A Hierarchy of Nondeterminism

We study three levels in a hierarchy of nondeterminism: A nondeterministic automaton $\cal A$ is determinizable by pruning (DBP) if we can obtain a deterministic automaton equivalent to $\cal A$ by removing some of its transitions. Then, $\cal A$ is good-for-games (GFG) if its nondeterministic choices can be resolved in a way that only depends on the past. Finally, $\cal A$ is semantically deterministic (SD) if different nondeterministic choices in $\cal A$ lead to equivalent states. Some applications of automata in formal methods require deterministic automata, yet in fact can use automata with some level of nondeterminism. For example, DBP automata are useful in the analysis of online algorithms, and GFG automata are useful in synthesis and control. For automata on finite words, the three levels in the hierarchy coincide. We study the hierarchy for B\"uchi, co-B\"uchi, and weak automata on infinite words. We show that the hierarchy is strict, study the expressive power of the different levels in it, as well as the complexity of deciding the membership of a language in a given level. Finally, we describe a probability-based analysis of the hierarchy, which relates the level of nondeterminism with the probability that a random run on a word in the language is accepting.Comment: 21 pages, 5 figure

A Hierarchy of Nondeterminism

We study three levels in a hierarchy of nondeterminism: A nondeterministic automaton $\cal A$ is determinizable by pruning (DBP) if we can obtain a deterministic automaton equivalent to $\cal A$ by removing some of its transitions. Then, $\cal A$ is good-for-games (GFG) if its nondeterministic choices can be resolved in a way that only depends on the past. Finally, $\cal A$ is semantically deterministic (SD) if different nondeterministic choices in $\cal A$ lead to equivalent states. Some applications of automata in formal methods require deterministic automata, yet in fact can use automata with some level of nondeterminism. For example, DBP automata are useful in the analysis of online algorithms, and GFG automata are useful in synthesis and control. For automata on finite words, the three levels in the hierarchy coincide. We study the hierarchy for B\"uchi, co-B\"uchi, and weak automata on infinite words. We show that the hierarchy is strict, study the expressive power of the different levels in it, as well as the complexity of deciding the membership of a language in a given level. Finally, we describe a probability-based analysis of the hierarchy, which relates the level of nondeterminism with the probability that a random run on a word in the language is accepting.Comment: 21 pages, 5 figure

ICAMs Are Not Obligatory for Functional Immune Synapses between Naive CD4 T Cells and Lymph Node DCs

Protective immune responses depend on the formation of immune synapses between T cells and antigen-presenting cells (APCs). The two main LFA-1 ligands, ICAM-1 and ICAM-2, are co-expressed on many cell types, including APCs and blood vessels. Although these molecules were suggested to be key players in immune synapses studied in vitro, their contribution to helper T cell priming in vivo is unclear. Here, we used transgenic mice and intravital imaging to examine the role of dendritic cell (DC) ICAM-1 and ICAM-2 in naive CD4 T cell priming and differentiation in skin-draining lymph nodes. Surprisingly, ICAM deficiency on endogenous CD40-stimulated lymph node DCs did not impair their ability to arrest and prime CD4 lymphocyte activation and differentiation into Th1 and Tfh effectors. Thus, functional T cell receptor (TCR)-specific helper T cell synapses with antigen-presenting DCs and subsequent proliferation and early differentiation into T effectors do not require LFA-1-mediated T cell adhesiveness to DC ICAMs