Skip to main content

A Calculator Language



Calc illustraties the basic facilities of TypePal


By developing a type checker for Calc, a tiny pocket calculator language, we illustrate elementary usage of TypePal. The full source code of Calc can be found at See Examples of Typecheckers for a list of all available type checker examples.

Syntax of Calc

module lang::calc::Syntax

extend lang::CommonLex;

start syntax Calc
= Decl+

syntax Decl
= "var" Id "=" Exp ";"

syntax Exp
= Id
| Integer
| Boolean
| bracket "(" Exp ")"
> left Exp "*" Exp
> left Exp "+" Exp
> "if" Exp "then" Exp "else" Exp ❶⓿

keyword Reserved
= "var" | "if" | "then" | "else"
  • ❶ For brevity we use some common lexical definitions in lang::CommonLex. They take, amongst others, care of whitespace, comments and definitions for integer and Boolean constants (Integer and Boolean) as well as identifiers (Id).
  • ❷ A Calc program consists of one or more declarations (Decl).
  • ❸ Each declaration introduces a new variable and binds it to the value of an expression.
  • ❹ An expression can be an identifier (that should have been introduced in a preceding declaration), or
  • ❺ an integer constant, or
  • ❻ a Boolean constant, or
  • ❼ be surrounded by parentheses, or
  • ❽ a multiplication (arguments should have the same type), or
  • ❾ an addition (arguments should have the same type), or
  • ❶⓿ a conditional expression (the first expression should have type Boolean, the other two should have the same type).

Typechecking Calc

Type checking Calc amounts to checking the following:

  • Every name that is used in an expression should have been defined in a declaration and has the type of the expression in that declaration.
  • Addition or multiplication on integers yields an integer type.
  • Addition or multiplication on Booleans yields a Boolean type.
  • The condition in a condional expression should be Boolean.
  • The then and else part of a conditional expression should have the same type; that type becomes the type of the conditional expression.

Let's now move to the Calc type checker described in lang::calc::Checker. We break it up in parts and describe the parts as we go. You should be explicitly aware of the fact that type checking consists of two phases:

  • Collecting facts and constraints: this is all about formulating constraints and attaching them to parts of the source program. In this phase types are unknown and constraints can only be created but not yet solved. All computations on types have therefore to be postponed until the solving phase. We will see how in a minute.
  • Solving constraints: constraints are solved (if possible) and types are assigned to parts of the source program.

Begin of type checker module for Calc

module lang::calc::Checker

import lang::calc::Syntax; // The Calc syntax
extend analysis::typepal::TypePal; // TypePal

We import the Calc syntax lang::calc::Syntax and then we extend the TypePal analysis::typepal::TypePal.

NOTE: There is a technical reason why we have to extend TypePal rather than import it. Some data types (e.g., AType) and functions (e.g., collect) are already defined in TypePal itself and they are being extended in each type checker.

Extend AType

TypePal has a built-in data type to represent types: AType. This data type can be extended to represent language-specific types.

data AType   
= boolType()
| intType()

str prettyAType(boolType()) = "bool";
str prettyAType(intType()) = "int";

The values in Calc are Booleans and integers and we create two AType constants boolType() and intType() to represent these types. prettyAType converts them back to a string representation; this is used in error reporting.

Collecting facts and constraints

Writing a type checker in TypePal amounts to collecting facts and constaints from a source program to be checked. This is achieved by the collect function that has the form

void collect(current: _syntactic pattern_, Collector c){ ... }`

where syntactic pattern corresponds with a rule from the language syntax, in this case lang::calc::Syntax. The Collector argument provides methods to add facts and constraints.

NOTE: One collect declaration is needed per alternative in the syntax but some rules (i.e., chain rules) are handled automatically.

NOTE: By convention the syntactic pattern is named current and the Collector is named c.

WARNING: Each collect declaration is responsible for calling collect on relevant subparts (a run-time error is given for a missing collect).

