26,059 research outputs found
Minimal cover-automata for finite languages
AbstractA cover-automaton A of a finite language L⊆Σ∗ is a finite deterministic automaton (DFA) that accepts all words in L and possibly other words that are longer than any word in L. A minimal deterministic finite cover automaton (DFCA) of a finite language L usually has a smaller size than a minimal DFA that accept L. Thus, cover automata can be used to reduce the size of the representations of finite languages in practice. In this paper, we describe an efficient algorithm that, for a given DFA accepting a finite language, constructs a minimal deterministic finite cover-automaton of the language. We also give algorithms for the boolean operations on deterministic cover automata, i.e., on the finite languages they represent
Canonical Algebraic Generators in Automata Learning
Many methods for the verification of complex computer systems require the
existence of a tractable mathematical abstraction of the system, often in the
form of an automaton. In reality, however, such a model is hard to come up
with, in particular manually. Automata learning is a technique that can
automatically infer an automaton model from a system -- by observing its
behaviour. The majority of automata learning algorithms is based on the
so-called L* algorithm. The acceptor learned by L* has an important property:
it is canonical, in the sense that, it is, up to isomorphism, the unique
deterministic finite automaton of minimal size accepting a given regular
language. Establishing a similar result for other classes of acceptors, often
with side-effects, is of great practical importance. Non-deterministic finite
automata, for instance, can be exponentially more succinct than deterministic
ones, allowing verification to scale. Unfortunately, identifying a canonical
size-minimal non-deterministic acceptor of a given regular language is in
general not possible: it can happen that a regular language is accepted by two
non-isomorphic non-deterministic finite automata of minimal size. In
particular, it thus is unclear which one of the automata should be targeted by
a learning algorithm. In this thesis, we further explore the issue and identify
(sub-)classes of acceptors that admit canonical size-minimal representatives.Comment: PhD thesi
Finite Automata for the Sub- and Superword Closure of CFLs: Descriptional and Computational Complexity
We answer two open questions by (Gruber, Holzer, Kutrib, 2009) on the
state-complexity of representing sub- or superword closures of context-free
grammars (CFGs): (1) We prove a (tight) upper bound of on
the size of nondeterministic finite automata (NFAs) representing the subword
closure of a CFG of size . (2) We present a family of CFGs for which the
minimal deterministic finite automata representing their subword closure
matches the upper-bound of following from (1).
Furthermore, we prove that the inequivalence problem for NFAs representing sub-
or superword-closed languages is only NP-complete as opposed to PSPACE-complete
for general NFAs. Finally, we extend our results into an approximation method
to attack inequivalence problems for CFGs
Dense Quantum Coding and a Lower Bound for 1-way Quantum Automata
We consider the possibility of encoding m classical bits into much fewer n
quantum bits so that an arbitrary bit from the original m bits can be recovered
with a good probability, and we show that non-trivial quantum encodings exist
that have no classical counterparts. On the other hand, we show that quantum
encodings cannot be much more succint as compared to classical encodings, and
we provide a lower bound on such quantum encodings. Finally, using this lower
bound, we prove an exponential lower bound on the size of 1-way quantum finite
automata for a family of languages accepted by linear sized deterministic
finite automata.Comment: 12 pages, 3 figures. Defines random access codes, gives upper and
lower bounds for the number of bits required for such (possibly quantum)
codes. Derives the size lower bound for quantum finite automata of the
earlier version of the paper using these result
Learn with SAT to Minimize B\"uchi Automata
We describe a minimization procedure for nondeterministic B\"uchi automata
(NBA). For an automaton A another automaton A_min with the minimal number of
states is learned with the help of a SAT-solver.
This is done by successively computing automata A' that approximate A in the
sense that they accept a given finite set of positive examples and reject a
given finite set of negative examples. In the course of the procedure these
example sets are successively increased. Thus, our method can be seen as an
instance of a generic learning algorithm based on a "minimally adequate
teacher" in the sense of Angluin.
We use a SAT solver to find an NBA for given sets of positive and negative
examples. We use complementation via construction of deterministic parity
automata to check candidates computed in this manner for equivalence with A.
Failure of equivalence yields new positive or negative examples. Our method
proved successful on complete samplings of small automata and of quite some
examples of bigger automata.
We successfully ran the minimization on over ten thousand automata with
mostly up to ten states, including the complements of all possible automata
with two states and alphabet size three and discuss results and runtimes;
single examples had over 100 states.Comment: In Proceedings GandALF 2012, arXiv:1210.202
On the (In)Succinctness of Muller Automata
There are several types of finite automata on infinite words, differing in their acceptance conditions. As each type has its own advantages, there is an extensive research on the size blowup involved in translating one automaton type to another.
Of special interest is the Muller type, providing the most detailed acceptance condition. It turns out that there is inconsistency and incompleteness in the literature results regarding the translations to and from Muller automata. Considering the automaton size, some results take into account, in addition to the number of states, the alphabet length and the number of transitions while ignoring the length of the acceptance condition, whereas other results consider the length of the acceptance condition while ignoring the two other parameters.
