An interpreter, a compiler, and a virtual machine
prerequisites: Basic experience with Coq or similar
1. Introduction
In this post, we show how to implement an interpreter and a compiler for a small arithmetic language, in the Coq Proof Assistant, along with a virtual machine for running the output of the compiler. We implement the language in Coq such that we can later prove an equivalence relation between evaluation with an interpreter and a compiler, in a followup blog post.
We start by introducing the small arithmetic language in Section 2. Having defined our language, we introduce an interpreter in Section 3, which implements the operational semantics of the language. Before we introduce the corresponding compiler, we first have to build a virtual machine in Section 4 onto which we can execute the bytecode output of the compiler. Finally, we introduce the compiler in Section 5 and conclude in Section 6.
2. Language
The first step towards implementing an interpreter is to define the language
which we want to interpret. Thus, we define a small arithmetic language with
which we can perform addition and multiplication of natural numbers. We define
the language as a formal grammar
by stating that a program consists of an expression, e
, which can either be:
 a literal,
Lit
, that takes a natural number,n
, or  an addition expression,
Plus
, that takes two arithmetic expressions,e
, or  a multiplication expression,
Mult
, that takes two arithmetic expressions,e
.
The above description yields the following grammar:
which we in turn can translate into an Inductive
type in Coq:
Here, we state that any instance of the type arithmetic_expression
is either a
literal, an addition expression, or a multiplication expression, as described
above. Furthermore, we can construct instances of this type by applying the
three constructors of the definition. For example, if we want to create an
instance of the arithmetic_expression
type that corresponds to the expression
\((2 + 1) \cdot 5\), we write the following:
where the keyword Compute
is simply used to evaluate the expression. Note that
we have not yet computed the result of the expression, as we have not yet
implemented the semantics of the language.
Now that we have defined the grammar of our arithmetic expression language, we are ready to define an interpreter for the language in the next section.
3. Interpreter
Having defined our language as an inductive algebraic data type, we can now introduce a function that, given an element of this type, recursively traverses the structure of such an element and returns the result of evaluating the arithmetic expression corresponding to the element. The semantics of our arithmetic language are pretty straight forward:
 A literal expression,
Lit n
, evaluates to the natural number,n
, and  an addition expression,
Plus e e
, evaluates to the sum of the two subexpressions,e
, and  a multiplication expression,
Mult e e
, evaluates to the product of the two subexpressions,e
.
If we translate the description above into Coq, we get the following Fixpoint
:
Here, the interpret
function pattern matches on the structure of the
arithmetic expression, e
, and recursively evaluates its subexpressions. If we
want to evaluate the expression \((2 \cdot 5) \cdot (1 + 3)\), we first
translate it into the arithmetic_expression
language and then pass it to the
interpret
function:
When evaluated, the above expression yields the result 40 : nat
as expected.
Thus, we have now implemented an interpreter – in six lines of code – which can
evaluate any expression of our arithmetic language. However, in order to
implement a corresponding compiler, we first have to take a look a virtual
machines.
4. Virtual machine
Before we can build a compiler for our arithmetic expression language, we first need a machine onto which we can execute the compiled source code. Thus, we construct a minimal stack machine with the following three bytecode instructions:
Furthermore, we define a bytecode program to be a list of byte code instructions:
and the data stack of our virtual machine to be a list of natural numbers:
With these three definitions taken care of, we move on to define the semantics
of the three byte code instructions and translate them into a corresponding
function. We define the instruction PUSH
to take a natural number which it
then pushes onto the stack, while the ADD
and MUL
instructions each pop the
two topmost elements of the stack and adds or multiplies them, respectively. For
the sake of simplicity, we define the effects of executing ADD
or MUL
on a
stack with less than two elements to be an unchanged stack. The above semantics
result in the function execute_bytecode_instruction
, which takes a
bytecode_instruction
and a data_stack
and returns a new data_stack
,
capturing the effect of evaluating a bytecode_instruction
with respect to a
given data_stack
:
An example application of execute_bytecode_instruction
multiplies the two
elements on top of the stack [5,3,2]
^{1}:
and returns the resulting stack [15,2]
.
The last step we need, in order to finish our virtual machine, is to wrap the
execute_bytecode_instruction
in a function that takes a whole bytecode program
and runs it on an initial data stack. This can be achieved by simply traversing
the list of bytecode instructions and executing them onebyone, like so:
Now we can execute a whole bytecode program on our virtual stack machine by
calling execute_bytecode_program
with a bytecode_program
and a data_stack
:
If we step through the execution of the above program, the major steps are as follows:
 First we push
2
and3
onto the empty stack, giving us the stack[3, 2]
, then  we pop the two elements and push their result onto the stack, resulting in the
stack
[5]
, again  we push a
5
onto the stack and multiply the two elements such that we get the stack[25]
, lastly  we push
1
onto the stack, at which point we have run our whole bytecode program and return the final stack,[1, 25]
.
With our virtual machine implemented and tested, we can finally move on to construct our compiler.
5. Compiler
Having introduced the needed set of bytecode instructions and seen how these can be executed on a virtual machine, we are ready to define a compiler that takes expressions of our arithmetic language as its input and generates a bytecode program as its output.
Returning to the constructors of our arithmetic_expression
language, we can
turn these into bytecode instructions:
 The literal constructor,
Lit n
, can be directly translated into the act of returning the bytecode program consisting of thePUSH n
instruction, while  the addition expression,
Plus e1 e2
, corresponds to the result of first compiling the second expression,e2
, followed by concatenating the result of compiling the first expression,e1
, and then concatenating the bytecode program consisting of theADD
instruction, lastly  the multiplication expression,
Mult e1 e2
, is identical to the compilation of the addition expression except for the use of theMUL
instruction rather than theADD
instruction.
This brings us to the following definition of our compiler:
which takes an arithmetic_expression
as its input and produces a
bytecode_program
as its output. If we want to compile an
arithmetic_expression
corresponding to \(5 + (3 \cdot 2)\), we pass it to
compile
like so:
which results in the following bytecode_program
output:
Note that because we are working with a stack machine, the outputted program both flattens and reverses the compiled expression.
Now that we have finally defined our compiler, we can demonstrate the equivalence relation between interpretation of an arithmetic expression and compilation of an arithmetic expression followed by execution of the compiled bytecode program:
6. Conclusion
In this post, we have shown how to implement an interpreter, a compiler, and a virtual machine for a small arithmetic language. Furthermore, we have also shown how the two types of evaluation relate to each other.
Lastly, the language has been implemented in the Coq Proof Assistant such that we can prove an equivalence relation between interpretation of an arithmetic expression and compilation followed by execution of an arithmetic expression, which is exactly the topic of the next blog post.

We use the notation
[1,2,3]
and(1 :: 2 :: 3 :: nil)
interchangeably to denote the content of alist
in Coq. ↩
Mathematics
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