:     mul
19:     pop 1
21:     pushv 0
23:     pushc 1
25:     sub
26:     pop 0
28:     jmp 8
30:     pushv 1
32:     ret
33:     pushc 4
35:     call 2
37:     print
38:     halt

Generating such VM code from an abstract syntax tree is straight-forward. In essence, we will write a predicate compilation/3 with clauses roughly of the form
compilation(functor(Arg1,Arg2,...,ArgN), State0, State) :-
        compilation(Arg1, State0, State1),
        compilation(Arg2, State1, State2),
        compilation(Argn, State_(N-1), State_N),
        vminstr(instruction_depending_on_functor, State_N, State).
Notice the naming convention S0, S1, S2, ..., S for successive states.

As we will see in the following, only a few predicates need to access and modify the state in this case. We will therefore use Prolog semicontext notation to implicitly thread the state through, yielding the more readable nonterminal compilation//1:
compilation(functor(Arg1,Arg2,...,ArgN)) -->

When compiling to VM code, we keep track of the state of the compilation process: We store all this in a quadruple s(Is, Fs, Vs, PC). The entry point for compilation is ast_vminstrs/2, which relates an abstract syntax tree to a list of virtual machine instructions. It starts with a fresh state, transforms it via compilation//1 and then extracts the accumulated instructions, also translating names to offsets in function calls:
ast_vminstrs(AST, VMs) :-
        phrase(compilation(AST), [S0], [S]),
        state_vminstrs(S, VMs).


state_vminstrs(s(Is0,Fs,_,_), Is) :-
        reverse([halt|Is0], Is1),
        maplist(resolve_calls(Fs), Is1, Is).

resolve_calls(Fs, I0, I) :-
        (   I0 = call(Name) ->
            memberchk(Name-Adr, Fs),
            I = call(Adr)
        ;   I = I0
To portably (i.e., without relying on a particular expansion method for DCGs) access and modify the state that is implicitly threaded through in DCG notation, we use the following two nonterminals:
state(S), [S] --> [S].

state(S0, S), [S] --> [S0].
Thus, state(S) can be read as "the current state is S", and state(S0, S) can be read as "the current state is S0, and henceforth it is S".

Here are auxiliary predicates to access and transform parts of the state:
current_pc(PC) --> state(s(_,_,_,PC)).

vminstr(I) -->
        state(s(Is,Fs,Vs,PC0), s([I|Is],Fs,Vs,PC)),
        { I =.. Ls,
          length(Ls, L),   % length of instruction including arguments
          PC #= PC0 + L }.

start_function(Name, Arg) -->
        state(s(Is,Fs,_,PC), s(Is,[Name-PC|Fs],[Arg-0],PC)).

num_variables(Num) -->
        { length(Vs, Num0),
          Num #= Num0 - 1 }.      % don't count parameter

variable_offset(Name, Offset) -->
        state(s(Is,Fs,Vs0,PC), s(Is,Fs,Vs,PC)),
        { (   memberchk(Name-Offset, Vs0) ->
              Vs = Vs0
          ;   Vs0 = [_-Curr|_],
              Offset #= Curr + 1,
              Vs = [Name-Offset|Vs0]
          ) }.
For example, start_function//2 records the offset (= current program counter) of the function to be defined and starts a new list of encountered variables, originally consisting only of the function's argument, whose offset in the stack frame is 0. Further variables are assigned ascending offsets as they are encountered. This is handled by variable_offset//2, which either determines a variable's offset from the list of encountered variables or, if it is new, registers it with a new offset computed by adding one to the offset of the variable registered previously.

