*Can Programming Be Liberated from the von Neumann Style? A Functional Style and Its Algebra of Programs. (Comm. ACM 21.8, Aug. 1978, 613-641)*he developed an amazing framework for functional programming, from theoretical foundations to implementation hints, e.g. for installation, user privileges, and system self-protection. In a nutshell, his FP system comprises

- a set O of objects (atoms or sequences)
- a set F of functions that map objects into objects (f : O |-> O}
- an operation,
*application*(very roughly, eval) - a set FF of functional forms, used to combine functions or objects to form new functions in F
- a set D of definitions that map names to functions in F

Def Innerproduct = (Insert +) o (ApplyToAll x) o Transposeand wanted to bring it to life - slightly adapted to Tcl style, especially by replacing the infix operator "o" with a Polish prefix style:

Def Innerproduct = {o {Insert +} {ApplyToAll *} Transpose}Unlike procs or lambdas, more like APL or RPN, this definition needs no variables - it declares (from right to left) what to do with the input; the result of each step is the input for the next step (to the left of it). In an RPN language, the example might look like this:

/Innerproduct {Transpose * swap ApplyToAll + swap Insert} defwhich has the advantage that execution goes from left to right, but requires some stack awareness (and some swaps to set the stack right ;^)Implementing

**Def**, I took an easy route by just creating a proc that adds an argument and leaves it to the "functional" to do the right thing (with some quoting heaven :-)

proc Def {name = functional} { proc $name x "\[$functional\] \$x" }For functional composition, where, say for two functions f and g,

[{o f g} $x] == [f [g $x]]again a proc is created that does the bracket nesting:

proc o args { set body return foreach f $args {append body " \[$f"} set name [info level 0] proc $name x "$body \$x [string repeat \] [llength $args]]" set name }Why Backus used

**Transpose**on the input, wasn't first clear to me, but as he (like we Tclers) represents a matrix as a list of rows, which are again lists (also known as vectors), it later made much sense to me. This code for transposing a matrix uses the fact that variable names can be any string, including those that look like integers, so the column contents are collected into variables named 0 1 2 ... and finally turned into the result list:

proc Transpose matrix { set cols [iota [llength [lindex $matrix 0]]] foreach row $matrix { foreach element $row col $cols { lappend $col $element } } set res {} foreach col $cols {lappend res [set $col]} set res }An integer range generator produces the variable names, e.g

iota 3 => {0 1 2} proc iota n { set res {} for {set i 0} {$i<$n} {incr i} {lappend res $i} set res } #-- This "functional form" is mostly called [map] in more recent FP: proc ApplyToAll {f list} { set res {} foreach element $list {lappend res [$f $element]} set res }...and

**Insert**is better known as

**fold**, I suppose. My oversimple implementation assumes that the operator is one that expr understands:

proc Insert {op arguments} {expr [join $arguments $op]}#-- Prefix multiplication comes as a special case of this:

interp alias {} * {} Insert *#-- Now to try out the whole thing:

Def Innerproduct = {o {Insert +} {ApplyToAll *} Transpose} puts [Innerproduct {{1 2 3} {6 5 4}}]which returns 28 just as Dr. Backus ordered (= 1*6 + 2*5 + 3*4). Ah, the joys of weekend Tcl'ing... - and belatedly, Happy Birthday, John! :)Another example, cooked up by myself this time, computes the average of a list. For this we need to implement the

**construction**operator, which is sort of inverse mapping - while mapping a function over a sequence of inputs produces a sequence of outputs of that function applied to each input, Backus' construction maps a sequence of functions over one input to produce a sequence of results of each function to that input, e.g.

