Algebraic Types

An algebraic type is a type that is an operand in an algebra of types.

In an algebra of types, arithmetic operators such as + and * are overloaded with domain-specific meanings, and types themselves, rather than instances of the types, are the operands. Using these operators, more complex types can be assembled from the more basic types. Such type algebras serve as the theoretical underpinnings of the type systems found in functional languages like Haskell, where a type algebra is used to declare Abstract Data Types.

Implementation by Serguey Zefirov

149 lines with little testing code and comments. That's why I like Tcl - complex language flow control construct in such a snippet.

Actually, it was my second attempt. First was a lot uglier and bigger.

# Algebraic types.
# Creating constructors and deconstruction by pattern matching.

namespace eval atypes {
    # List of defined constructors.
    # Actually, it is an map from constructor names to their typenames,
    # and used to keep track of constructor name uniqueness.
    array set defcons {}
    # Create algebraic type.
    # Argument 'typename' is useless, but FP languages need it.
    # 'args' argument is a list of constructor
    # descriptions. They are in the form
    # ConstructorName ?param? ?param ...?.
    # ConstructorName should be unique.
    # At the end we create a bunch of commands.
    proc atype {typename args} {
        variable defcons
        # Defining constructors for type.
        foreach c $args {
            # Split constructor definition into name and args
            set cvars [lrange $c 1 end]
            set constrname [lindex $c 0]
            if {![string is upper [string range $constrname 0 0]]} {
                error "Incorrect constructor name '$constrname' for type $typename"
            if {[info exists defcons($constrname)]} {
                error "Constructor $constrname (type $typename) was already defined for type $defcons($constrname)"
            # We ought to create constructor header and
            # construction part. Constructor header is a list
            # of variables.
            set construction [list $constrname]
            foreach cv $cvars {
                if {[llength $cv]!=1} {
                    error "Improper argument name $cv for $constrname (type $typename)"
                lappend construction "\[set $cv\]"
            set construction [join $construction " "]
            proc ::$constrname $cvars "return \[list $construction\]"
    # Match a constructed value against pattern(s).
    # Pattern can be:
    # $<varname> - if element starts from $, then [string range 1 end]
    #              is considered variable name to bind.
    # _ - for dummy variable (expression is matched but dropped from
    #     assignment)
    # {ConstrName ?pattern pattern ...?} - match of constructed expression
    proc match {expr patterns} {
        foreach {p pcode} $patterns {
            # Try to match
            foreach {ok bindlist} [matchbind $expr $p] break
            if {$ok} {
                foreach {var val} $bindlist {
                    uplevel 1 [list set $var $val]
                set rcode [catch {uplevel 1 $pcode} result]
                return -code $rcode $result
        error "$expr does not match patterns $patterns"
    # Auxillary function.
    # Actually tries to match expression and pattern.
    # Returns [list 1 bindlist] for success and [list 0 {}] for failure.
    proc matchbind {expr pattern} {
        # pattern might be a '_':
        if {[string equal _ $pattern]} {
            return {1 {}}
        # pattern might be a $var:
        if {[string equal \$ [string range $pattern 0 0]]} {
            return [list 1 [list [string range $pattern 1 end] $expr]]
        # Complex structural match case.
        # First, 'shape' of pattern binding and
        # expression should match:
        if {[llength $expr]!=[llength $pattern]} {
            return {0 {}}
        # Split pattern and expression into respective constructors
        # and arguments.
        set subps  [lrange $pattern 1 end]
        set pattconstr [lindex $pattern 0]
        set subexprs [lrange $expr 1 end]
        set exprconstr [lindex $expr 0]
        # Second, head of expression and pattern should be equal.
        if {![string equal $pattconstr $exprconstr]} {
            return {0 {}}
        # Okay, then we should match every expression with
        # every pattern, going recursively, if needed.
        set binds {}        ;# bind list
        foreach p $subps e $subexprs {
            # Match subexpression with subpattern
            foreach {ok bindadd} [matchbind $e $p] break
            if {!$ok} {
                # If not matched - fail.
                return {0 {}}
            # Otherwise - grow bind pars list.
            set binds [concat $binds $bindadd]
        # Return success
        return [list 1 $binds]
    # Export such a useful command:
    namespace export atype match
catch {rename match {}}
catch {rename atype {}}
namespace import atypes::*

# tests:
if 1 {
    # Maybe type (single element list)
    atype Maybe {Nothing} {Just a}
    # List as it should be:
    atype List {Nil} {List head tail}
    # Verifying construction:
    puts [Nothing]
    puts [Just "hello, world!"]
    puts [List ? [List ! Nil]]
    # Verifying matching.
    proc testmatch {e} {
        match $e {
            {Just $x} {puts "Just x branch: $x"}
            {Nothing} {puts "Nothing branch"}
            {List $head {List $head2 $tail2}} {
                puts "Complex list branch."
                puts "head '$head', head2 '$head2', tail2 '$tail2'"
            {List $head $tail} {
                puts "List branch: head '$head' tail '$tail'"
            {generate error} {
                puts "We will generate an error"
                error "Error was generated"
            _ {
                puts "Unknown expression $e"
    testmatch [Just "what?"]
    testmatch [Nothing]
    testmatch [List ? [List ! Nil]]
    testmatch [List ? Nil]
    testmatch ???
    testmatch "generate error"

You may see that we testing a match with arbitrary string in

testmatch "generate error"

This is perfectly Ok and I mean it from the start. "Everything is a string", isn't it?

