Computer Science

In this work, we affirm the conjecture proposed by Gabriele Fici and Filippo Mignosi at the 10th Conference on Combinatorics on Words.

In this first paper, we look at the following question: are the properties of the Fibonacci tree still true if we consider a finitely generated tree by the same rules but rooted at a black node? The direct answer is no, but new properties arise, a bit more complex than in the case of a tree rooted at a white node, but still of interest.

Several abstract machines that operate on symbolic input alphabets have been proposed in the last decade, for example, symbolic automata or lattice automata. Applications of these types of automata include software security analysis and natural language processing. While these models provide means to describe words over infinite input alphabets, there is no considerable work on symbolic output (as present in transducers) alphabets, or even abstraction (widening) thereof. Furthermore, established approaches for transforming, for example, minimizing or reducing, finite-state machines that produce output on states or transitions are not applicable. A notion of equivalence of this type of machines is needed to make statements about whether or not transformations maintain the semantics. We present abstract transducers as a new form of finite-state transducers. Both their input alphabet and the output alphabet is composed of abstract words, where one abstract word represents a set of concrete words. The mapping between these representations is described by abstract word domains. By using words instead of single letters, abstract transducers provide the possibility of lookaheads to decide on state transitions to conduct. Since both the input symbol and the output symbol on each transition is an abstract entity, abstraction techniques can be applied naturally. We apply abstract transducers as the foundation for sharing task artifacts for reuse in context of program analysis and verification, and describe task artifacts as abstract words. A task artifact is any entity that contributes to an analysis task and its solution, for example, candidate invariants or source code to weave.

We present abstraction-refinement algorithms for model checking safety properties of timed automata. The abstraction domain we consider abstracts away zones by restricting the set of clock constraints that can be used to define them, while the refinement procedure computes the set of constraints that must be taken into consideration in the abstraction so as to exclude a given spurious counterexample. We implement this idea in two ways: an enumerative algorithm where a lazy abstraction approach is adopted, meaning that possibly different abstract domains are assigned to each exploration node; and a symbolic algorithm where the abstract transition system is encoded with Boolean formulas.

We consider notions of freeness and ambiguity for the acceptance probability of Moore-Crutchfield Measure Once Quantum Finite Automata (MO-QFA). We study the distribution of acceptance probabilities of such MO-QFA, which is partly motivated by similar freeness problems for matrix semigroups and other computational models. We show that determining if the acceptance probabilities of all possible input words are unique is undecidable for 32 state MO-QFA, even when all unitary matrices and the projection matrix are rational and the initial configuration is defined over real algebraic numbers. We utilize properties of the skew field of quaternions, free rotation groups, representations of tuples of rationals as a linear sum of radicals and a reduction of the mixed modification Post's correspondence problem.

We introduce the notion of adaptive synchronisation for pushdown automata, in which there is an external observer who has no knowledge about the current state of the pushdown automaton, but can observe the contents of the stack. The observer would then like to decide if it is possible to bring the automaton from any state into some predetermined state by giving inputs to it in an \emph{adaptive} manner, i.e., the next input letter to be given can depend on how the contents of the stack changed after the current input letter. We show that for non-deterministic pushdown automata, this problem is 2-EXPTIME-complete and for deterministic pushdown automata, we show EXPTIME-completeness. To prove the lower bounds, we first introduce (different variants of) subset-synchronisation and show that these problems are polynomial-time equivalent with the adaptive synchronisation problem. We then prove hardness results for the subset-synchronisation problems. For proving the upper bounds, we consider the problem of deciding if a given alternating pushdown system has an accepting run with at most k leaves and we provide an n O( k 2 ) time algorithm for this problem.

An infinite sequence over a finite alphabet {\Sigma} of symbols is called normal iff the limiting frequency of every finite string w exists and equals |{\Sigma}|^{|w|}. A celebrated theorem by Agafonov states that a sequence {\alpha} is normal iff every finite-state selector. Normality is generalised to arbitrary probability maps \mu: {\alpha} is is \mu-distributed if, for every finite string w, the limiting frequency of w in {\alpha} exists and equals \mu(w). Unlike normality, \mu-distributedness is not preserved by finite-state selectors for all probability maps \mu. This raises the question of how to characterize the probability maps \mu for which \mu-distributedness is preserved across finite-state selection, or equivalently, by selection by programs using constant space. We prove the following result: for any finite or countably infinite alphabet {\Sigma}, every finite-state selector over {\Sigma} selects a \mu-distributed sequence from every \mu-distributed sequence {\alpha} iff \mu is induced by a Bernoulli distribution on {\Sigma}, that is a probability distribution on the alphabet extended to words by taking the product. The primary -- and remarkable -- consequence of our main result is a complete characterization of the set of probability maps, on finite and infinite alphabets, for which Agafonov-type results hold. The main positive takeaway is that (the appropriate generalization of) Agafonov's Theorem holds for Bernoulli distributions (rather than just equidistributions) on both finite and countably infinite alphabets. As a further consequence, we obtain a result in the area of symbolic dynamical systems: the shift-invariant measures {\nu} on {\Sigma}^{\omega} such that any finite-state selector preserves the property of genericity for {\mu}, are exactly the positive Bernoulli measures.

We develop an algebraic language theory based on the notion of an Eilenberg--Moore algebra. In comparison to previous such frameworks the main contribution is the support for algebras with infinitely many sorts and the connection to logic in form of so-called `definable algebras'.

We investigate the state complexity of the star of symmetrical differences using modifiers and monsters. A monster is an automaton in which every function from states to states is represented by at least one letter. A modifier is a set of functions allowing one to transform a set of automata into one automaton. These recent theoretical concepts allow one to find easily the desired state complexity. We then exhibit a witness with a constant size alphabet.

We study alternating automata with qualitative semantics over infinite binary trees: alternation means that two opposing players construct a decoration of the input tree called a run, and the qualitative semantics says that a run of the automaton is accepting if almost all branches of the run are accepting. In this paper we prove a positive and a negative result for the emptiness problem of alternating automata with qualitative semantics. The positive result is the decidability of the emptiness problem for the case of Büchi acceptance condition. An interesting aspect of our approach is that we do not extend the classical solution for solving the emptiness problem of alternating automata, which first constructs an equivalent non-deterministic automaton. Instead, we directly construct an emptiness game making use of imperfect information. The negative result is the undecidability of the emptiness problem for the case of co-Büchi acceptance condition. This result has two direct consequences: the undecidability of monadic second-order logic extended with the qualitative path-measure quantifier, and the undecidability of the emptiness problem for alternating tree automata with non-zero semantics, a recently introduced probabilistic model of alternating tree automata.