Andreas Krebs

• Medical Doctor at University of Zurich

The notion of a difference hierarchy, first introduced by Hausdorff, plays an important role in many areas of mathematics, logic and theoretical computer science such as descriptive set theory, complexity theory, and the theory of regular languages and automata. Lattice theoretically, the difference hierarchy over a distributive lattice stratifies...

Kernelization – a mathematical key concept for provably effective polynomial-time preprocessing of NP-hard problems – plays a central role in parameterized complexity and has triggered an extensive line of research. This is in part due to a lower bounds framework that allows to exclude polynomial-size kernels under the assumption of [Formula: see t...

It is an open problem whether definability in Propositional Dynamic Logic (PDL) on forests is decidable. Based on an algebraic characterization by Boja\'nczyk, et. al.,(2012) in terms of forest algebras, Straubing (2013) described an approach to PDL based on a k-fold iterated distributive law. A proof that all languages satisfying such a k-fold ite...

We investigate the feasibility of a topological method for proving separations of non-uniform circuit classes. Thereto, we chose a rather simple class of circuits: non-uniform constant size circuit classes with gate types underlying certain restrictions. In particular, we consider gate types admitting for a description through regular and commutati...

We study two extensions of FO2[<], first-order logic interpreted in finite words, in which formulas are restricted to use only two variables. We adjoin to this language two-variable atomic formulas that say, `the letter a appears between positions x and y' and `the factor u appears between positions x and y'. These are, in a sense, the simplest pro...

The notion of a difference hierarchy, first introduced by Hausdorff, plays an important role in many areas of mathematics, logic and theoretical computer science such as descriptive set theory, complexity theory, and the theory of regular languages and automata. From a lattice theoretic point of view, the difference hierarchy over a bounded distrib...

We build a notion of algebraic recognition for visibly pushdown languages by finite algebraic objects. These come with a typical Eilenberg relationship, now between classes of visibly pushdown languages and classes of finite algebras. Building on that algebraic foundation, we further construct a topological object with one purpose being the possibi...

The aim of this study is to understand the inherent expressive power of CTL operators. We investigate the complexity of model checking for all CTL fragments with one CTL operator and arbitrary Boolean operators. This gives us a fingerprint of each CTL operator. The comparison between the fingerprints yields a hierarchy of the operators that mirrors...

It is an open problem whether definability in Propositional Dynamic Logic (PDL) on forests is decidable. Based on an algebraic characterization by Boja&nacute;czyk, et. al., (2012) in terms of forest algebras, Straubing (2013) described an approach to PDL based on a k-fold iterated distributive law. A proof that all languages satisfying such a k-fo...

Kernelization—a mathematical key concept for provably effective polynomial-time preprocessing of NP-hard problems—plays a central role in parameterized complexity and has triggered an extensive line of research. In this paper we consider a restricted yet natural variant of kernelization, namely strict kernelization, where one is not allowed to incr...

Cost register automata (CRAs) are one-way finite automata whose transitions have the side effect that a register is set to the result of applying a state-dependent semiring operation to a pair of registers. Here it is shown that CRAs over the tropical semiring \((\mathbb {N}\cup \{\infty \},\min ,+)\) can simulate polynomial time computation, provi...

We give an algebraic characterization, based on the bilateral semidirect product of finite monoids, of the quantifier alternation hierarchy in two-variable first-order logic on finite words. As a consequence, we obtain a new proof that this hierarchy is strict. Moreover, by application of the theory of finite categories, we are able to make our cha...

The Parikh automaton model equips a finite automaton with integer registers and imposes a semilinear constraint on the set of their final settings. Here the theories of typed monoids and of rational series are used to characterize the language classes that arise algebraically. Complexity bounds are derived, such as containment of the unambiguous Pa...

In this paper we relate two generalisations of the finite monoid recognisers of automata theory for the study of circuit complexity classes: Boolean spaces with internal monoids and typed monoids. Using the setting of stamps, this allows us to generalise a number of results from algebraic automata theory as it relates to Büchi's logic on words. We...

this book summarize the indication and landmarks for infiltrations based on clinical or ultrasound approoaches

Bounded context switching (BCS) is an under-approximate method for finding violations to safety properties in shared memory concurrent programs. Technically, BCS is a reachability problem that is known to be NP-complete. Our contribution is a parameterized analysis of BCS. The first result is an algorithm that solves BCS when parameterized by the n...

In this work, we study the power of bounded width branching programs by comparing them with bounded width skew circuits. It is well known that branching programs of bounded width have the same power as skew circuit of bounded width. The naive approach converts a BP of width w to a skew circuit of width \(w^2\). We improve this bound and show that B...