I eliminated use of lassign so it doesn't need Tclx anymore. So I (and you) can use it on my (or your) Linux notebook without any upgrades.

Take a look at Simple BDD for demonstration of capabilities.

I've used algebraic types for expression manipulation. An example could be found here: Expression Tree Package.

NEM notes that this code appears to be written by SZ. I've also had a go at implementing algebraic pattern matching while playing with Monadic TOOT, and you can see the code on that page. Haskell is full of interesting ideas...

Implementation by NEM

NEM 2009-05-15: Here is a version of algebraic data-types that can handle matching multiple values simultaneously:

# datatype.tcl --
#       Algebraic datatypes and pattern matching in Tcl.

package require Tcl         8.5
package provide datatype    0.1

namespace eval ::datatype {
    namespace export define match matches
    namespace ensemble create

    # Datatype definitions
    proc define {type = args} { 
        set ns [uplevel 1 { namespace current }]
        foreach cons [split [join $args] |] {
            set name [lindex $cons 0]
            set args [lrange $cons 1 end]
            proc $ns\::$name $args [format {
                lreplace [info level 0] 0 0 %s
            } [list $name]]
        return $type

    # Pattern matching
    # matches pattern value envVar --
    #   Returns 1 if value matches pattern, else 0
    #   Binds match variables in envVar
    proc matches {pattern value envVar} {
        upvar 1 $envVar env
        if {[var? $pattern]} { return [bind env $pattern $value] }
        if {[llength $pattern] != [llength $value]} { return 0 }
        if {[lindex $pattern 0] ne [lindex $value 0]} { return 0 }
        foreach pat [lrange $pattern 1 end] val [lrange $value 1 end] {
            if {![matches $pat $val env]} { return 0 }
        return 1
    # A variable starts with lower-case letter or _. _ is a wildcard.
    proc var? term { string match {[a-z_]*} $term }
    proc bind {envVar var value} {
        upvar 1 $envVar env
        if {![info exists env]} { set env [dict create] }
        if {$var eq {_}} { return 1 }
        dict set env $var $value
        return 1
    proc match args {
        #puts "MATCH: $args"
        set values [lrange $args 0 end-1]
        set choices [lindex $args end]
        append choices \n [list return -code error -level 2 "no match for $values"]
        set f [list values $choices [namespace current]]
        lassign [apply $f $values] env body
        #puts "RESULT: $env -> $body"
        dict for {k v} $env { upvar 1 $k var; set var $v }
        catch { uplevel 1 $body } msg opts
        dict incr opts -level
        return -options $opts $msg
    proc case args {
        upvar 1 values values
        set patterns [lrange $args 0 end-2]
        set body [lindex $args end]
        set env [dict create]
        if {[llength $patterns] != [llength $values]} { return }
        foreach pattern $patterns value $values {
            if {![matches $pattern $value env]} { return }
        return -code return [list $env $body]
    proc default body { return -code return [list {} $body] }

As an example of use, here is an implementation of insertion into a Red-Black tree, as described at [L1 ] (based on the Haskell code there):

datatype define Color = R | B
datatype define Tree  = E | T color left val right

# balance :: Color -> Tree a -> a -> Tree a -> Tree a
proc balance {color left val right} {
    datatype match $color $left $val $right {
        case B [T R [T R a x b] y c] z d -> { T R [T B $a $x $b] $y [T B $c $z $d] }
        case B [T R a x [T R b y c]] z d -> { T R [T B $a $x $b] $y [T B $c $z $d] }
        case B a x [T R [T R b y c] z d] -> { T R [T B $a $x $b] $y [T B $c $z $d] }
        case B a x [T R b y [T R c z d]] -> { T R [T B $a $x $b] $y [T B $c $z $d] }
        case col a x b                   -> { T $col $a $x $b }
# insert :: Ord a => a -> Tree a -> Tree a
proc insert {x s} {
    datatype match [ins $x $s] {
        case [T _ a y b]  -> { T B $a $y $b }
# ins :: Ord a => a -> Tree a -> Tree a
proc ins {x s} {
    datatype match $s {
        case E               -> { T R E $x E }
        case [T col a y b]   -> {
            if {$x < $y} { return [balance $col [ins $x $a] $y $b] }
            if {$x > $y} { return [balance $col $a $y [ins $x $b]] }
            return $s
# Test on random numbers:
set tree [E]
set i 0
while {[incr i] < 20} {
    set n [expr {int(rand()*100)}]
    set tree [insert $n $tree]
puts $tree


  • Assumes each element is a well-formed list
  • Assumes string equality comparison
  • No way to match a literal string beginning with a lower-case letter or underscore

I believe these can all be solved by treating each type itself as an ensemble and having per-type equal and match "methods" (with built-ins for strings and numbers).


The Algebra of Data, and the Calculus of Mutation , 2009-04-18
What the Heck are Algebraic Data Types? ( for Programmers ) , Daniel Eklund, 2011-12
The Algebra of Algebraic Data Types, Part 1 , Chris Taylor, 2013-02-10
The Derivative of a Regular Type is its Type of One-Hole Contexts