arXiv subjects

Martin Hofmann

Publications and source records attributed to Martin Hofmann.

30 records · Page 2Linked to original sources

arXiv2014

Certification for mu-calculus with winning strategies

We define memory-efficient certificates for $μ$-calculus model checking problems based on the well-known correspondence of the $μ$-calculus model checking with winning certain parity games. Winning strategies can independently checked, in low polynomial time, by observing that there is no reachable strongly connected component in the graph of the parity game whose largest priority is odd. Winning strategies are computed by fixpoint iteration following the naive semantics of $μ$-calculus. We instrument the usual fixpoint iteration of $μ$-calculus model checking so that it produces evidence in the form of a winning strategy; these winning strategies can be computed in polynomial time in $|S|$ and in space $O(|S|^2 |ϕ|^2)$, where $|S|$ is the size of the state space and $|ϕ|$ the length of the formula $ϕ$\@. The main technical contribution here is the notion and algebra of partial winning strategies. On the technical level our work can be seen as a new, simpler, and immediate constructive proof of the correspondence between $μ$-calculus and parity games.

cs.LO↗

arXiv2013

Power of Nondetreministic JAGs on Cayley graphs

The Immerman-Szelepcsenyi Theorem uses an algorithm for co-st- connectivity based on inductive counting to prove that NLOGSPACE is closed un- der complementation. We want to investigate whether counting is necessary for this theorem to hold. Concretely, we show that Nondeterministic Jumping Graph Autmata (ND-JAGs) (pebble automata on graphs), on several families of Cayley graphs, are equal in power to nondeterministic logspace Turing machines that are given such graphs as a linear encoding. In particular, it follows that ND-JAGs can solve co-st-connectivity on those graphs. This came as a surprise since Cook and Rackoff showed that deterministic JAGs cannot solve st-connectivity on many Cayley graphs due to their high self-similarity (every neighbourhood looks the same). Thus, our results show that on these graphs, nondeterminism provably adds computational power. The families of Cayley graphs we consider include Cayley graphs of abelian groups and of all finite simple groups irrespective of how they are presented and graphs corresponding to groups generated by various product constructions, in- cluding iterated ones. We remark that assessing the precise power of nondeterministic JAGs and in par- ticular whether they can solve co-st-connectivity on arbitrary graphs is left as an open problem by Edmonds, Poon and Achlioptas. Our results suggest a positive answer to this question and in particular considerably limit the search space for a potential counterexample.

cs.CC↗

arXiv2013

Device-independent entanglement quantification and related applications

We present a general method to quantify both bipartite and multipartite entanglement in a device-independent manner, meaning that we put a lower bound on the amount of entanglement present in a system based on observed data only but independently of any quantum description of the employed devices. Some of the bounds we obtain, such as for the Clauser-Horne-Shimony-Holt Bell inequality or the Svetlichny inequality, are shown to be tight. Besides, device-independent entanglement quantification can serve as a basis for numerous tasks. We show in particular that our method provides a rigorous way to construct dimension witnesses, gives new insights into the question whether bound entangled states can violate a Bell inequality, and can be used to construct device independent entanglement witnesses involving an arbitrary number of parties.

quant-ph↗

arXiv2013

On the Reflection Type Decomposition of the Adjoint Reduced Phase Space of a Compact Semisimple Lie group

We consider a system with symmetries whose configuration space is a compact Lie group, acted upon by inner automorphisms. The classical reduced phase space of this system decomposes into connected components of orbit type subsets. To investigate hypothetical quantum effects of this decomposition one has to construct the associated costratification of the Hilbert space of the quantum system in the sense of Huebschmann. In the present paper, instead of the decomposition by orbit types, we consider the related decomposition by reflection types (conjugacy classes of reflection subgroups). These two decompositions turn out to coincide e.g. for the classical groups SU(n) and Sp(n). We derive defining relations for reflection type subsets in terms of irreducible characters and discuss how to obtain from that the corresponding costratification of the Hilbert space of the system. To illustrate the method, we give explicit results for some low rank classical groups.

math-ph↗

arXiv2012

Abstract Effects and Proof-Relevant Logical Relations

We introduce a novel variant of logical relations that maps types not merely to partial equivalence relations on values, as is commonly done, but rather to a proof-relevant generalisation thereof, namely setoids. The objects of a setoid establish that values inhabit semantic types, whilst its morphisms are understood as proofs of semantic equivalence. The transition to proof-relevance solves two well-known problems caused by the use of existential quantification over future worlds in traditional Kripke logical relations: failure of admissibility, and spurious functional dependencies. We illustrate the novel format with two applications: a direct-style validation of Pitts and Stark's equivalences for "new" and a denotational semantics for a region-based effect system that supports type abstraction in the sense that only externally visible effects need to be tracked; non-observable internal modifications, such as the reorganisation of a search tree or lazy initialisation, can count as `pure' or `read only'. This `fictional purity' allows clients of a module soundly to validate more effect-based program equivalences than would be possible with traditional effect systems.