We study several classical decision problems on finite automata under the (Strong) Exponential Time Hypothesis. We focus on three types of problems: universality, equivalence, and emptiness of intersection. All these problems are known to be CoNP-hard for nondeterministic finite automata, even when restricted to unary input alphabets. A different t...

The problem of determining whether several finite automata accept a word in common is closely related to the well-studied membership problem in transformation monoids. We raise the issue of limiting the number of final states in the automata intersection problem. For automata with two final states, we show the problem to be ⊕ L-complete or NP-compl...

Kernelization -- the mathematical key concept for provably effective polynomial-time preprocessing of NP-hard problems -- plays a central role in parameterized complexity and has triggered an extensive line of research. This is in part due to a lower bounds framework that allows to exclude polynomial-size kernels under the assumption that NP is not...

A proof system for a language L is a function f such that Range(f) is exactly L. In this article, we look at proof systems from a circuit complexity point of view and study proof systems that are computationally very restricted. The restriction we study is proof systems that can be computed by bounded fanin circuits of constant depth (NC⁰) or of O(...

In the last two decades visibly pushdown languages (VPLs) have found many applications in diverse areas such as formal verification and processing of XML documents. Recently, there has been a significant interest in studying quantitative versions of finite-state systems as well as visibly pushdown systems. In this work, we take forward this study f...

Logical formulas are naturally decomposed into their subformulas and circuits into their layers. How are these decompositions expressed in a purely language-theoretical setting? We address that question, and in doing so, introduce a product directly on languages that parallels formula composition. This framework makes an essential use of languages...

We study several classical decision problems on finite automata under the (Strong) Exponential Time Hypothesis. We focus on three types of problems: universality, equivalence, and emptiness of intersection. All these problems are known to be CoNP-hard for nondeterministic finite automata, even when restricted to unary input alphabets. A different t...

We study an extension of FO2[<], first-order logic interpreted in finite words, in which formulas are restricted to use only two variables. We adjoin to this language two-variable atomic formulas that say, 'the letter a appears between positions x and y'. This is, in a sense, the simplest property that is not expressible using only two variables. W...

We study an extension of FO^2[<], first-order logic interpreted in finite words, in which formulas are restricted to use only two variables. We adjoin to this language two-variable atomic formulas that say, `the letter a appears between positions x and y'. This is, in a sense, the simplest property that is not expressible using only two variables....

We investigate in a method for proving lower bounds for abstract circuit classes. A well established method to characterize varieties of regular languages are identities. We use a recently established generalization of these identities to non-regular languages by Gehrke, Grigorieff, and Pin: so called equations, which are capable of describing arbi...

We investigate in a method for proving separation results for abstract classes of languages. A well established method to characterize varieties of regular languages are identities. We use a recently established generalization of these identities to non-regular languages by Gehrke, Grigorieff, and Pin: so called equations, which are capable of desc...

We propose to study value automata with filters, a natural generalization of regular cost automata to nondeterminism. Models such as weighted automata and Parikh automata appear naturally as specializations. Results on the expressiveness of this model offer a general understanding of the behavior of the models that arise as special cases. A landsca...

The aim of this study is to understand the inherent expressive power of CTL operators. We investigate the complexity of model checking for all CTL fragments with one CTL operator and arbitrary Boolean operators. This gives us a fingerprint of each CTL operator. The comparison between the fingerprints yields a hierarchy of the operators that mirrors...

We introduce two variants of computation tree logic CTL based on team semantics: an asynchronous one and a synchronous one. For both variants we investigate the computational complexity of the satisfiability as well as the model checking problem. The satisfiability problem is shown to be EXPTIME-complete. Here it does not matter which of the two se...

Low circuit complexity classes and regular languages exhibit very tight interactions that shade light on their respective expressiveness. We propose to study these interactions at a functional level, by investigating the deterministic rational transductions computable by constant-depth, polysize circuits. To this end, a circuit framework of indepen...

We extend the familiar program of understanding circuit complexity in terms of regular languages to visibly counter languages. Like the regular languages, the visibly counter languages are \(\mathrm {NC}^{1}\)- complete. We investigate what the visibly counter languages in certain constant depth circuit complexity classes are. We have initiated thi...

In this work, we study the power of bounded width branching programs by comparing them with bounded width skew circuits. It is well known that branching programs of bounded width have the same power as skew circuit of bounded width. The naive approach converts a BP of width w to a skew circuit of width \(w^2\). We improve this bound and show that B...

