Computer Science

This paper is concerned with learners who aim to learn patterns in infinite binary sequences: shown longer and longer initial segments of a binary sequence, they either attempt to predict whether the next bit will be a 0 or will be a 1 or they issue forecast probabilities for these events. Several variants of this problem are considered. In each case, a no-free-lunch result of the following form is established: the problem of learning is a formidably difficult one, in that no matter what method is pursued, failure is incomparably more common that success; and difficult choices must be faced in choosing a method of learning, since no approach dominates all others in its range of success. In the simplest case, the comparison of the set of situations in which a method fails and the set of situations in which it succeeds is a matter of cardinality (countable vs. uncountable); in other cases, it is a topological matter (meagre vs. co-meagre) or a hybrid computational-topological matter (effectively meagre vs. effectively co-meagre).

Labelling-based formal argumentation relies on labelling functions that typically assign one of 3 labels to indicate either acceptance, rejection, or else undecided-to-be-either, to each argument. While a classical labelling-based approach applies globally uniform conditions as to how an argument is to be labelled, they can be determined more locally per argument. Abstract dialectical frameworks (ADF) is a well-known argumentation formalism that belongs to this category, offering a greater labelling flexibility. As the size of an argumentation increases in the numbers of arguments and argument-to-argument relations, however, it becomes increasingly more costly to check whether a labelling function satisfies those local conditions or even whether the conditions are as per the intention of those who had specified them. Some compromise is thus required for reasoning about a larger argumentation. In this context, there is a more recently proposed formalism of may-must argumentation (MMA) that enforces still local but more abstract labelling conditions. We identify how they link to each other in this work. We prove that there is a Galois connection between them, in which ADF is a concretisation of MMA and MMA is an abstraction of ADF. We explore the consequence of abstract interpretation at play in formal argumentation, demonstrating a sound reasoning about the judgement of acceptability/rejectability in ADF from within MMA. As far as we are aware, there is seldom any work that incorporates abstract interpretation into formal argumentation in the literature, and, in the stated context, this work is the first to demonstrate its use and relevance.

We introduce the concept of access-based intuitionistic knowledge which relies on the intuition that agent i knows φ if i has found access to a proof of φ . Basic principles are distribution and factivity of knowledge as well as □φ→ K i φ and K i (φ∨ψ)→( K i φ∨ K i ψ) , where □φ reads ` φ is proved'. The formalization extends a family of classical modal logics designed in [Lewitzka 2015, 2017, 2019] as combinations of IPC and CPC and as systems for the reasoning about proof, i.e. intuitionistic truth. We adopt a formalization of common knowledge from [Lewitzka 2011] and interpret it here as access-based common knowledge. We compare our proposal with recent approaches to intuitionistic knowledge [Artemov and Protopopescu 2016; Lewitzka 2017, 2019] and bring together these different concepts in a unifying semantic framework based on Heyting algebra expansions.

Active learning of timed languages is concerned with the inference of timed automata from observed timed words. The agent can query for the membership of words in the target language, or propose a candidate model and verify its equivalence to the target. The major difficulty of this framework is the inference of clock resets, central to the dynamics of timed automata, but not directly observable. Interesting first steps have already been made by restricting to the subclass of event-recording automata, where clock resets are tied to observations. In order to advance towards learning of general timed automata, we generalize this method to a new class, called reset-free event-recording automata, where some transitions may reset no clocks. This offers the same challenges as generic timed automata while keeping the simpler framework of event-recording automata for the sake of readability. Central to our contribution is the notion of invalidity, and the algorithm and data structures to deal with it, allowing on-the-fly detection and pruning of reset hypotheses that contradict observations, a key to any efficient active-learning procedure for generic timed automata.

The correctness of networks is often described in terms of the individual data flow of components instead of their global behavior. In software-defined networks, it is far more convenient to specify the correct behavior of packets than the global behavior of the entire network. Petri nets with transits extend Petri nets and Flow-LTL extends LTL such that the data flows of tokens can be tracked. We present the tool AdamMC as the first model checker for Petri nets with transits against Flow-LTL. We describe how AdamMC can automatically encode concurrent updates of software-defined networks as Petri nets with transits and how common network specifications can be expressed in Flow-LTL. Underlying AdamMC is a reduction to a circuit model checking problem. We introduce a new reduction method that results in tremendous performance improvements compared to a previous prototype. Thereby, AdamMC can handle software-defined networks with up to 82 switches.

In relational approach to general rough sets, ideas of directed relations are supplemented with additional conditions for multiple algebraic approaches in this research paper. The relations are also specialized to representations of general parthood that are upper-directed, reflexive and antisymmetric for a better behaved groupoidal semantics over the set of roughly equivalent objects by the first author. Another distinct algebraic semantics over the set of approximations, and a new knowledge interpretation are also invented in this research by her. Because of minimal conditions imposed on the relations, neighborhood granulations are used in the construction of all approximations (granular and pointwise). Necessary and sufficient conditions for the lattice of local upper approximations to be completely distributive are proved by the second author. These results are related to formal concept analysis. Applications to student centered learning and decision making are also outlined.

A number of categories is presented that are algebraically complete and cocomplete, i.e., every endofunctor has an initial algebra and a terminal coalgebra. For all finitary (and, more generally, all precontinuous) set functors the initial algebra and terminal coalgebra are proved to carry a canonical partial order with the same ideal CPO-completion. And they also both carry a canonical ultrametric with the same Cauchy completion.

We extend the formalisation of confluence results in Kleene algebras to a formalisation of coherent proofs by confluence. To this end, we introduce the structure of modal higher-dimensional globular Kleene algebra, a higher-dimensional generalisation of modal and concurrent Kleene algebra. We give a calculation of a coherent Church-Rosser theorem and Newman's lemma in higher-dimensional Kleene algebras. We interpret these results in the context of higher-dimensional rewriting systems described by polygraphs.

We develop algebraic models of simple type theories, laying out a framework that extends universal algebra to incorporate both algebraic sorting and variable binding. Examples of simple type theories include the unityped and simply-typed λ -calculi, the computational λ -calculus, and predicate logic. Simple type theories are given models in presheaf categories, with structure specified by algebras of polynomial endofunctors that correspond to natural deduction rules. Initial models, which we construct, abstractly describe the syntax of simple type theories. Taking substitution structure into consideration, we further provide sound and complete semantics in structured cartesian multicategories. This development generalises Lambek's correspondence between the simply-typed λ -calculus and cartesian-closed categories, to arbitrary simple type theories.

In the present paper, we develop the algorithmic correspondence theory for hybrid logic with binder. We define the class of Sahlqvist inequalities, each inequality of which is shown to have a first-order frame correspondent effectively computable by an algorithm ALBA.