We establish a full picture of the translations to and from Muller automata, enhancing known results and adding new ones. Overall, Muller automata can be considered less succinct than parity, Rabin, and Streett automata: translating nondeterministic Muller automata to the other nondeterministic types involves a polynomial size blowup, while the other way round is exponential; translating between the deterministic versions is exponential in both directions; and translating nondeterministic automata of all types to deterministic Muller automata is doubly exponential, as opposed to a single exponent in the translations to the other deterministic types
Canonical Algebraic Generators in Automata Learning
Many methods for the verification of complex computer systems require the existence of a tractable mathematical abstraction of the system, often in the form of an automaton. In reality, however, such a model is hard to come up with, in particular manually. Automata learning is a technique that can automatically infer an automaton model from a system -- by observing its behaviour. The majority of automata learning algorithms is based on the so-called L* algorithm. The acceptor learned by L* has an important property: it is canonical, in the sense that, it is, up to isomorphism, the unique deterministic finite automaton of minimal size accepting a given regular language. Establishing a similar result for other classes of acceptors, often with side-effects, is of great practical importance. Non-deterministic finite automata, for instance, can be exponentially more succinct than deterministic ones, allowing verification to scale. Unfortunately, identifying a canonical size-minimal non-deterministic acceptor of a given regular language is in general not possible: it can happen that a regular language is accepted by two non-isomorphic non-deterministic finite automata of minimal size. In particular, it thus is unclear which one of the automata should be targeted by a learning algorithm. In this thesis, we further explore the issue and identify (sub-)classes of acceptors that admit canonical size-minimal representatives.
In more detail, the contributions of this thesis are three-fold.
First, we expand the automata (learning) theory of Guarded Kleene Algebra with Tests (GKAT), an efficiently decidable logic expressive enough to model simple imperative programs. In particular, we present GL*, an algorithm that learns the unique size-minimal GKAT automaton for a given deterministic language, and prove that GL* is more efficient than an existing variation of L*. We implement both algorithms in OCaml, and compare them on example programs.
Second, we present a category-theoretical framework based on generators, bialgebras, and distributive laws, which identifies, for a wide class of automata with side-effects in a monad, canonical target models for automata learning. Apart from recovering examples from the literature, we discover a new canonical acceptor of regular languages, and present a unifying minimality result.
Finally, we show that the construction underlying our framework is an instance of a more general theory. First, we see that deriving a minimal bialgebra from a minimal coalgebra can be realized by applying a monad on a category of subobjects with respect to an epi-mono factorisation system. Second, we explore the abstract theory of generators and bases for algebras over a monad: we discuss bases for bialgebras, the product of bases, generalise the representation theory of linear maps, and compare our ideas to a coalgebra-based approach
State-deterministic Finite Automata with Translucent Letters and Finite Automata with Nondeterministically Translucent Letters
Deterministic and nondeterministic finite automata with translucent letters
were introduced by Nagy and Otto more than a decade ago as Cooperative
Distributed systems of a kind of stateless restarting automata with window size
one. These finite state machines have a surprisingly large expressive power:
all commutative semi-linear languages and all rational trace languages can be
accepted by them including various not context-free languages. While the
nondeterministic variant defines a language class with nice closure properties,
the deterministic variant is weaker, however it contains all regular languages,
some non-regular context-free languages, as the Dyck language, and also some
languages that are not even context-free. In all those models for each state,
the letters of the alphabet could be in one of the following categories: the
automaton cannot see the letter (it is translucent), there is a transition
defined on the letter (maybe more than one transitions in nondeterministic
case) or none of the above categories (the automaton gets stuck by seeing this
letter at the given state and this computation is not accepting).
State-deterministic automata are recent models, where the next state of the
computation determined by the structure of the automata and it is independent
of the processed letters. In this paper our aim is twofold, on the one hand, we
investigate state-deterministic finite automata with translucent letters. These
automata are specially restricted deterministic finite automata with
translucent letters.
In the other novel model we present, it is allowed that for a state the set
of translucent letters and the set of letters for which transition is defined
are not disjoint. One can interpret this fact that the automaton has a
nondeterministic choice for each occurrence of such letters to see them (and
then erase and make the transition) or not to see that occurrence at that time.
Based on these semi-translucent letters, the expressive power of the automata
increases, i.e., in this way a proper generalization of the previous models is
obtained.Comment: In Proceedings AFL 2023, arXiv:2309.0112
Interference Automata
We propose a computing model, the Two-Way Optical Interference Automata
(2OIA), that makes use of the phenomenon of optical interference. We introduce
this model to investigate the increase in power, in terms of language
recognition, of a classical Deterministic Finite Automaton (DFA) when endowed
with the facility of optical interference. The question is in the spirit of
Two-Way Finite Automata With Quantum and Classical States (2QCFA) [A. Ambainis
and J. Watrous, Two-way Finite Automata With Quantum and Classical States,
Theoretical Computer Science, 287 (1), 299-311, (2002)] wherein the classical
DFA is augmented with a quantum component of constant size. We test the power
of 2OIA against the languages mentioned in the above paper. We give efficient
2OIA algorithms to recognize languages for which 2QCFA machines have been shown
to exist, as well as languages whose status vis-a-vis 2QCFA has been posed as
open questions. Finally we show the existence of a language that cannot be
recognized by a 2OIA but can be recognized by an space Turing machine.Comment: 19 pages. A preliminary version appears under the title "On a Model
of Computation based on Optical Interference" in Proc. of the 16-th
Australasian Workshop on Combinatorial Algorithms (AWOCA'05), pp. 249-26
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