We give a method for specifying ultrafilter equations and identify their projections on the set of profinite words. Let B be the set of languages captured by first-order sentences using unary predicates for each letter, arbitrary uniform unary numerical predicates and a predicate for the length of a word. We illustrate our methods by giving ultrafi...

We obtain results within the area of dense completeness, which describes a close relation between families of formal languages and complexity classes. Previously we were able show that this relation exists between counter languages and \(\mathbf {NL}\) but not between the regular languages and \(\mathbf {NC^1}\). We narrow the gap between the regul...

We introduce two variants of computation tree logic CTL based on team semantics: an asynchronous one and a synchronous one. For both variants we investigate the computational complexity of the satisfiability as well as the model checking problem. The satisfiability problem is shown to be EXPTIME-complete. Here it does not matter which of the two se...

The aim of this study is to understand the inherent expressive power of CTL operators. We investigate the complexity of model checking for all CTL fragments with one CTL operator and arbitrary Boolean operators. This gives us a fingerprint of each CTL operator. The comparison between the fingerprints yields a hierarchy of the operators that mirrors...

We examine visibly counter languages, which are languages recognized by visibly counter automata (a.k.a. input driven counter automata). We are able to effectively characterize the visibly counter languages in AC0 and show that they are contained in FO[+].

We describe a new way to construct finite geometric objects. For every $k$ we obtain a symmetric configuration $\mathcal{E }(k-1)$ with $k$ points on a line. In particular, we have a constructive existence proof for such configurations. The method is very simple and purely geometric. It also produces interesting periodic matrices.

We extend the # operator in a natural way and derive a new type of counting complexity. While #\(\mathcal C\) classes (where \(\mathcal C\) is some circuit-based class like NC 1) only count proofs for acceptance of some input in circuits, one can also count proofs for rejection. The here proposed Zap-\(\mathcal C\) complexity classes implement this...

We give a method for specifying ultrafilter equations and identify their projections on the set of profinite words. Let ℬ be the set of languages captured by first-order sentences using unary predicates for each letter, arbitrary uniform unary numerical predicates and a predicate for the length of a word. We illustrate our methods by giving profini...

We examine languages of unranked forests definable using the temporal operators EF and EX. We characterize the languages definable in this logic, and various fragments thereof, using the syntactic forest algebras introduced by Bojanczyk and Walukiewicz. Our algebraic characterizations yield efficient algorithms for deciding when a given language of...

Ultrasound (US) has become a useful tool in the detection of early disease, differential diagnosis, guidance of treatment decisions and treatment monitoring of rheumatoid arthritis (RA). In 2008, the Swiss Sonography in Arthritis and Rheumatism (SONAR) group was established to promote the use of US in inflammatory arthritis in clinical practice. A...

The Parikh automaton model equips a finite automaton with integer registers and imposes a semilinear constraint on the set of their final settings. Here the theory of typed monoids is used to characterize the language classes that arise algebraically. Complexity bounds are derived, such as containment of the unambiguous Parikh automata languages in...

A proof system for a language L is a function f such that Range(f) is exactly L. In this paper, we look at proof systems from a circuit complexity point of view and study proof systems that are computationally very restricted. The restriction we study is: they can be computed by bounded fanin circuits of constant depth (NC0), or of O(loglogn) depth...

Injections of joint and of periarticular structures (e. g. tendon sheath) are frequent procedures in daily practice. In this review we summarize recent guidelines of injection technique, as we presented at the practical workshops oft he «Medidays» congress at the University Hospital Zurich. Furthermore we discuss drugs, indications, complications,...

We describe a new way to construct finite geometric objects. For every k we obtain a symmetric configuration E(k-1) with k points on a line. In particular, we have a constructive existence proof for such configurations. The method is very simple and purely geometric. It also produces interesting periodic matrices.

We give a #NC 1 upper bound for the problem of counting accepting paths in any fixed visibly pushdown automaton. Our algorithm involves a non-trivial adaptation of the arithmetic formula evaluation algorithm of Buss, Cook, Gupta and Ramachandran (SIAM J. Comput. 21:755–780, 1992). We also show that the problem is #NC 1 hard. Our results show that t...

The effect of severely tightening the uniformity of Boolean circuit families is investigated. The impact on NC 1 and its subclasses is shown to depend on the characterization chosen for the class, while classes such as P appear to be more robust. Tightly uniform subclasses of NC 1 whose separation may be within reach of current techniques emerge.