We can now define compilation//1:
compilation(nop) --> [].
compilation(print(P)) -->
compilation(sequence(A,B)) -->
compilation(call(Name,Arg)) -->
compilation(function(Name,Arg,Body)) -->
        start_function(Name, Arg),
compilation(if(Cond,Then,Else)) -->
        { Cond = bin(Op,A,B) },
        condition(Op, Adr1),
compilation(assign(Var,Expr)) -->
        variable_offset(Var, Offset),
compilation(while(Cond,Body)) -->
        { Cond = bin(Op,A,B) },
        condition(Op, Break),
compilation(return(Expr)) -->
compilation(bin(Op,A,B)) -->
        { op_vminstr(Op, VI) },
compilation(n(N)) -->
compilation(v(V)) -->
        variable_offset(V, Offset),

op_vminstr(+, add).
op_vminstr(-, sub).
op_vminstr(*, mul).
op_vminstr(/, div).

condition(=, Adr) --> vminstr(jne(Adr)).
condition(<, Adr) --> vminstr(jge(Adr)).
condition(>, Adr) --> vminstr(jle(Adr)).

Notice how we benefit from logical variables in several cases: For example, when alloc is emitted, we do not yet know how much space must be allocated. Nevertheless, we add the instruction to the sequence of virtual machine instructions, and instantiate its argument later.

By introducing a call_n instruction that discards the n most recently allocated local variables before calling a given function, we could add tail call optimisation and, more generally, stack trimming to the VM: If a variable isn't needed after a function call, its stack space can be reclaimed before the call.

To keep the generated VM code compact and easily accessible in other programming languages, we relate the mnemonic virtual machine instructions to lists of integers:
vminstrs_ints([])     --> [].
vminstrs_ints([I|Is]) -->

vminstr_ints(halt)      --> [0].
vminstr_ints(alloc(A))  --> [1,A].
vminstr_ints(pushc(C))  --> [2,C].
vminstr_ints(pushv(V))  --> [3,V].
vminstr_ints(pop(V))    --> [4,V].
vminstr_ints(add)       --> [5].
vminstr_ints(sub)       --> [6].
vminstr_ints(mul)       --> [7].
vminstr_ints(div)       --> [8].
vminstr_ints(jmp(Adr))  --> [9,Adr].
vminstr_ints(jne(Adr))  --> [10,Adr].
vminstr_ints(jge(Adr))  --> [11,Adr].
vminstr_ints(jle(Adr))  --> [12,Adr].
vminstr_ints(call(Adr)) --> [13,Adr].
vminstr_ints(print)     --> [14].
vminstr_ints(ret)       --> [15].

States in virtual machines

Let us now implement the VM we devised. Its state is determined by: Using Haskell, we can use a quadruple to represent this state:
type State = ([Int], [Int], Int, Int)
The function step, given the integer code of a VM instruction and the VM's current state, computes and returns the VM's new state:
step :: Int -> State -> State
step instr (stack,instrs,pc,fp) =
    case instr of
      1 -> ((replicate next 0) ++ stack, instrs, pc2, fp + next)
      2 -> (next:stack, instrs, pc2, fp+1)
      3 -> ((stack!!(fp - next)):stack, instrs, pc2, fp + 1)
      4 -> (tail $ set_nth stack (fp - next) first, instrs, pc2, fp1)
      5 -> ((second+first):drop2, instrs, pc1, fp1)
      6 -> ((second-first):drop2, instrs, pc1, fp1)
      7 -> ((second*first):drop2, instrs, pc1, fp1)
      8 -> ((div second first):drop2, instrs, pc1, fp1)
      9 -> (stack, instrs, next, fp)
      10 -> if second /= first then (drop2, instrs, next, fp2)
            else (drop2, instrs, pc2, fp2)
      11 -> if second >= first then (drop2, instrs, next, fp2)
            else (drop2, instrs, pc2, fp2)
      12 -> if second <= first  then (drop2, instrs, next, fp2)
            else (drop2, instrs, pc2, fp2)
      13 -> ([first,fp,pc2] ++ tail stack, instrs, next, 0)
      15 -> let fp' = stack !! (fp + 1)
                pc' = stack !! (fp + 2)
              (first : drop (fp+3) stack, instrs, pc', fp')
    where next = instrs !! (pc+1)
          first = head stack
          second = stack !! 1
          drop2 = drop 2 stack
          fp1 = fp - 1
          fp2 = fp - 2
          pc1 = pc + 1
          pc2 = pc + 2