[f,g](x) == <f(x),g(x)>Of course I can't use circumfix brackets as operator name, so let's call it

*constr*:

proc constr args { set functions [lrange $args 0 end-1] set x [lindex $args end] set res {} foreach f $functions {lappend res [eval $f [list $x]]} set res } #-- Testing: Def mean = {o {Insert /} {constr {Insert +} llength}} puts [mean {1 2 3 4 5}]which returns correctly 3. However, as integer division takes place, it would be better to make that

proc double x {expr {double($x)}} Def mean = {o {Insert /} {constr {Insert +} dlength}} Def dlength = {o double llength} puts [mean {1 2 3 4}]giving the correct result 2.5. However, the auxiliary definition for

*dlength*cannot be inlined into the definition of

*mean*- so this needs more work... But this version, that maps

*double*first, works:

Def mean = {o {Insert /} {constr {Insert +} llength} {ApplyToAll double}}One more experiment, just to get the feel:

Def hypot = {o sqrt {Insert +} {ApplyToAll square}} Def square = {o {Insert *} {constr id id}} proc sqrt x {expr {sqrt($x)}} proc id x {set x} puts [hypot {3 4}]which gives 5.0. Compared to an RPN language,

*hypot*would be

/hypot {dup * swap dup * + sqrt} defwhich is shorter and simpler, but meddles more directly with the stack.An important functional form is the conditional, which at Backus looks like

p1 -> f; p2 -> g; hmeaning, translated to Tcl,

if {[p1 $x]} then {f $x} elseif {[p2 $x]} then {g $x} else {h $x}Let's try that, rewritten Polish-ly to:

cond p1 f p2 g h proc cond args { set body "" foreach {condition function} [lrange $args 0 end-1] { append body "if {\[$condition \$x\]} {$function \$x} else" } append body " {[lindex $args end] \$x}" set name [info level 0] proc $name x $body set name } #-- Testing, with [K] in another role as Konstant function :) Def abs = {cond {> 0} -- id} proc > {a b} {expr {$a>$b}} proc < {a b} {expr {$a<$b}} proc -- x {expr -$x} puts [abs -42],[abs 0],[abs 42] Def sgn = {cond {< 0} {K 1} {> 0} {K -1} {K 0}} proc K {a b} {set a} puts [sgn 42]/[sgn 0]/[sgn -42] #--Another famous toy example, reading a file's contents: Def readfile = {o 1 {constr read close} open} #--where Backus' '''selector''' (named just as integer) is here: proc 1 x {lindex $x 0}

IL: I just started reading Backus's paper last night, and it hurts! I'd envisioned a similar alternative to modern languages a while ago; except my ideas which i though revolutionary had been documented in 1978. I see a lot of similarities between unix philosophy and Backus's core premise (summarized poorly by myself here): modern languages and tools need to angle towards simpler interfaces, not bloated systems of apis.I think even in his examples perhaps his functional forms are still too basic, though I like some ideas, like his sequences, which like you mention, are very similar to tcl's treatment of lists. The introduction mentions he was tasked with developing a language around these ideas, what happened to it?RS: The first implementation was simply calls "FP" ([2]). From 1989 on, there was a successor "FL" ([3]). The latest chip off that tree seems to be "NGL" ([4]).NEM: The style of programming by composing functions, without mentioning any variables, is sometimes referred to as "Point Free Style". A discussion at [5] has some links to material of related interest. Incidentally, does anyone know where this use of the term "point" to mean argument/name/variable comes from? e.g. "point-free", "fixpoint" (in relation to the Y-combinator) etc. -- RS 2009-06-04: http://haskell.org/haskellwiki/Pointfree discusses the term.IL: why work when you can research fascinating languages? :) Following the wiki links, the J language also appears to be a member in the philosophy, and according to the limited propaganda on the site enjoys appreciation at the least, as well as a 64 bit implementation! "J" ([6])

See also Tacit programming for another chapter to this story... e.g.

Def mean = fork /. sum llength

IL, what is the difference between an interface and an API? Is an API not an interface? -MoritzMoritz, sorry, bad choice of terms. I guess I meant flexible and expressive language design vs. a more traditional approach with a look that resembles the look of C++, and but has a very large runtime download... (but I'm not naming names!)