We introduce dense completeness, which gives tighter connection between formal language classes and complexity classes than the usual notion of completeness. A family of formal languages \(\mathcal F\) is densely complete in a complexity class \(\mathcal C\) iff \({\mathcal F}\subseteq{\mathcal C}\) and for each \(C \in{\mathcal C}\) there is an \(...

We study the streaming complexity of the membership problem of 1-turn-Dyck2 and Dyck2 when there are a few errors in the input string. 1-turn-Dyck2 with errors: We prove that there exists a randomized one-pass algorithm that given x checks whether there exists a string x' in 1-turn-Dyck2 such that x is obtained by flipping at most $k$ locations of...

We characterize the languages in the individual levels of the quantifier alternation hierarchy of first-order logic with two variables by identities. This implies decidability of the individual levels. More generally we show that the two-sided semidirect product of a decidable variety with the variety J is decidable.

We study the streaming complexity of the membership problem of \(1\mbox{-turn-}\mbox{\sf Dyck}_2\) and \(\mbox{\sf Dyck}_2\) when there are a few errors in the input string. \(1\mbox{-turn-}\mbox{\sf Dyck}_2\) with errors: We prove that there exists a randomized one-pass algorithm that given x checks whether there exists a string \(x' \in 1\mbox{-t...

Based on different concepts to obtain a finer notion of language recognition via finite monoids we develop an algebraic structure called typed monoid. This leads to an algebraic description of regular and non regular languages. We obtain for each language a unique minimal recognizing typed monoid, the typed syntactic monoid. We prove an Eilenberg-...

Education and training in musculoskeletal ultrasound (MSUS) comprises attendance at theoretical and practical courses and independent study. Web-based learning as a novel teaching method has previously been described. The present study summarizes normal and pathological findings in a web-based approach using widely accepted guidelines. In a prospec...

Education and training in musculoskeletal ultrasound (MSUS) comprises attendance at theoretical and practical courses and independent study. Web-based learning as a novel teaching method has previously been described. The present study summarizes normal and pathological findings in a web-based approach using widely accepted guidelines. In a prospec...

We give a #NC 1 upper bound for the problem of counting accepting paths in any fixed visibly pushdown automaton. Our algorithm involves a non-trivial adaptation of the arithmetic formula evaluation algorithm of Buss, Cook, Gupta, Ramachandran ([9]). We also show that the problem is #NC 1 hard. Our results show that the difference between #BWBP and...

In this paper we consider the class of all regular languages definable by the extended majority quantifier and the order predicate but using only two variables. The main part of the paper is the presentation of a geometric method which is used to show that a given regular language cannot be defined by such formulas. Applying this method we can give...

Motivated by the open question whether \(\mbox{TC{$^0$}}=\mbox{NC{$^1$}}\) we consider the case of linear size TC0. We use the connections between circuits, logic, and algebra, in particular the characterization of \(\mbox{TC{$^0$}}\) in terms of finitely typed monoids. Applying algebraic methods we show that the word problem for finite non-solvabl...

We show that a maximal partial plane of order 6 with 31 lines and a maximal pure partial plane of order 6 with 25 lines can be constructed from the icosahedron and the Petersen graph.

Following recent works connecting two-variable logic to circuits and monoids, we establish, for numerical predicate sets satisfying a certain closure property, a one-to-one correspondence between \(FO[<,\ensuremath{\mathfrak{P}}]\)-uniform linear circuits, two-variable formulae with \(\ensuremath{\mathfrak{P}}\) predicates, and weak block products...

We characterize the languages in TC0 = L(Maj[<,Bit]) and L(Maj[<]) as inverse morphic images of certain groups. Necessarily these are infinite, since nonregular sets are concerned. To limit the power of these infinite algebraic objects, we equip them with a finite type set and introduce the notion of a finitely typed (infinite) monoid. Following th...

We study languages with bounded communication complexity in the multiparty "input on the forehead model" with worst-case partition. In the two-party case, languages with bounded complexity are exactly those recognized by programs over commutative monoids [19]. This can be used to show that these languages all lie in shallow ACC0. In contrast, we us...

We study languages with bounded communication complexity in the multiparty “input on the forehead model” with worst-case partition. In the two-party case, languages with bounded complexity are exactly those recognized by programs over commutative monoids [19]. This can be used to show that these languages all lie in shallow ACC0. In contrast, we us...

Estimates of locations are never certain, especially in indoors environments, and it is useful not only to determine an estimate of measurement variables but also to know its uncertainty. In addition, location information is gathered at different places, devices, and sensors. This leads to the problem of transmitting uncertain location estimates ef...