[FOM] Formal grammar question
a.hazen at philosophy.unimelb.edu.au
Tue Nov 8 22:12:50 EST 2005
Stupid, stupid, me!
In yesterday's post I asked about the formal grammar (or
computational complexity) aspects of parenthesis-free, "Polish,"
Well-formedness of Polish propositional formulas, like that of
formulas in traditional (parentheses and infix operators) notation,
is recognizable by a push-down automaton making a single, one-way,
pass through the candidate string. For simplicity, assume that all
operators are binary: then a well-formed formula will contain one
more atom than operator, and no initial segment of it will contain
more atoms than operators. So, starting from the left, the
automaton will "push" (add 1 to its pushdown stack) when it gets to
an operator, "pop" (subtract 1) when it gets to an atom, and accept
iff it reaches the right end with 1 in its stack, never having read
an atom with 0 in ist stack.
This can be converted straightforwardly into a definition of WFF
in terms of the basis of Goodman and Quine's 1947 article.
(I've been trying to figure out whether there is something of
mathematical interest in their article, given that the philosophical
project that motivated them has since been generally dismissed --
even by Quine in later writings -- as misguided. It comes as a
relief that they at least aren't depending on special features of the
(Thanks to Charles Silver for off-forum note on this.)
University of Melbourne
More information about the FOM