Existential width and universal width respectively quantify the amount of nondeterminism and parallelism present in an alternating finite automaton (AFA). These measures can be defined as either worst case measures, known as maximal existential and universal widths, or as the best case measures, referred to as optimal existential and universal widths. We investigate upper bounds on the finite optimal widths of AFAs and construct several examples of AFAs that attain finite optimal widths strictly larger than any achievable finite maximal width. We establish an upper bound on the finite optimal universal width of unary universal finite automata (UFAs) and derive a partial upper bound on the finite optimal existential width of unary nondeterministic finite automata (NFAs).
We consider shuffle ideals and their finite bases, both represented by regular expressions. We introduce an operation that, given a regular expression α∈_fin representing a finite language, returns an expression for the shuffle ideal . This operation induces a new combinatorial class, which we call _SI . For both classes, we define the Antimirov automaton and provide results on their state complexity in the worst as well as the average case. For α∈_fin , in the worst case, the number of states is at most |α |_ Σ , the number of letters in α . The asymptotic average is, however, bounded by |α |_ Σ /2 . For τ∈_SI , in the worst case, the number of states is at most |τ |_ Σ ^⋆ , the number of occurrences of Σ ^⋆ in τ . On the other hand, the asymptotic average number of states is bounded by |τ |_ Σ ^⋆/2 .
We investigate the state complexity of the boundary operation on the classes of closed and ideal languages. We establish a tight upper bound of (n+2)2^n-3+2 for prefix-closed languages, n(n-1)/2+2 for suffix-closed languages, and n+2 for factor- and subword-closed languages. To describe witnesses, we use a binary alphabet for prefix-closed languages, and a ternary alphabet for the remaining classes. In the binary case, we derive a lower bound of n(n-1)/2 for suffix-closed languages, and a tight upper bound of n+1 for factor- and subword-closed languages. The state complexity of the boundary of unary closed languages is n. Since the boundary of a language coincides with that of its complement, and the complement of a closed language is an ideal language, all our results for closed languages extend to the classes of right, left, two-sided, and all-sided ideals.
Guo et al. proved that the number of palindromes occurring in the set of all conjugates of a word is bounded above by two. In this paper, we show that for an antimorphic involution θ , the number of θ -palindromes occurring in the set of all θ -conjugates of a word is bounded above by six.
Context-conditional grammars have been introduced by Masopust and Meduna about 20 years ago, when also first few descriptional complexity results have been obtained. These grammars generalize both semi-conditional and generalized forbidding grammars. We extend these studies considerably and derive some descriptional complexity results which are new characterizations of the class of recursively enumerable languages by parsimonious resources. As context-conditional grammars offer a broad variety of natural descriptional complexity parameters, we will focus on the number of nonterminals, the degree, the index, and the number of conditional rules, which can be refined by additionally counting the non-simple rules and the non-semi-conditional rules. Most parameters are known from the literature studying descriptional complexity aspects of semi-conditional grammars, apart from the index which is the pair of numbers formed by the maximum cardinality of any permitting context set and the maximum cardinality of any forbidden context set. We also introduce a novel way of illustrating the proofs that eases the reader to keep track of the various case distinctions.
