Narrow Results

A theory of representation of semiautomata by canonical words and equivalences was developed in [7]. That work was motivated by trace-assertion specifications of software modules, but its focus was entirely on the underlying mathematical model. In the present paper we extend that theory to automata with Moore and Mealy outputs, and show how to apply the extended theory to the specification of modules. In particular, we present a unified view of the trace-assertion methodology, as guided by our theory. We illustrate this approach, and some specific issues, using several nontrivial examples. We include a discussion of finite versus infinite modules, methods of error handling, some awkward features of the trace-assertion method, and a comparison to specifications by automata. While specifications by trace assertions and automata are equivalent in power, there are cases where one approach appears to be more natural than the other. We conclude that, for certain types of system modules, formal specification by automata, as opposed to informal state machines, is not only possible, but practical.

We consider the equivalence problem for labeled Markov chains (LMCs), where each state is labeled with an observation. Two LMCs are equivalent if every finite sequence of observations has the same probability of occurrence in the two LMCs. We show that equivalence can be decided in polynomial time, using a reduction to the equivalence problem for probabilistic automata, which is known to be solvable in polynomial time. We provide an alternative algorithm to solve the equivalence problem, which is based on a new definition of bisimulation for probabilistic automata. We also extend the technique to decide the equivalence of weighted probabilistic automata.

Then, we consider the equivalence problem for labeled Markov decision processes (LMDPs), which asks given two LMDPs whether for every scheduler (i.e. way of resolving the nondeterministic decisions) for each of the processes, there exists a scheduler for the other process such that the resulting LMCs are equivalent. The decidability of this problem remains open. We show that the schedulers can be restricted to be observation-based, but may require infinite memory.

Simulations of weighted tree automata (wta) are considered. It is shown how such simulations can be decomposed into simpler functional and dual functional simulations also called forward and backward simulations. In addition, it is shown in several cases (fields, commutative rings, NOETHERIAN semirings, semiring of natural numbers) that all equivalent wta M and N can be joined by a finite chain of simulations. More precisely, in all mentioned cases there is a single wta that simulates both M and N. Those results immediately yield decidability of equivalence provided that the semiring is finitely (and effectively) presented.

The decidability of equivalence for three important classes of tree transducers is discussed. Each class can be obtained as a natural restriction of deterministic macro tree transducers (MTTs): (1) no context parameters, i.e., top-down tree transducers, (2) linear size increase, i.e., MSO definable tree transducers, and (3) monadic input and output ranked alphabets. For the full class of mtts, decidability of equivalence remains a long-standing open problem.

We consider one of the weakest variants of cost register automata over a tropical semiring, namely copyless cost register automata over $\mathbb{N}$ with updates using $min$ and increments. We show that this model can simulate, in some sense, the runs of counter machines with zero-tests. We deduce that a number of problems pertaining to that model are undecidable, namely equivalence, upperboundedness, and semilinearity. In particular, the undecidability of equivalence disproves a conjecture of Alur et al. from 2012. To emphasize how weak these machines are, we also show that they can be expressed as a restricted form of linearly-ambiguous weighted automata.

A language over an alphabet $\mathsf{B}=\mathsf{A}\cup \overline{\mathsf{A}}$ of opening ($\mathsf{A}$) and closing ($\overline{\mathsf{A}}$) brackets, is balanced if it is a subset of the Dyck language ${𝔻}_{\mathsf{B}}$ over $\mathsf{B}$, and it is well-formed if all words are prefixes of words in ${𝔻}_{\mathsf{B}}$. We show that well-formedness of a context-free language is decidable in polynomial time, and that the longest common reduced suffix can be computed in polynomial time. With this at a hand we decide for the class 2-$\mathsf{TW}$ of non-linear tree transducers with output alphabet $\mathsf{B}$ whether or not the output language is balanced.

We show that the equivalence, unambiguity, and sequentiality problems are decidable for finitely ambiguous max-plus tree automata. A max-plus tree automaton is a weighted tree automaton over the max-plus semiring. A max-plus tree automaton is called finitely ambiguous if the number of accepting runs on every tree is bounded by a global constant and it is called unambiguous if there exists at most one accepting run on every tree. For the equivalence problem, we show that for two finitely ambiguous max-plus tree automata, it is decidable whether both assign the same weight to every tree. For the unambiguity and sequentiality problems, we show that for every finitely ambiguous max-plus tree automaton, both the existence of an equivalent unambiguous automaton and the existence of an equivalent deterministic automaton are decidable.

Some special sets of ω-words, namely, the sets that can be obtained by catenating nonempty finite sets of words, an infinite number of times, are considered. A condition for the equality of two such sets of ω-words is obtained.