Check Declaration

A declaration introduces a new name and associates the type of the righthand side expression with it.

void collect(current: (Decl) `var <Id name> = <Exp exp> ;`, Collector c){
c.define("<name>", variableId(), current, defType(exp));
collect(exp, c);

Here we define name as a variable and define its type as the same type as exp. Let's digest this:

  • c.define(...) calls the define function in the Collector and adds this definition to the internal state of the Collector.
  • name is a parse tree for the lexical syntax rule Id. "<name>" reduces this tree to a string and this is how the variable can be used later on.
  • variableId() defines that the role of this name (IdRole) is a variable. variableId is a built-in role IdRole.
  • current is the parse tree of the whole declaration, we use it as unique identification of the definition of name.
  • defType is used to define the type of name and it comes in several flavours. Here, we use another parse tree fragment exp as argument. defType(exp) should be read as the same type as exp and the effect is that name is defined having the same type as exp. This implies that the type of name can only be known when the type of exp is known.

Check Exp: Id

An expression consisting of a single identifier, refers to a name introduced in another declaration and gets the type introduced in that declaration.

void collect(current: (Exp) `<Id name>`, Collector c){
c.use(name, {variableId()});

An expression consisting of a single identifier represents a use of that identifier. c.use(name, roles) records this. There are then two possibilities:

  • a matching define is found for one of the given roles: use and definition are connected to each other.
  • no matching define is found and an error is reported.

NOTE: In larger languages names may be defined in different scopes. Scopes do not play a role in Cal.

NOTE: We do not enforce define-before-use in this example, but see XXX how to achieve this.


There is a "TODO" in the documentation source: :add reference for XXX (((TODO:add reference for XXX)))

Check Exp: Boolean and Integer constants

void collect(current: (Exp) `<Boolean boolean>`, Collector c){
c.fact(current, boolType());

void collect(current: (Exp) `<Integer integer>`, Collector c){
c.fact(current, intType());

When encountering a Boolean or integer constant we record their type using c.fact(current, _its type_).

NOTE: The second argument of fact maybe an AType, an arbitrary parse tree (in which case the type of that tree will be used), or a function that returns a type.

Check Exp: parentheses

void collect(current: (Exp) `( <Exp e> )`, Collector c){
c.fact(current, e);
collect(e, c);

The type of an expression enclosed by parentheses is the same as the same of the enclosed expression.

A final, essential, step is to collect constraints from the subpart e.

Check Exp: addition

Addition of integers given an integer result and addition of Booleans gives a Boolean result.

void collect(current: (Exp) `<Exp e1> + <Exp e2>`, Collector c){
c.calculate("addition", current, [e1, e2],
AType (Solver s) {
switch(<s.getType(e1), s.getType(e2)>){
case <intType(), intType()>: return intType();
case <boolType(), boolType()>: return boolType();
default:, "`+` not defined for %t and %t", e1, e2));
collect(e1, e2, c);

Checking addition is more complex since we need the type of sub expressions e1 and e2 in order to do our job. The type of an arbitrary parse tree can be computed using

c.calculate(_informative label_, current, _list of dependencies_, AType(Solver s) { ... }.


  • informative label is a label that can be used in error messages and log files.
  • current is the parse tree for which we are computing a type.
  • list of dependencies is a list of other parse trees whose type is needed in the computation of the type of current.
  • AType(Solver s) { ... } performs the type computation:
    • it is only called when the types of all dependencies are known.
    • it has a Solver as argument: a Solver manages the constraint solving process and is aware of all facts and solved constraints so far and knows, for instance, about the existence of type information for some parse tree.
    • it either returns an AType or reports an error.

The above code could be paraphrased as follows: The type of current can only be computed when the types of e1 and e2 are known. When known, get their types (using s.getType) and distinguish cases: two integer arguments give an integer result type and two Boolean arguments a Boolean result type; otherwise report an error.

A final, essential, step is to collect constraints from the subparts e1 and e2.

