Unambiguous Languages Exhaust the Index Hierarchy

This work is a study of the expressive power of unambiguity in the case of automata over infinite trees. An automaton is called unambiguous if it has at most one accepting run on every input, the language of such an automaton is called an unambiguous language. It is known that not every regular language of infinite trees is unambiguous. Except that, very little is known about which regular tree languages are unambiguous. This paper answers the question whether unambiguous languages are of bounded complexity among all regular tree languages. The notion of complexity is the canonical one, called the (parity or Rabin/Mostowski) index hierarchy. The answer is negative, as exhibited by a family of examples of unambiguous languages the cannot be recognised by any alternating parity tree automata of bounded range of priorities. Hardness of the examples is based on the theory of signatures, previously studied by Walukiewicz. The technical core of the article is a definition of the canonical signatures together with a parity game that compares signatures of a given pair of parity games (of the same index)

On the Complexity of Branching Games with Regular Conditions

Infinite duration games with regular conditions are one of the crucial tools in the areas of verification and synthesis. In this paper we consider a branching variant of such games - the game contains branching vertices that split the play into two independent sub-games. Thus, a play has the form of~an~infinite tree. The winner of the play is determined by a winning condition specified as a set of infinite trees. Games of this kind were used by Mio to provide a game semantics for the probabilistic mu-calculus. He used winning conditions defined in terms of parity games on trees. In this work we consider a more general class of winning conditions, namely those definable by finite automata on infinite trees. Our games can be seen as a branching-time variant of the stochastic games on graphs. We address the question of determinacy of a branching game and the problem of computing the optimal game value for each of the players. We consider both the stochastic and non-stochastic variants of the games. The questions under consideration are parametrised by the family of strategies we allow: either mixed, behavioural, or pure. We prove that in general, branching games are not determined under mixed strategies. This holds even for topologically simple winning conditions (differences of two open sets) and non-stochastic arenas. Nevertheless, we show that the games become determined under mixed strategies if we restrict the winning conditions to open sets of trees. We prove that the problem of comparing the game value to a rational threshold is undecidable for branching games with regular conditions in all non-trivial stochastic cases. In the non-stochastic cases we provide exact bounds on the complexity of the problem. The only case left open is the 0-player stochastic case, i.e. the problem of computing the measure of a given regular language of infinite trees

How Deterministic are Good-For-Games Automata?

In GFG automata, it is possible to resolve nondeterminism in a way that only depends on the past and still accepts all the words in the language. The motivation for GFG automata comes from their adequacy for games and synthesis, wherein general nondeterminism is inappropriate. We continue the ongoing effort of studying the power of nondeterminism in GFG automata. Initial indications have hinted that every GFG automaton embodies a deterministic one. Today we know that this is not the case, and in fact GFG automata may be exponentially more succinct than deterministic ones. We focus on the typeness question, namely the question of whether a GFG automaton with a certain acceptance condition has an equivalent GFG automaton with a weaker acceptance condition on the same structure. Beyond the theoretical interest in studying typeness, its existence implies efficient translations among different acceptance conditions. This practical issue is of special interest in the context of games, where the Buchi and co-Buchi conditions admit memoryless strategies for both players. Typeness is known to hold for deterministic automata and not to hold for general nondeterministic automata. We show that GFG automata enjoy the benefits of typeness, similarly to the case of deterministic automata. In particular, when Rabin or Streett GFG automata have equivalent Buchi or co-Buchi GFG automata, respectively, then such equivalent automata can be defined on a substructure of the original automata. Using our typeness results, we further study the place of GFG automata in between deterministic and nondeterministic ones. Specifically, considering automata complementation, we show that GFG automata lean toward nondeterministic ones, admitting an exponential state blow-up in the complementation of a Streett automaton into a Rabin automaton, as opposed to the constant blow-up in the deterministic case

Uniformisation Gives the Full Strength of Regular Languages

Given R a binary relation between words (which we treat as a language over a product alphabet AxB), a uniformisation of it is another relation L included in R which chooses a single word over B, for each word over A whenever there exists one. It is known that MSO, the full class of regular languages, is strong enough to define a uniformisation for each of its relations. The quest of this work is to see which other formalisms, weaker than MSO, also have this property. In this paper, we solve this problem for pseudo-varieties of semigroups: we show that no nonempty pseudo-variety weaker than MSO can provide uniformisations for its relations

A Characterisation of Pi^0_2 Regular Tree Languages

We show an algorithm that for a given regular tree language L decides if L is in Pi^0_2, that is if L belongs to the second level of Borel Hierarchy. Moreover, if L is in Pi^0_2, then we construct a weak alternating automaton of index (0, 2) which recognises L. We also prove that for a given language L, L is recognisable by a weak alternating (1, 3)-automaton if and only if it is recognisable by a weak non-deterministic (1, 3)-automaton

