Python implementation of OCA-L* for active learning of deterministic real-time one-counter automata and Python implementation of OCA-L* and MinOCA for active learning of visibly one-counter automata.

We give an active learning algorithm for deterministic one-counter automata (DOCA) where the learner can ask the teacher membership and minimal equivalence queries. The algorithm called OL∗learns a DOCA in time polynomial in the size of the smallest DOCA, recognising the target language.All existing algorithms for learning DOCA, even for the subclasses of deterministic real-time one-counter automata (DROCA) and visibly one-counter automata (VOCA), in the worst case, run in exponential time with respect to the size of the DOCA under learning. Furthermore, previous learning algorithms are "grey-box" algorithms relying on an additional query type - counter value query - where the teacher returns the counter value reached on reading a given word. In contrast, our algorithm is a "black-box" algorithm.It is known that the minimisation of VOCA is NP-hard. However, OL∗can be used for approximate minimisation of DOCA. In this case, the output size is at most polynomial in the size of a minimal DOCA.

Abstract In this paper, we introduce a novel method for active learning of deterministic real-time one-counter automata (droca). The existing techniques for learning a droca rely on observing the behaviour of the droca up to exponentially large counter values. Our algorithm eliminates this need and requires only a polynomial number of queries. Additionally, our method differs from existing techniques as we learn a minimal counter-synchronous droca, resulting in much smaller counter-examples on equivalence queries. Learning a minimal counter-synchronous droca cannot be done in polynomial time unless $$\mathsf {P = NP}$$ P = NP , even in the case of visibly one-counter automata. We use a SAT solver to overcome this difficulty. The solver is used to compute a minimal separating DFA from a given set of positive and negative samples. We prove that the equivalence of two counter-synchronous drocas can be checked significantly faster than that of general drocas. For visibly one-counter automata, we have discovered an even faster algorithm for equivalence checking. We implemented the proposed learning algorithm and tested it on randomly generated drocas. Our evaluations show that the proposed method outperforms the existing techniques on the test set.

We introduce weighted one-deterministic-counter automata (odca). These are weighted one-counter automata (oca) with the property of counter-determinacy, meaning that all paths labelled by a given word starting from the initial configuration have the same counter-effect. Weighted odcas are a strict extension of weighted visibly ocas, which are weighted ocas where the input alphabet determines the actions on the counter. We present a novel problem called the co-VS (complement to a vector space) reachability problem for weighted odcas over fields, which seeks to determine if there exists a run from a given configuration of a weighted odca to another configuration whose weight vector lies outside a given vector space. We establish two significant properties of witnesses for co-VS reachability: they satisfy a pseudo-pumping lemma, and the lexicographically minimal witness has a special form. It follows that the co-VS reachability problem is in 𝖯. These reachability problems help us to show that the equivalence problem of weighted odcas over fields is in 𝖯 by adapting the equivalence proof of deterministic real-time ocas [Stanislav Böhm and Stefan Göller, 2011] by Böhm et al. This is a step towards resolving the open question of the equivalence problem of weighted ocas. Finally, we demonstrate that the regularity problem, the problem of checking whether an input weighted odca over a field is equivalent to some weighted automaton, is in 𝖯. We also consider boolean odcas and show that the equivalence problem for (non-deterministic) boolean odcas is in PSPACE, whereas it is undecidable for (non-deterministic) boolean ocas.

We propose a seamless integration of the block product operation to the recently developed algebraic framework for regular languages of countable words. A simple but subtle accompanying block product principle has been established. Building on this, we generalize the well-known algebraic characterizations of first-order logic (resp. first-order logic with two variables) in terms of strongly (resp. weakly) iterated block products. We use this to arrive at a complete analogue of Schiitzenberger-McNaughton-Papert theorem for countable words. We also explicate the role of block products for linear temporal logic by formulating a novel algebraic characterization of a natural fragment.

We study the class of languages of finitely-labelled countable linear orderings definable in two-variable first-order logic. We give a number of characterisations, in particular an algebraic one in terms of circle monoids, using equations. This generalises the corresponding characterisation, namely variety DA, over finite words to the countable case. A corollary is that the membership in this class is decidable: for instance given an MSO formula it is possible to check if there is an equivalent two-variable logic formula over countable linear orderings. In addition, we prove that the satisfiability problems for two-variable logic over arbitrary, countable, and scattered linear orderings are NEXPTIME-complete.

We consider first-order logic with monoidal quantifiers over words. We show that all languages with a neutral letter, definable using the addition predicate are also definable with the order predicate as the only numerical predicate. Let S be a subset of monoids. Let L be the logic closed under quantification over the monoids in S. Then we prove that L[<;,+] and L[<;] define the same neutral letter languages. Our result can be interpreted as the Crane Beach conjecture to hold for the logic L[<;,+]. As a consequence we get the result of Roy and Straubing that FO+MOD[<;,+] collapses to FO+MOD[<;]. For cyclic groups, we answer an open question of Roy and Straubing, proving that MOD[<;,+] collapses to MOD[<;]. Our result also shows that multiplication as a numerical predicate is necessary for Barrington's theorem to hold and also to simulate majority quantifiers. All these results can be viewed as separation results for highly uniform circuit classes. For example we separate FO[<;,+]-uniform CC0 from FO[<;,+]-uniform ACC0.