ω-automaton

About: ω-automaton is a research topic. Over the lifetime, 2299 publications have been published within this topic receiving 68468 citations. The topic is also known as: stream automaton & ω-automata.

5′ → 3′ Watson-Crick Automata With Several Runs

Abstract: 5′ → 3′ WK-automata are Watson-Crick automata whose two heads start on opposite ends of the input word and always run in opposite directions. One full reading in both directions is called a run. We prove that the expressive power of these automata increases with every additional run that they can make, both for deterministic and non-deterministic machines. This defines two incomparable infinite hierarchies of language classes between the regular and the context-sensitive languages. These hierarchies are complemented with classes defined by several restricted variants of 5′ → 3′ WK-automata like stateless automata. Finally we show that several standard problems are undecidable for languages accepted by 5′ → 3′ WK-automata in only one run, for example the emptiness and the finiteness problems.

Measures of Nondeterminism in Finite Automata

Abstract: While deterministic finite automata seem to be well understood, surprisingly many important problems concerning nondeterministic finite automata (nfa's) remain open. One such problem area is the study of different measures of nondeterminism in finite automata. Our results are: 1. There is an exponential gap in the number of states between unambiguous nfa's and general nfa's. Moreover, deterministic communication complexity provides lower bounds on the size of unambiguous nfa's. 2. For an nfa A we consider the complexity measures adviceA(n) as the number of advice bits, ambigA(n) as the number of accepting computations, and lea fA(n) as the number of computations for worst case inputs of size n. These measures are correlated as follows (assuming that the nfa A has at most one "terminally rejecting" state): adviceA(n); ambigA(n) ≤ leafA(n) ≤ O(adviceA(n) ċ ambigA(n)). 3. leafA(n) is always either a constant, between linear and polynomial in n, or exponential in n. 4. There is a language for which there is an exponential size gap between nfa's with exponential leaf number/ambiguity and nfa's with polynomial leaf number/ambiguity. There also is a family of languages KONm2 such that there is an exponential size gap between nfa's with polynomial leaf number/ambiguity and nfa's with ambiguity m.

Unitarity in one dimensional nonlinear quantum cellular automata

Abstract: Unitarity of the global evolution is an extremely stringent condition on finite state models in discrete spacetime. Quantum cellular automata, in particular, are tightly constrained. In previous work we proved a simple No-go Theorem which precludes nontrivial homogeneous evolution for linear quantum cellular automata. Here we carefully define general quantum cellular automata in order to investigate the possibility that there be nontrivial homogeneous unitary evolution when the local rule is nonlinear. Since the unitary global transition amplitudes are constructed from the product of local transition amplitudes, infinite lattices require different treatment than periodic ones. We prove Unitarity Theorems for both cases, expressing the equivalence in 1+1 dimensions of global unitarity and certain sets of constraints on the local rule, and then show that these constraints can be solved to give a variety of multiparameter families of nonlinear quantum cellular automata. The Unitarity Theorems, together with a Surjectivity Theorem for the infinite case, also imply that unitarity is decidable for one dimensional cellular automata.

Profile trees for Büchi word automata, with application to determinization

Abstract: The determinization of Buchi automata is a celebrated problem, with applications in synthesis, probabilistic verification, and multi-agent systems. Since the 1960s, there has been a steady progress of constructions: by McNaughton, Safra, Piterman, Schewe, and others. Despite the proliferation of solutions, they are all essentially ad-hoc constructions, with little theory behind them other than proofs of correctness. Since Safra, all optimal constructions employ trees as states of the deterministic automaton, and transitions between states are defined operationally over these trees. The operational nature of these constructions complicates understanding, implementing, and reasoning about them, and should be contrasted with complementation, where a solid theory in terms of automata run dags underlies modern constructions.In 2010, we described a profile-based approach to Buchi complementation, where a profile is simply the history of visits to accepting states. We developed a structural theory of profiles and used it to describe a complementation construction that is deterministic in the limit. Here we extend the theory of profiles to prove that every run dag contains a profile tree with at most a finite number of infinite branches. We then show that this property provides a theoretical grounding for a new determinization construction where macrostates are doubly preordered sets of states. In contrast to extant determinization constructions, transitions in the new construction are described declaratively rather than operationally.