The logical strength of Büchi's decidability theorem

We study the strength of axioms needed to prove various results related to automata on infinite words and Büchi's theorem on the decidability of the MSO theory of (N, less_or_equal). We prove that the following are equivalent over the weak second-order arithmetic theory RCA: 1. Büchi's complementation theorem for nondeterministic automata on infinite words, 2. the decidability of the depth-n fragment of the MSO theory of (N, less_or_equal), for each n greater than 5, 3. the induction scheme for Sigma^0_2 formulae of arithmetic. Moreover, each of (1)-(3) is equivalent to the additive version of Ramsey's Theorem for pairs, often used in proofs of (1); each of (1)-(3) implies McNaughton's determinisation theorem for automata on infinite words; and each of (1)-(3) implies the "bounded-width" version of König's Lemma, often used in proofs of McNaughton's theorem

Modeling Oncogenic Signaling in Colon Tumors by Multidirectional Analyses of Microarray Data Directed for Maximization of Analytical Reliability

Clinical progression of colorectal cancers (CRC) may occur in parallel with distinctive signaling alterations. We designed multidirectional analyses integrating microarray-based data with biostatistics and bioinformatics to elucidate the signaling and metabolic alterations underlying CRC development in the adenoma-carcinoma sequence.Studies were performed on normal mucosa, adenoma, and carcinoma samples obtained during surgery or colonoscopy. Collections of cryostat sections prepared from the tissue samples were evaluated by a pathologist to control the relative cell type content. The measurements were done using Affymetrix GeneChip HG-U133plus2, and probe set data was generated using two normalization algorithms: MAS5.0 and GCRMA with least-variant set (LVS). The data was evaluated using pair-wise comparisons and data decomposition into singular value decomposition (SVD) modes. The method selected for the functional analysis used the Kolmogorov-Smirnov test. Expressional profiles obtained in 105 samples of whole tissue sections were used to establish oncogenic signaling alterations in progression of CRC, while those representing 40 microdissected specimens were used to select differences in KEGG pathways between epithelium and mucosa. Based on a consensus of the results obtained by two normalization algorithms, and two probe set sorting criteria, we identified 14 and 17 KEGG signaling and metabolic pathways that are significantly altered between normal and tumor samples and between benign and malignant tumors, respectively. Several of them were also selected from the raw microarray data of 2 recently published studies (GSE4183 and GSE8671).Although the proposed strategy is computationally complex and labor–intensive, it may reduce the number of false results

Comparative kinome analysis to identify putative colon tumor biomarkers

Kinase domains are the type of protein domain most commonly found in genes associated with tumorigenesis. Because of this, the human kinome (the protein kinase component of the genome) represents a promising source of cancer biomarkers and potential targets for novel anti-cancer therapies. Alterations in the human colon kinome during the progression from normal colon (NC) through adenoma (AD) to adenocarcinoma (AC) were investigated using integrated transcriptomic and proteomic datasets. Two hundred thirty kinase genes and 42 kinase proteins showed differential expression patterns (fold change ≥ 1.5) in at least one tissue pair-wise comparison (AD vs. NC, AC vs. NC, and/or AC vs. AD). Kinases that exhibited similar trends in expression at both the mRNA and protein levels were further analyzed in individual samples of NC (n = 20), AD (n = 39), and AC (n = 24) by quantitative reverse transcriptase PCR. Individual samples of NC and tumor tissue were distinguishable based on the mRNA levels of a set of 20 kinases. Altered expression of several of these kinases, including chaperone activity of bc1 complex-like (CABC1) kinase, bromodomain adjacent to zinc finger domain protein 1B (BAZ1B) kinase, calcium/calmodulin-dependent protein kinase type II subunit delta (CAMK2D), serine/threonine-protein kinase 24 (STK24), vaccinia-related kinase 3 (VRK3), and TAO kinase 3 (TAOK3), has not been previously reported in tumor tissue. These findings may have diagnostic potential and may lead to the development of novel targeted therapeutic interventions for colorectal cancer

System size and centrality dependence of the balance function in A+A collisions at sqrt[sNN]=17.2 GeV

Electric charge correlations were studied for p+p, C+C, Si+Si, and centrality selected Pb+Pb collisions at sqrt[sNN]=17.2 GeV with the NA49 large acceptance detector at the CERN SPS. In particular, long-range pseudorapidity correlations of oppositely charged particles were measured using the balance function method. The width of the balance function decreases with increasing system size and centrality of the reactions. This decrease could be related to an increasing delay of hadronization in central Pb+Pb collisions

System size and centrality dependence of the balance function in A + A collisions at sqrt s NN = 17.2 GeV

Electric charge correlations were studied for p+p, C+C, Si+Si and centrality selected Pb+Pb collisions at sqrt s_NN = 17.2$ GeV with the NA49 large acceptance detector at the CERN-SPS. In particular, long range pseudo-rapidity correlations of oppositely charged particles were measured using the Balance Function method. The width of the Balance Function decreases with increasing system size and centrality of the reactions. This decrease could be related to an increasing delay of hadronization in central Pb+Pb collisions