How unprovable is Rabin's decidability theorem?

We study the strength of set-theoretic axioms needed to prove Rabin's theorem on the decidability of the MSO theory of the infinite binary tree. We first show that the complementation theorem for tree automata, which forms the technical core of typical proofs of Rabin's theorem, is equivalent over the moderately strong second-order arithmetic theory $\mathsf{ACA}_0$ to a determinacy principle implied by the positional determinacy of all parity games and implying the determinacy of all Gale-Stewart games given by boolean combinations of ${\bf \Sigma^0_2}$ sets. It follows that complementation for tree automata is provable from $\Pi^1_3$- but not $\Delta^1_3$-comprehension. We then use results due to MedSalem-Tanaka, M\"ollerfeld and Heinatsch-M\"ollerfeld to prove that over $\Pi^1_2$-comprehension, the complementation theorem for tree automata, decidability of the MSO theory of the infinite binary tree, positional determinacy of parity games and determinacy of $\mathrm{Bool}({\bf \Sigma^0_2})$ Gale-Stewart games are all equivalent. Moreover, these statements are equivalent to the $\Pi^1_3$-reflection principle for $\Pi^1_2$-comprehension. It follows in particular that Rabin's decidability theorem is not provable in $\Delta^1_3$-comprehension.Comment: 21 page

Aspects of Diffeomorphism and Conformal invariance in classical Liouville theory

The interplay between the diffeomorphism and conformal symmetries (a feature common in quantum field theories) is shown to be exhibited for the case of black holes in two dimensional classical Liouville theory. We show that although the theory is conformally invariant in the near horizon limit, there is a breaking of the diffeomorphism symmetry at the classical level. On the other hand, in the region away from the horizon, the conformal symmetry of the theory gets broken with the diffeomorphism symmetry remaining intact.Comment: Accepted in Euro. Phys. Letters., Title changed, abstract modified, major modifications made in the pape

Deciding Quantifier-Free Presburger Formulas Using Parameterized Solution Bounds

Given a formula in quantifier-free Presburger arithmetic, if it has a satisfying solution, there is one whose size, measured in bits, is polynomially bounded in the size of the formula. In this paper, we consider a special class of quantifier-free Presburger formulas in which most linear constraints are difference (separation) constraints, and the non-difference constraints are sparse. This class has been observed to commonly occur in software verification. We derive a new solution bound in terms of parameters characterizing the sparseness of linear constraints and the number of non-difference constraints, in addition to traditional measures of formula size. In particular, we show that the number of bits needed per integer variable is linear in the number of non-difference constraints and logarithmic in the number and size of non-zero coefficients in them, but is otherwise independent of the total number of linear constraints in the formula. The derived bound can be used in a decision procedure based on instantiating integer variables over a finite domain and translating the input quantifier-free Presburger formula to an equi-satisfiable Boolean formula, which is then checked using a Boolean satisfiability solver. In addition to our main theoretical result, we discuss several optimizations for deriving tighter bounds in practice. Empirical evidence indicates that our decision procedure can greatly outperform other decision procedures.Comment: 26 page

Decidability of quantified propositional intuitionistic logic and S4 on trees

Quantified propositional intuitionistic logic is obtained from propositional intuitionistic logic by adding quantifiers \forall p, \exists p over propositions. In the context of Kripke semantics, a proposition is a subset of the worlds in a model structure which is upward closed. Kremer (1997) has shown that the quantified propositional intuitionistic logic H\pi+ based on the class of all partial orders is recursively isomorphic to full second-order logic. He raised the question of whether the logic resulting from restriction to trees is axiomatizable. It is shown that it is, in fact, decidable. The methods used can also be used to establish the decidability of modal S4 with propositional quantification on similar types of Kripke structures.Comment: v2, 9 pages, corrections and additions; v1 8 page

Anonymous quantum communication

We present the first protocol for the anonymous transmission of a quantum state that is information-theoretically secure against an active adversary, without any assumption on the number of corrupt participants. The anonymity of the sender and receiver is perfectly preserved, and the privacy of the quantum state is protected except with exponentially small probability. Even though a single corrupt participant can cause the protocol to abort, the quantum state can only be destroyed with exponentially small probability: if the protocol succeeds, the state is transferred to the receiver and otherwise it remains in the hands of the sender (provided the receiver is honest).Comment: 11 pages, to appear in Proceedings of ASIACRYPT, 200

Model-based meta-analysis of salbutamol pharmacokinetics and practical implications for doping control.

Salbutamol was included in the prohibited list of the World Anti-Doping Agency (WADA) in 2004. Although systemic intake is banned, inhalation for asthma is permitted but with dosage restrictions. The WADA established a urinary concentration threshold to distinguish accordingly prohibited systemic self-administration from therapeutic prescription by inhalation. This study aimed at evaluating the ability of the WADA threshold to differentiate salbutamol therapeutic use from violation of antidoping rules. Concentration-time profile of salbutamol in plasma and its excretion in urine was characterized through a model-based meta-analysis of individual and aggregate data collected after administration of a large range of doses following different modes of administration and under a variety of conditions. The developed model adequately fitted salbutamol plasma and urine concentration-time profiles of the 13 selected studies. Model-based simulations confirmed that a wide range of salbutamol urine concentrations might be measured after drug intake. Although violation of the WADA Code can be strongly suspected in individuals showing very high salbutamol urine concentrations, uncertainty remains for values close to the WADA threshold as they can be compatible with both permitted therapeutic use and violation. Although not entirely discriminant, the current WADA rule is globally supported by our appraisal. It could be further improved by a slight and reasonable adjustment of inhaled daily dosages allowed for therapeutic use. Our model might help antidoping experts in the evaluation of suspected doping cases through confronting the athlete's urine measurements with their allegations about salbutamol treatment

Mean-payoff Automaton Expressions

Quantitative languages are an extension of boolean languages that assign to each word a real number. Mean-payoff automata are finite automata with numerical weights on transitions that assign to each infinite path the long-run average of the transition weights. When the mode of branching of the automaton is deterministic, nondeterministic, or alternating, the corresponding class of quantitative languages is not robust as it is not closed under the pointwise operations of max, min, sum, and numerical complement. Nondeterministic and alternating mean-payoff automata are not decidable either, as the quantitative generalization of the problems of universality and language inclusion is undecidable. We introduce a new class of quantitative languages, defined by mean-payoff automaton expressions, which is robust and decidable: it is closed under the four pointwise operations, and we show that all decision problems are decidable for this class. Mean-payoff automaton expressions subsume deterministic mean-payoff automata, and we show that they have expressive power incomparable to nondeterministic and alternating mean-payoff automata. We also present for the first time an algorithm to compute distance between two quantitative languages, and in our case the quantitative languages are given as mean-payoff automaton expressions

Noncommuting Electric Fields and Algebraic Consistency in Noncommutative Gauge theories

We show that noncommuting electric fields occur naturally in $\theta$-expanded noncommutative gauge theories. Using this noncommutativity, which is field dependent, and a hamiltonian generalisation of the Seiberg-Witten Map, the algebraic consistency in the lagrangian and hamiltonian formulations of these theories, is established. A comparison of results in different descriptions shows that this generalised map acts as canonical transformation in the physical subspace only. Finally, we apply the hamiltonian formulation to derive the gauge symmetries of the action.Comment: 16 pages, LaTex, considerably expanded version with a new section on `Gauge symmetries'; To appear in Phys. Rev.