The Concepts and Confusions of Pre-fix, In-fix, Post-fix and Fully Functional Notations
Xah Lee, 2006-03-15 Let me summarize: The LISP notation, is a functional notation, and is not a so-called pre-fix notation or algebraic notation. Algebraic notations have the concept of operators, meaning, symbols placed around arguments. In algebraic in-fix notation, different symbols have different stickiness levels defined for them. e.g. “3+2*5>7” means “(3+(2*5))>7”. The stickiness of operator symbols are normally called “Operator Precedence”. It is done by giving a order specification for the symbols, or equivalently, give each symbol a integer index, so that for example if we have “a⊗b⊙c”, we can unambiguously understand it to mean one of “(a⊗b)⊙c” or “a⊗(b⊙c)”. In a algebraic post-fix notation known as Polish Notation, there needs not to have the concept of Operator Precedence. For example, the in-fix notation “(3+(2*5))>7” is written as “3 2 5 * + 7 >”, where the operation simply evaluates from left to right. Similarly, for a pre-fix notation syntax, the evaluation goes from right to left, as in “> 7 + * 5 2 3”. While functional notations, do not employ the concept of Operators, because there is no operators. Everything is a syntactically a “function”, written as f(a,b,c...). For example, the same expression above is written as “>( +(3, *(2,5)), 7)” or “greaterThan( plus(3, times(2,5)), 7)”. For lisps in particular, their fully functional notation is historically termed sexp (short for S-Expression, where S stands for Symbolic). It is sometimes known as Fully Parenthesized Notation. For example, in lisp it would be (f a b c ...). In the above example it is: “(> (+ 3 (* 2 5)) 7)”. The common concepts of “pre-fix, post-fix, in-fix” are notions in algebraic notations only. Because in Full Functional Notation, there is no concept of where one places the “operator” or function. There is always just a single position given with explicitly enclosed arguments. Another way to see that lisp notation are not “pre” anything, is by realizing that the “head” f in (f a b c) can be defined to be placed anywhere. e.g. (a b c f) or even (a f b c), and it's still not pre- or in- or post- anything. For example, in the language Mathematica, f(a b c) would be written as f[a,b,c] where the argument enclosure symbols is the square bracket instead of parenthesis, and argument separator is comma instead of space, and the function symbol (or head) is placed in outside and in front of the argument enclosure symbols. The reason for the misconception that lisp notations are “pre-fix” is because the head appears before the enclosed arguments. Such “pre-fix” has no signifance in Full Functional Notation systems and can only engender confusion in the Algebraic Pre-fix Notation systems where the term has significance. 2000-02-21 The common name for the lisp way is Fully Parenthesized Notation. This syntax is the most straightforward to represent a tree, but it's not the only choice. For example, one could have Fully Parenthesized Notation by simply moving the semantics of the first element to the last. You write (arg1 arg2 ... f) instead of the usual (f arg1 arg2). Like wise, you can essentially move f anywhere and still make sense. In Mathematica, they put the f in front of the paren, and use square brackets instead. e.g. f[a,b,c], Sin[3], Map[f,list] ... etc. The f in front of parent makes better conventional sense until f is itself a list which then we'll see things like f[a,b][c, g[3,h]] etc. It's worse when there are arbitrary nesting of heads. A pre-fix notation in Mathematica is represented as [EMAIL PROTECTED] Essentially, a pre-fix notation in this context limits it to uses for function that has only one argument. More example: [EMAIL PROTECTED]@[EMAIL PROTECTED] is equivalent to “f[a[b[c]]]” or in lispy “(f (a (b c)))”. A post-fix notation is similar. In Mathematica it is, e.g. “c//b//a//f”. For example “List[1,2,3]//Sin” is syntactically equivalent to “Sin[List[1,2,3]]” or [EMAIL PROTECTED],2,3]”. (and they are semantically equivalent to “Map[Sin, List[1,2,3]]” in Mathematica) For in-fix notation, the function symbol is placed between its arguments. In Mathematica, the generic form for in-fix notation is by sandwiching the tilde symbol around the function name. e.g. “Join[List[1,2],List[3,4]]” can be written as “List[1,2] ~Join~ List[3,4]”. In general, when we say C is a in-fix notation language, we don't mean it's strictly in-fix but the situation is one-size-fits-all for convenience. Things like “i++”, “++i”, “for(;;)”, 0x123, “sprint(...%s...,...)”, ... are syntax whimsies. (that is, a ad hoc syntax soup) In Mathematica for example, there is quite a lot syntax sugars beside the above mentioned systimatic constructs. For instance, Plus[a,b,c] can be written in the following ways: “(a+b)+c” or “a+b+c” or “(a+b)~Plus~c” The gist being that certain functions such as Plus is assigned a special symbol '+' with a particular syntax form to emulate the irregular and inefficient but nevertheless well-understood conventional notation. For another example: Times[a,b] can be also written as “a*b” or just “a b”. Mathematica also have C language's convention of “i++”, “++i”, “i+=1” for examples. As a side note, the Perl mongers are proud of their slogan of There Are More Than One Way To Do It in their gazillion ad hoc syntax sugars but unaware that in functional languages (such as Mathematica, Haskell, Lisp) that there are consistent and generalized constructs that can generate far far more syntax variations than the ad hoc prefixed Perl both in theory AND in practice. (in lisps, their power syntax variation comes in the guise of macros.) And, more importantly, Perlers clamor about Perl's “expressiveness” more or less on the useless syntax level but don't realize that semantic expression is what's really important. ---- This post is archived at: http://xahlee.org/UnixResource_dir/writ/notations.html Xah [EMAIL PROTECTED] ∑ http://xahlee.org/ -- http://mail.python.org/mailman/listinfo/python-list