set_nth :: [a] -> Int -> a -> [a]
set_nth (x:xs) n a
    | n == 0 = a:xs
    | otherwise = x:(set_nth xs (n - 1) a)
We execute a list of integer codes by continually transforming the state until a halt instruction is reached:
exec :: State -> IO ()
exec s0@(stack,instrs,pc,fp) =
    let instr = instrs !! pc
      case instr of
        0 -> return ()
        14 -> do putStr $ (show $ head stack) ++ "\n"
                 exec (tail stack, instrs, pc + 1, fp - 1)
        otherwise -> exec $ step instr s0

main :: IO ()
main =
    do prog <- getLine
       let ints = read prog::[Int]
           s0 = ([],ints,0,0)
       exec s0

This code is available as vm.hs. Since we used lists to represent the stack and set of instructions, indexing is inefficient. To remedy this, we could use an array at least for the set of instructions. The stack, however, needs to grow and shrink. We could artificially fix its size, or resize on demand, and still use a Haskell array. Instead, let us formulate the program in a language with efficient array operations at its core: J, a successor of APL. In the J version (vm.ijs), we represent the VM's state using an array of four boxed vectors. Using the power conjunction ^:, we produce the limit of repeated applications of step:
st =. 3 : '> 0 { y'
is =. 3 : '> 1 { y'
pc =. 3 : '> 2 { y'
fp =. 3 : '> 3 { y'

next =. (>:&pc { is)

print =. 3 : 'y (1!:2) 2'

adv =. 3 : '(2 }.st y); (is y); (2+pc y); ((fp y) - 2)'
jmp =. 3 : '(2 }.st y); (is y); (next y); ((fp y) - 2)'

i1  =: 3 : '(((next y) # 0),st y); (is; 2&+&pc; next+fp) y'
i2  =: 3 : '((next,st); is ; 2&+&pc; >:&fp) y'
i3  =: 3 : '((((fp-next) { st),st); is; 2&+&pc; >:&fp) y'
i4  =: 3 : '(}.({.st y) ((fp-next)y) } st y); (is; 2&+&pc; <:&fp) y'
i5  =: 3 : '((+/1 0 { st y),2}.st y); (is; >:&pc; <:&fp) y'
i6  =: 3 : '((-/1 0 { st y),2}.st y); (is; >:&pc; <:&fp) y'
i7  =: 3 : '((*/1 0 { st y),2}.st y); (is; >:& pc; <:&fp) y'
i8  =: 3 : '((<.%/1 0 { st y),2}.st y); (is; >:&pc; <:&fp) y'
i9  =: 3 : '(st; is; next; fp) y'
i10 =: 3 : '(adv ` jmp @. (-.=/1 0 {st y)) y'
i11 =: 3 : '(adv ` jmp @. (>:/1 0 {st y)) y'
i12 =: 3 : '(adv ` jmp @. (<:/1 0 {st y)) y'
i13 =: 3 : '(((({.&st), fp, 2&+&pc),}.& st) y); (is y); (next y);0'
i14 =: 3 : '(print {. st y)]((}.&st); is; (>:&pc); (<:&fp)) y'
i15 =: 3 : 0
   fp1 =. (>:&fp { st) y
   pc1 =. (2&+&fp { st) y
   (({. st y),(3+fp y)}. st y) ; (is y); pc1 ; fp1

step =: 3 : 0
   instr =. (pc { is) y
   (]`i1`i2`i3`i4`i5`i6`i7`i8`i9`i10`i11`i12`i13`i14`i15  @. instr) y

state0 =:  ($0); instrs; 0; 0

step ^: _ state0

Source files

The source files are:
A transcript showing how to invoke these programs: log.txt

You can use SWI Prolog to try the Prolog code.

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