The degrees of nondeterminism in pushdown automata

Measuring nondeterminism in pushdown automata

“…(This definition is equivalent to the one in [20], which is a corrected version of the definition in [22].) This measure also equals the number of tokens needed to convert a CFL to a DCFL when inserted into the strings of the language at strategic points [22,15,20]. …”

“…However, this would require proving that there cannot be any coupling or entanglement between the two parts of the constructed string (the part over the alphabet and the part in K * ), that is, it would require proving that the Turing machine computation cannot contain information that would allow the PDA to distinguish between strings of the form a i b i and a i b 2i deterministically when i is large. 15 To circumvent this problem, we shall require that all inserts, instead of merely being in a * b * $, are in fact in K 0 . This restriction does not introduce any additional nondeterminism since K 0 is a deterministic CFL.…”

“…Thus, K * is a ∅-padded language. 15 For instance, if the substring c i d i e i occurred earlier in the encoding of a Turing machine computation which verifies that all three exponents are equal, then the PDA could deterministically use its stack to match a subsequent a i against e i , match b i against d i , and then, if any b's remain, match them against c i . To prove that this cannot happen, we would have to prove that the information which the PDA writes on its stack while reading c i d i e i must be consumed while checking the validity of the Turing machine computation, and therefore cannot also be used to assist in recognizing a string in K. We claim that the input to is a proper prefix of ua p b p , and hence that the inputs to 1 and 2 are the same since all of our PDAs are normal.…”

“…As another example, consider the following question. Nasyrov has extended the "repairable" version of the depth measure (the number of tokens that must be inserted into strings of a context-free language to make the language deterministic) into a dynamic measure of nondeterminism by counting the number of tokens as a function of the length of the string [14,15]. Are there any relationships between these two dynamic measures, the minmax measure and Nasyrov's extension of repairability?…”

“…However, for precisely this reason, the complexity hierarchy of context-free languages under this measure collapses into just three classes: deterministic context-free languages, finite unions of deterministic context-free languages, and all context-free languages. 2 Indeed, Vermeir and Savitch [22] and Nasyrov [15] both cite the inability of this dynamic measure to produce a hierarchy of more than three levels as a justification for focusing attention on the static measure of nondeterminism (which Nasyrov generalizes to produce a dynamic measure [14,15]), or on a dynamic measure of the amount of ambiguity rather than of nondeterminism in a pushdown automaton [15]. While we share the view that the inability to produce a meaningful hierarchy is a critical weakness of this measure, we believe that this is a defect in the definition of the dynamic measure used in [22], not in the concept of a dynamic measure.…”