Thinking in States
Motivation
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([ELs1], E, Ls2) :
list1_element_list2(Ls1, E, Ls2).
list1_element_list2([LLs1], E, [LLs2]) :
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 [ELs1], 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
metapredicate 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,aab,b+ab,c), (ac,aac,b,c+ac), (ba,a+ba,bba,c),
(bc,a,bbc,c+bc), (ca,a+ca,b,cca), (cb,a,b+cb,ccb)]
in
[(a',b',c')  (transfer,a',b',c') < ss, transfer > 0]
We can test this function interactively:
Main> successors (8,0,0)
[(3,5,0),(5,0,3)]
Let us now determine whether we can, starting from the initial
state, actually reach the target state. We try breadthfirst
search, a complete and spaceinefficient 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)]
True
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, aab, b+ab, c, path ++ [FromTo A B]),
(ac, aac, b, c+ac, path ++ [FromTo A C]),
(ba, a+ba, bba, c, path ++ [FromTo B A]),
(bc, a, bbc, c+bc, path ++ [FromTo B C]),
(ca, a+ca, b, cca, path ++ [FromTo C A]),
(cb, a, b+cb, ccb, path ++ [FromTo C B])]
in
[((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,BCBF),
AF1 #= AF  Move,
BF1 #= BF + Move },
[from_to(AID,BID)],
moves([jug(AID,AF1),jug(BID,BF1)Jugs2]).
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",
8puzzles, 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)
return(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 (interp.pl) 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(n1);
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(n1);
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)),
return(n(1)),
sequence(assign(c, call(catalan, bin(, v(n), n(1)))),
sequence(assign(r, bin(/,
bin(*,
bin(*,
n(2),
bin(+,
bin(*,
n(2),
v(n)),
n(1))),
v(c)),
bin(+, v(n), n(2)))),
return(v(r)))))),
print(call(catalan, n(4))))
To interpret such programs, we have to keep track of the
state of computation. It consists of:
 the binding environment for variables
 all encountered function definitions.
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 straightforward, 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 sideeffect: 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) :
env_new(Env),
interpret(AST, Env, _).
We can run our simple example program like this:
? string_ast("catalan (n) { if (n == 0) { return 1; } else { c = catalan(n1);
r = 2*(2*n + 1)*c / (n + 2); return r; } } print catalan(4);", AST), run(AST).
42
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 stackbased and have the
following instructions:
Instruction 
Effect 

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;
}
return(f);
}
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
straightforward. 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_(N1), 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)) >
compilation(Arg1),
compilation(Arg2),
:
:
compilation(Argn),
vminstr(instruction_depending_on_functor).
When compiling to VM code, we keep track of the state of
the compilation process:
 VM instructions emitted so far
 offsets of encountered function definitions
 offsets of variables encountered in the current function
 offset of next instruction ("program counter").
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) :
initial_state(S0),
phrase(compilation(AST), [S0], [S]),
state_vminstrs(S, VMs).
initial_state(s([],[],[],0)).
state_vminstrs(s(Is0,Fs,_,_), Is) :
reverse([haltIs0], Is1),
maplist(resolve_calls(Fs), Is1, Is).
resolve_calls(Fs, I0, I) :
( I0 = call(Name) >
memberchk(NameAdr, 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([IIs],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,[NamePCFs],[Arg0],PC)).
num_variables(Num) >
state(s(_,_,Vs,_)),
{ 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(NameOffset, Vs0) >
Vs = Vs0
; Vs0 = [_Curr_],
Offset #= Curr + 1,
Vs = [NameOffsetVs0]
) }.
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(P),
vminstr(print).
compilation(sequence(A,B)) >
compilation(A),
compilation(B).
compilation(call(Name,Arg)) >
compilation(Arg),
vminstr(call(Name)).
compilation(function(Name,Arg,Body)) >
vminstr(jmp(Skip)),
start_function(Name, Arg),
vminstr(alloc(NumVars)),
compilation(Body),
num_variables(NumVars),
current_pc(Skip).
compilation(if(Cond,Then,Else)) >
{ Cond = bin(Op,A,B) },
compilation(A),
compilation(B),
condition(Op, Adr1),
compilation(Then),
vminstr(jmp(Adr2)),
current_pc(Adr1),
compilation(Else),
current_pc(Adr2).
compilation(assign(Var,Expr)) >
variable_offset(Var, Offset),
compilation(Expr),
vminstr(pop(Offset)).
compilation(while(Cond,Body)) >
current_pc(Head),
{ Cond = bin(Op,A,B) },
compilation(A),
compilation(B),
condition(Op, Break),
compilation(Body),
vminstr(jmp(Head)),
current_pc(Break).
compilation(return(Expr)) >
compilation(Expr),
vminstr(ret).
compilation(bin(Op,A,B)) >
compilation(A),
compilation(B),
{ op_vminstr(Op, VI) },
vminstr(VI).
compilation(n(N)) >
vminstr(pushc(N)).
compilation(v(V)) >
variable_offset(V, Offset),
vminstr(pushv(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([IIs]) >
vminstr_ints(I),
vminstrs_ints(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:
 values on the stack
 list of instructions to execute
 pc: program counter (offset into list of instructions)
 fp: frame pointer (offset into stack).
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 > ((secondfirst):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)
in
(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
in
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 : '((((fpnext) { st),st); is; 2&+&pc; >:&fp) y'
i4 =: 3 : '(}.({.st y) ((fpnext)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:
 jugs.hs: Water jugs puzzle in Haskell.
 jugs2.hs: Water jugs puzzle in Haskell, different state representation.
 jugs.pl: Water jugs puzzle in Prolog.
 interp.pl: Interpreter and compiler
for a simple programming language, written in Prolog.
 vm.hs: Interpreter for the abstract
machine code generated by the compiler, written in Haskell.
 vm.ijs: Interpreter for the abstract
machine code, written in J.
A transcript showing how to invoke these programs:
log.txt
You can use SWI Prolog to
try the Prolog code.
More about Prolog
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