I know some basic logic symbols, but i'm not sure what this formula means:

Fragment: http://i35.tinypic.com/14jt1n9.png

Full paper: http://www.newton.ac.uk/preprints/NI07003.pdf

in particular, what is the subscript "1<=i

and the N with the A superscript. the multiple typefaces of "A".

this is beyond the basic and/or/exists/forall that i know in basic logic

is there a statement of this formula and its conditions in english words?

Thanks!

**Updated: oops my bad, I should've read the part where they defined it!**