Computer Science

The abelian critical exponent of an infinite word w is defined as the maximum ratio between the exponent and the period of an abelian power occurring in w . It was shown by Fici et al. that the set of finite abelian critical exponents of Sturmian words coincides with the Lagrange spectrum. This spectrum contains every large enough positive real number. We construct words whose abelian critical exponents fill the remaining gaps, that is, we prove that for each nonnegative real number θ there exists an infinite word having abelian critical exponent θ . We also extend this result to the k -abelian setting.

This work introduces a concept of explanations with respect to the violation of safe behaviours within software defined networks (SDNs) expressible in NetKAT. The latter is a network programming language that is based on a well-studied mathematical structure, namely, Kleene Algebra with Tests (KAT). Amongst others, the mathematical foundation of NetKAT gave rise to a sound and complete equational theory. In our setting, a safe behaviour is characterised by a NetKAT policy which does not enable forwarding packets from ingress to an undesirable egress. Explanations for safety violations are derived in an equational fashion, based on a modification of the existing NetKAT axiomatisation.

This work introduces a concept of explanations with respect to the violation of safe behaviours within software defined networks (SDNs) expressible in NetKAT. The latter is a network programming language based on a well-studied mathematical structure, namely, Kleene Algebra with Tests (KAT). Amongst others, the mathematical foundation of NetKAT gave rise to a sound and complete equational theory. In our setting, a safe behaviour is characterised by a NetKAT policy, or program, which does not enable forwarding packets from an ingress i to an undesirable egress e. We show how explanations for safety violations can be derived in an equational fashion, according to a modification of the existing NetKAT axiomatisation. We propose an approach based on the Maude system for actually computing the undesired behaviours witnessing the forwarding of packets from i to e as above. SDN-SafeCheck is a tool based on Maude equational theories satisfying important properties such as Church-Rosser and termination. SDN-SafeCheck automatically identifies all the undesired behaviours leading to e, covering forwarding paths up to a user specified size.

Many of the numerous automaton models proposed in the literature can be regarded as a finite automaton equipped with an additional storage mechanism. In this thesis, we focus on two such models, namely the finite automata over groups and the homing vector automata. A finite automaton over a group G is a nondeterministic finite automaton equipped with a register that holds an element of the group G . The register is initialized to the identity element of the group and a computation is successful if the register is equal to the identity element at the end of the computation after being multiplied with a group element at every step. We investigate the language recognition power of finite automata over integer and rational matrix groups and reveal new relationships between the language classes corresponding to these models. We examine the effect of various parameters on the language recognition power. We establish a link between the decision problems of matrix semigroups and the corresponding automata. We present some new results about valence pushdown automata and context-free valence grammars. We also propose the new homing vector automaton model, which is a finite automaton equipped with a vector that can be multiplied with a matrix at each step. The vector can be checked for equivalence to the initial vector and the acceptance criterion is ending up in an accept state with the value of the vector being equal to the initial vector. We examine the effect of various restrictions on the model by confining the matrices to a particular set and allowing the equivalence test only at the end of the computation. We define the different variants of the model and compare their language recognition power with that of the classical models.

We consider extensions of monadic second order logic over ω -words, which are obtained by adding one language that is not ω -regular. We show that if the added language L has a neutral letter, then the resulting logic is necessarily undecidable. A corollary is that the ω -regular languages are the only decidable Boolean-closed full trio over ω -words.

A theorem of Nekrashevych and Sidki shows the Mealy Automata structures one can place on Z^m are parametrized by a family of matrices (called "1/2-integral") and a choice of residuation vector e in Z^m. While the impact of the chosen matrix is well understood, the impact of the residuation vector on the resulting structure is seemingly sporadic. In this paper we characterize the impact of the residuation vector e by recognizing an initial structure when e is the first standard basis vector. All other choices of e extend this initial structure by adding "fractional elements" in a way we make precise.

Form validators based on regular expressions are often used on digital forms to prevent users from inserting data in the wrong format. However, writing these validators can pose a challenge to some users. We present FOREST, a regular expression synthesizer for digital form validations. FOREST produces a regular expression that matches the desired pattern for the input values and a set of conditions over capturing groups that ensure the validity of integer values in the input. Our synthesis procedure is based on enumerative search and uses a Satisfiability Modulo Theories (SMT) solver to explore and prune the search space. We propose a novel representation for regular expressions synthesis, multi-tree, which induces patterns in the examples and uses them to split the problem through a divide-and-conquer approach. We also present a new SMT encoding to synthesize capture conditions for a given regular expression. To increase confidence in the synthesized regular expression, we implement user interaction based on distinguishing inputs. We evaluated FOREST on real-world form-validation instances using regular expressions. Experimental results show that FOREST successfully returns the desired regular expression in 72% of the instances and outperforms REGEL, a state-of-the-art regular expression synthesizer.

Computer Science students, in general, find Automata Theory difficult and mostly unrelated to their area of study. To mitigate these perceptions, FSM, a library to program state machines and grammars, was developed to bring programming to the Automata Theory classroom. The results of the library's maiden voyage at Seton Hall University had a positive impact on students, but the students found the library difficult to use due to the error messages generated. These messages were generated by the host language meaning that students needed to be familiar with the library's implementation to make sense of them. This article presents the design of and results obtained from using an error-messaging system tailor-made for FSM. The effectiveness of the library was measured by both a control group study and a survey. The results strongly suggest that the error-messaging system has had a positive impact on students' attitude towards automata theory, towards programming in FSM, and towards FSM error messages. The consequence has been a marked improvement on students' ability to implement algorithms developed as part of constructive proofs by making the debugging of FSM programs easier.

A popular method for solving reachability in timed automata proceeds by enumerating reachable sets of valuations represented as zones. A naïve enumeration of zones does not terminate. Various termination mechanisms have been studied over the years. Coming up with efficient termination mechanisms has been remarkably more challenging when the automaton has diagonal constraints in guards. In this paper, we propose a new termination mechanism for timed automata with diagonal constraints based on a new simulation relation between zones. Experiments with an implementation of this simulation show significant gains over existing methods.

Cyber-physical systems come with increasingly complex architectures and failure modes, which complicates the task of obtaining accurate system reliability models. At the same time, with the emergence of the (industrial) Internet-of-Things, systems are more and more often being monitored via advanced sensor systems. These sensors produce large amounts of data about the components' failure behaviour, and can, therefore, be fruitfully exploited to learn reliability models automatically. This paper presents an effective algorithm for learning a prominent class of reliability models, namely fault trees, from observational data. Our algorithm is evolutionary in nature; i.e., is an iterative, population-based, randomized search method among fault-tree structures that are increasingly more consistent with the observational data. We have evaluated our method on a large number of case studies, both on synthetic data, and industrial data. Our experiments show that our algorithm outperforms other methods and provides near-optimal results.