Definition talk:Balanced String

How generic is this? It appears that it is assumed that all alphabets of prop/pred calc contain parentheses, and just one style of them. In the current context, with the particular treatments that have been implemented on this site, it is indeed the case that we have $($ and $)$ - but different treatments use different brackets (e.g. 1996: H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability uses $[$ and $]$).

I appreciate that I am one of those tedious mathematically-mediocre pedants desperately in need of a length of rope and a sturdy tree-branch, but I thought I'd mention this. --prime mover (talk) 22:19, 9 November 2012 (UTC)

You could simply delete $($ and $)$... I came up with the very definition myself to make referring to it less ad hoc. --Lord_Farin (talk) 22:21, 9 November 2012 (UTC)

... and are we sure there is only one style of brackets in any given alphabet? Maybe we can formalise this to a "balanceable pair" of symbols, or something, and make this a definition of general formal systems. Then the proof would be "Parentheses in PropLog are balanceable pairs". Maybe. Dunno. Just a thought. --prime mover (talk) 22:31, 9 November 2012 (UTC)

You could take this even further. I don't see merit at the current point; it'd only obfuscate what we're trying to express. As a possible generalisation, consider additive homomorphisms from the Kleene closure of the alphabet to $\Z$; mapping all but two to zero, and those two to $\pm 1$ brings us to a balanced string as one in the kernel of that homomorphism. --Lord_Farin (talk) 22:38, 9 November 2012 (UTC)

Outside my limited experience, but if so, then yeah right ... --prime mover (talk) 22:39, 9 November 2012 (UTC)