Alternation-Trading Proofs, Linear Programming, and Lower Bounds

A fertile area of recent research has demonstrated concrete polynomial time lower bounds for solving natural hard problems on restricted computational models. Among these problems are Satisfiability, Vertex Cover, Hamilton Path, Mod6-SAT, Majority-of-Majority-SAT, and Tautologies, to name a few. The proofs of these lower bounds follow a certain proof-by-contradiction strategy that we call alternation-trading. An important open problem is to determine how powerful such proofs can possibly be. We propose a methodology for studying these proofs that makes them amenable to both formal analysis and automated theorem proving. We prove that the search for better lower bounds can often be turned into a problem of solving a large series of linear programming instances. Implementing a small-scale theorem prover based on this result, we extract new human-readable time lower bounds for several problems. This framework can also be used to prove concrete limitations on the current techniques.Comment: To appear in STACS 2010, 12 page

Two-Way Automata Making Choices Only at the Endmarkers

The question of the state-size cost for simulation of two-way nondeterministic automata (2NFAs) by two-way deterministic automata (2DFAs) was raised in 1978 and, despite many attempts, it is still open. Subsequently, the problem was attacked by restricting the power of 2DFAs (e.g., using a restricted input head movement) to the degree for which it was already possible to derive some exponential gaps between the weaker model and the standard 2NFAs. Here we use an opposite approach, increasing the power of 2DFAs to the degree for which it is still possible to obtain a subexponential conversion from the stronger model to the standard 2DFAs. In particular, it turns out that subexponential conversion is possible for two-way automata that make nondeterministic choices only when the input head scans one of the input tape endmarkers. However, there is no restriction on the input head movement. This implies that an exponential gap between 2NFAs and 2DFAs can be obtained only for unrestricted 2NFAs using capabilities beyond the proposed new model. As an additional bonus, conversion into a machine for the complement of the original language is polynomial in this model. The same holds for making such machines self-verifying, halting, or unambiguous. Finally, any superpolynomial lower bound for the simulation of such machines by standard 2DFAs would imply LNL. In the same way, the alternating version of these machines is related to L =? NL =? P, the classical computational complexity problems.Comment: 23 page

Bounded Model Checking for Asynchronous Hyperproperties

Many types of attacks on confidentiality stem from the nondeterministic nature of the environment that computer programs operate in (e.g., schedulers and asynchronous communication channels). In this paper, we focus on verification of confidentiality in nondeterministic environments by reasoning about asynchronous hyperproperties. First, we generalize the temporal logic A-HLTL to allow nested trajectory quantification, where a trajectory determines how different execution traces may advance and stutter. We propose a bounded model checking algorithm for A-HLTL based on QBF-solving for a fragment of the generalized A-HLTL and evaluate it by various case studies on concurrent programs, scheduling attacks, compiler optimization, speculative execution, and cache timing attacks. We also rigorously analyze the complexity of model checking for different fragments of A-HLTL.Comment: 34 page

Parameterised Counting in Logspace

Logarithmic space bounded complexity classes such as L and NL play a central role in space bounded computation. The study of counting versions of these complexity classes have lead to several interesting insights into the structure of computational problems such as computing the determinant and counting paths in directed acyclic graphs. Though parameterised complexity theory was initiated roughly three decades ago by Downey and Fellows, a satisfactory study of parameterised logarithmic space bounded computation was developed only in the last decade by Elberfeld, Stockhusen and Tantau (IPEC 2013, Algorithmica 2015). In this paper, we introduce a new framework for parameterised counting in logspace, inspired by the parameterised space bounded models developed by Elberfeld, Stockhusen and Tantau (IPEC 2013, Algorithmica 2015). They defined the operators para_W and para_? for parameterised space complexity classes by allowing bounded nondeterminism with multiple-read and read-once access, respectively. Using these operators, they characterised the parameterised complexity of natural problems on graphs. In the spirit of the operators para_W and para_? by Stockhusen and Tantau, we introduce variants based on tail-nondeterminism, para_{W[1]} and para_{?tail}. Then, we consider counting versions of all four operators applied to logspace and obtain several natural complete problems for the resulting classes: counting of paths in digraphs, counting first-order models for formulas, and counting graph homomorphisms. Furthermore, we show that the complexity of a parameterised variant of the determinant function for (0,1)-matrices is #para_{?tail} L-hard and can be written as the difference of two functions in #para_{?tail} L. These problems exhibit the richness of the introduced counting classes. Our results further indicate interesting structural characteristics of these classes. For example, we show that the closure of #para_{?tail} L under parameterised logspace parsimonious reductions coincides with #para_? L, that is, modulo parameterised reductions, tail-nondeterminism with read-once access is the same as read-once nondeterminism. Initiating the study of closure properties of these parameterised logspace counting classes, we show that all introduced classes are closed under addition and multiplication, and those without tail-nondeterminism are closed under parameterised logspace parsimonious reductions. Also, we show that the counting classes defined can naturally be characterised by parameterised variants of classes based on branching programs in analogy to the classical counting classes. Finally, we underline the significance of this topic by providing a promising outlook showing several open problems and options for further directions of research

Automata for Unordered Trees

International audienceWe present a framework for defining automata for unordereddata trees that is parametrized by the way in which multisets of children nodes are described. Presburger tree automata and alternatingPresburger tree automata are particular instances. We establish the usual equivalence in expressiveness of tree automata and MSO for the automata defined inour framework.We then investigate subclasses of automata for unordered treesfor which testing language equivalence is in P-time. For this we start from automata in our framework that describe multisets of childrenby finite automata, and propose two approaches of how todo this deterministically. We show that a restriction to confluent horizontal evaluation leads to polynomial-time emptiness and universality, but still suffers fromcoNP-completeness of the emptiness of binary intersections. Finally, efficient algorithms can be obtained by imposing an order of horizontal evaluation globally for all automata in the class. Depending onthe choice of the order, we obtain different classes of automata, eachof which has the same expressiveness as Counting MSO

Distributed Graph Automata and Verification of Distributed Algorithms

Combining ideas from distributed algorithms and alternating automata, we introduce a new class of finite graph automata that recognize precisely the languages of finite graphs definable in monadic second-order logic. By restricting transitions to be nondeterministic or deterministic, we also obtain two strictly weaker variants of our automata for which the emptiness problem is decidable. As an application, we suggest how suitable graph automata might be useful in formal verification of distributed algorithms, using Floyd-Hoare logic.Comment: 26 pages, 6 figures, includes a condensed version of the author's Master's thesis arXiv:1404.6503. (This version of the article (v2) is identical to the previous one (v1), except for minor changes in phrasing.