Nondeterminism and boolean operations in pda's

Detlef Wotschke
1978 Journal of computer and system sciences (Print)  
There are nondeterministic context-free languages that cannot be expressed as a Boolean formula over deterministic context-free languages. The closure of the contextfree languages under intersection does not yield closure under complementation. The question: "How much more powerful is nondeterminism than determinism I" has intrigued almost everyone in the area of theoretical computer science for many years. A classical problem in this context is whether deterministic and nondeterministic
more » ... ounded automata are equally powerful [8, 131. A more recent problem is whether iVP = ?P [l, 9, lo]. These problems are intrinsically difficult and have withstood many attempts despite tremendous efforts. For pushdown automata (pda's) it has, of course, been known for quite some time that nondeterministic pda's can do more than deterministic pda's [3, 61. But very little is known on how much more powerful nondeterminism is in pda's than determinism. We will shed some light onto this question by investigating Boolean operations on deterministic context-free languages. It is well known that applying intersection as an operation to the deterministic contextfree languages leads already outside of the context-free languages, e.g., {U%V / n > I}. Hence, an arbitrary finite number of Boolean (union, intersection, complementation) operations applied to the deterministic context-free languages seems to be a powerful sequence of operations. So the immediate question arises whether nondeterminism in a pda can be expressed as a Boolean formula over the deterministic subparts, or in other words, whether the context-free languages are contained, and if so, then properly, in the Boolean closure of the deterministic context-free languages. If the answer were affirmative then context-free languages could be parsed in linear time. We will show, however, that there are nondeterministic context-free languages that cannot be represented as a Boolean expression of deterministic context-free languages. This is a nontrivial extension of the result in [6] which states that there are context-free languages that cannot be expressed as the finite union of deterministic context-free
doi:10.1016/0022-0000(78)90030-2 fatcat:gj4dxcykc5dhbaly6mcyo5gmzu