cs.PL↗

arXiv2012

Learn with SAT to Minimize Büchi Automata

We describe a minimization procedure for nondeterministic Büchi automata (NBA). For an automaton A another automaton A_min with the minimal number of states is learned with the help of a SAT-solver. This is done by successively computing automata A' that approximate A in the sense that they accept a given finite set of positive examples and reject a given finite set of negative examples. In the course of the procedure these example sets are successively increased. Thus, our method can be seen as an instance of a generic learning algorithm based on a "minimally adequate teacher" in the sense of Angluin. We use a SAT solver to find an NBA for given sets of positive and negative examples. We use complementation via construction of deterministic parity automata to check candidates computed in this manner for equivalence with A. Failure of equivalence yields new positive or negative examples. Our method proved successful on complete samplings of small automata and of quite some examples of bigger automata. We successfully ran the minimization on over ten thousand automata with mostly up to ten states, including the complements of all possible automata with two states and alphabet size three and discuss results and runtimes; single examples had over 100 states.

cs.FL↗

arXiv2012

On the Hitting Probability of Max-Stable Processes

The probability that a max-stable process η in C[0, 1] with identical marginal distribution function F hits x \in R with 0 < F (x) < 1 is the hitting probability of x. We show that the hitting probability is always positive, unless the components of η are completely dependent. Moreover, we consider the event that the paths of standard MSP hit some x \in R twice and we give a sufficient condition for a positive probability of this event.

math.PR↗

arXiv2012

The multivariate Piecing-Together approach revisited

The univariate Piecing-Together approach (PT) fits a univariate generalized Pareto distribution (GPD) to the upper tail of a given distribution function in a continuous manner. A multivariate extension was established by Aulbach et al. (2012a): The upper tail of a given copula C is cut off and replaced by a multivariate GPD-copula in a continuous manner, yielding a new copula called a PT-copula. Then each margin of this PT-copula is transformed by a given univariate distribution function. This provides a multivariate distribution function with prescribed margins, whose copula is a GPD-copula that coincides in its central part with C. In addition to Aulbach et al. (2012a), we achieve in the present paper an exact representation of the PT-copula's upper tail, giving further insight into the multivariate PT approach. A variant based on the empirical copula is also added. Furthermore our findings enable us to establish a functional PT version as well.

math.PR↗

arXiv2011

Sojourn Times and the Fragility Index

We investigate the sojourn time above a high threshold of a continuous stochastic process Y on [0,1]. It turns out that the limit, as the threshold increases, of the expected sojourn time given that it is positive, exists if the copula process corresponding to Y is in the functional domain of attraction of of an extreme value process. This limit coincides with the limit of the fragility index corresponding to finite (n-)dimensional distributions of Y as n and the threshold increase. If the process is in a certain neighborhood of a generalized Pareto process, then we can replace the constant threshold by a general threshold function and we can compute the asymptotic sojourn time distribution. An extreme value process is a prominent example. Given that there is an exceedance at some t_0 above the threshold, we can also compute the asymptotic distribution of the time cluster length, which the process spends above the threshold function.

math.PR↗

arXiv2011

On Max-Stable Processes and the Functional D-Norm

We introduce a functional domain of attraction approach for stochastic processes, which is more general than the usual one based on weak convergence. The distribution function G of a continuous max-stable process on [0,1] is introduced and it is shown that G can be represented via a norm on functional space, called D-norm. This is in complete accordance with the multivariate case and leads to the definition of functional generalized Pareto distributions (GPD) W. These satisfy W=1+log(G) in their upper tails, again in complete accordance with the uni- or multivariate case. Applying this framework to copula processes we derive characterizations of the domain of attraction condition for copula processes in terms of tail equivalence with a functional GPD. δ-neighborhoods of a functional GPD are introduced and it is shown that these are characterized by a polynomial rate of convergence of functional extremes, which is well-known in the multivariate case.

math.PR↗

arXiv2010

Bounded Linear Logic, Revisited

We present QBAL, an extension of Girard, Scedrov and Scott's bounded linear logic. The main novelty of the system is the possibility of quantifying over resource variables. This generalization makes bounded linear logic considerably more flexible, while preserving soundness and completeness for polynomial time. In particular, we provide compositional embeddings of Leivant's RRW and Hofmann's LFPL into QBAL.

cs.LO↗

arXiv2010

Optical Scattering Lengths in Large Liquid-Scintillator Neutrino Detectors

For liquid-scintillator neutrino detectors of kiloton scale, the transparency of the organic solvent is of central importance. The present paper reports on laboratory measurements of the optical scattering lengths of the organic solvents PXE, LAB, and Dodecane which are under discussion for next-generation experiments like SNO+, Hanohano, or LENA. Results comprise the wavelength range from 415 to 440nm. The contributions from Rayleigh and Mie scattering as well as from absorption/re-emission processes are discussed. Based on the present results, LAB seems to be the preferred solvent for a large-volume detector.

physics.ins-det↗