Hi Alexandre,
> However now that you have written this down, I think your interpretation r
> SSI => r[+] SSI and r[*] SSI is also sound, but I have no proof for this.
I tried to prove this, but failed, and then found a counterexample: e.g. if L = a[+];(!a)[+];a[+], then L is si but both L[+] and L[*] are not si. I believe if one in addition requires that L is prefix-closed or suffix-closed, then the result would hold (I have an idea how to prove this).
> > > I would suggest that at the SERE level the rule for the n-ary
> concatenation
> > > r1;r2;r3;...;rn
> > > could be extended so that the entire concatenation is SSI iff each
> > > ri is SSI and at most one of them rejects the empty word.
Again, this seems wrong, with the following counterexample:
If K=[*0] | ((!a)[+] ; a[+]) and L=a[+] then K contains the empty word, K and L are si, but K;L is not si: it accepts !a;a;a but rejects !a;a.
I think the result would hold if one replaces the assumption that K or L contains the empty word by K being prefix-closed or L being suffix-closed.
> > > Transposed to {s}|=>t, that rule would state that {s}|=>t is SSI iff
> > > - s and t are both SSI, and
> > > - s recognizes the empty word
Again, I think we could require either s being prefix-closed or t being suffix-closed. Note the symmetry with s;t :-)
Regards,
Victor.