Check Exp: multiplication

void collect(current: (Exp) `<Exp e1> * <Exp e2>`, Collector c){
c.calculate("multiplication", current, [e1, e2],
AType (Solver s) {
switch(<s.getType(e1), s.getType(e2)>){
case <intType(), intType()>: return intType();
case <boolType(), boolType()>: return boolType();
default:, "`*` not defined for %t and %t", e1, e2));
collect(e1, e2, c);

Checking multiplication follows exactly the same pattern as checking addition. Even in this simple example we see repetition of code that could be factored out.

Check Exp: conditional expression

void collect(current: (Exp) `if <Exp cond> then <Exp e1> else <Exp e2>`, Collector c){
c.calculate("if Exp", current, [cond, e1, e2],
AType(Solver s){
s.requireEqual(cond, boolType(),
error(cond, "Condition should be Boolean, found %t", cond));
s.requireEqual(e1, e2,
error(current, "Equal types required, found %t and %t", e1, e2));
return s.getType(e1);
collect(cond, e1, e2, c);

Checking a conditional expression amounts to checking that the condition has type Boolean and that the then and else branch have the same type (which also becomes the type of the conditional expression as a whole). In the above code we see s.requireEqual(arg1, arg2, message). Here arg1 and arg2 may either be a subtree (in which case its type is used) or an AType. requireEqual requires that both types are equal or reports an error.

A final, essential, step is to collect constraints from the subparts cond, e1 and e2.

Getting started

module lang::calc::Test

extend lang::calc::Checker;
extend analysis::typepal::TestFramework; // TypePal test utilities
import ParseTree; // In order to parse tests

We need three ingredients for testing:

  • lang::calc::Checker, the Calc type checker we have just completed.
  • analysis::typepal::TestFramework, the TypePal test framework which enables test automation.
  • ParseTree that provides parsing functionality that is needed to parse Calc source text.

Manual testing

TModel calcTModelFromTree(Tree pt){
return collectAndSolve(pt);

Given a parse tree pt for a Calc program, we apply collect and solve to it. This creates a Collector, uses it to collect constraints from pt and then creates a Solver to solve all constraints. The result is a TModel. calcTModelFromTree will also be used during automated testing.

TModel calcTModelFromStr(str text){
pt = parse(#start[Calc], text).top;
return calcTModelFromTree(pt);

In order to obtain a parse tree, we need to parse it as shown above.

TModels contain much information but here we are only interested in the messages that have been generated during constraint solving. They are contained in the messsages field of a TModel.

With the above machinery in place we can perform some experiments:

  • calcTModelFromStr("var x = 3;"),messages gives no messages.
  • calcTModelFromStr("var x = 3+true;").messsages reports that addition of integer and Boolean is illegal.

Automated testing

bool calcTests() {
return runTests([|project://typepal-examples/src/lang/calc/tests.ttl|],
#Calc, calcTModelFromTree);

TypePal's testing framework enables scripting of tests. Each script is written in TTL (TypePal Testing Language). Use runtests(_list of TTL files_, _start non-terminal_, _function to create a TModel_) to execute the scripts.

Here is a sample script:

test Ok1 [[ var x = 1; ]]

test Ok2 [[ var x = 1; var y = x + 2; ]]

test Undef1 [[ var x = y; ]]
expect { "Undefined variable `y`" }

test Add1 [[ var x = 1 + 2; ]]

test Add2 [[ var x = true + false; ]]

test Add3 [[ var x = 1 + true; ]]
expect { "`+` not defined for `int` and `bool`" }

test If1 [[ var x = if true then 1 else 2;
var y = x + 1;]]
test If2 [[ var x = if true then 1 else false; ]]
expect { "Equal types required, found `int` and `bool`" }

test If3 [[ var x = if 1 then 2 else 3; ]]
expect { "Condition should be Boolean, found `int`" }

and here is the produced output:

Test Ok1: true
Test Ok2: true
Test Undef1: true
Test Add1: true
Test Add2: true
Test Add3: true
Test If1: true
Test If2: true
Test If3: true
Test summary: 9 tests executed, 9 succeeded, 0 failed