61 research outputs found
Two-Way Visibly Pushdown Automata and Transducers
Automata-logic connections are pillars of the theory of regular languages.
Such connections are harder to obtain for transducers, but important results
have been obtained recently for word-to-word transformations, showing that the
three following models are equivalent: deterministic two-way transducers,
monadic second-order (MSO) transducers, and deterministic one-way automata
equipped with a finite number of registers. Nested words are words with a
nesting structure, allowing to model unranked trees as their depth-first-search
linearisations. In this paper, we consider transformations from nested words to
words, allowing in particular to produce unranked trees if output words have a
nesting structure. The model of visibly pushdown transducers allows to describe
such transformations, and we propose a simple deterministic extension of this
model with two-way moves that has the following properties: i) it is a simple
computational model, that naturally has a good evaluation complexity; ii) it is
expressive: it subsumes nested word-to-word MSO transducers, and the exact
expressiveness of MSO transducers is recovered using a simple syntactic
restriction; iii) it has good algorithmic/closure properties: the model is
closed under composition with a unambiguous one-way letter-to-letter transducer
which gives closure under regular look-around, and has a decidable equivalence
problem
A Generalised Twinning Property for Minimisation of Cost Register Automata
Weighted automata (WA) extend finite-state automata by associating with transitions weights from a semiring S, defining functions from words to S. Recently, cost register automata (CRA) have been introduced as an alternative model to describe any function realised by a WA by means of a deterministic machine. Unambiguous WA over a monoid (M, â) can equivalently be described by cost register automata whose registers take their values in M, and are updated by operations of the form x: = y â c, with c â M. This class is denoted by CRAâc(M).
We introduce a twinning property and a bounded variation property parametrised by an integer k, such that the corresponding notions introduced originally by Choffrut for finite-state transducers are obtained for k = 1. Given an unambiguous weighted automaton W over an infinitary group (G, â) realizing some function f, we prove that the three following properties are equivalent: i) W satisfies the twinning property of order k, ii) f satisfies the k-bounded variation property, and iii) f can be described by a CRAâc(G) with at most k registers.
In the spirit of tranducers, we actually prove this result in a more general setting by considering machines over the semiring of finite sets of elements from (G, â): the three properties are still equivalent for such finite-valued weighted automata, that is the ones associating with words subsets of G of cardinality at most â, for some natural â. Moreover, we show that if the operation â of G is commutative and computable, then one can decide whether a WA satisfies the twinning property of order k. As a corollary, this allows to decide the register minimisation problem for the class CRAâc(G).
Last, we prove that a similar result holds for finite-valued finite-state transducers, and that the register minimisation problem for the class CRA.c (B*) is Pspace-complete
Aperiodic String Transducers
Regular string-to-string functions enjoy a nice triple characterization
through deterministic two-way transducers (2DFT), streaming string transducers
(SST) and MSO definable functions. This result has recently been lifted to FO
definable functions, with equivalent representations by means of aperiodic 2DFT
and aperiodic 1-bounded SST, extending a well-known result on regular
languages. In this paper, we give three direct transformations: i) from
1-bounded SST to 2DFT, ii) from 2DFT to copyless SST, and iii) from k-bounded
to 1-bounded SST. We give the complexity of each construction and also prove
that they preserve the aperiodicity of transducers. As corollaries, we obtain
that FO definable string-to-string functions are equivalent to SST whose
transition monoid is finite and aperiodic, and to aperiodic copyless SST
Certainly Unsupervisable States
This paper proposes an abstraction method for compositional synthesis. Synthesis is a method to automatically compute a control program or supervisor that restricts the behaviour of a given system to ensure safety and liveness. Compositional synthesis uses repeated abstraction and simplification to combat the state-space explosion problem for large systems. The abstraction method proposed in this paper finds and removes the so-called certainly unsupervisable states. By removing these states at an early stage, the final state space can be reduced substantially. The paper describes an algorithm with cubic time complexity to compute the largest possible set of removable states. A practical example demonstrates the feasibility of the method to solve real-world problems
The Impatient May Use Limited Optimism to Minimize Regret
Discounted-sum games provide a formal model for the study of reinforcement
learning, where the agent is enticed to get rewards early since later rewards
are discounted. When the agent interacts with the environment, she may regret
her actions, realizing that a previous choice was suboptimal given the behavior
of the environment. The main contribution of this paper is a PSPACE algorithm
for computing the minimum possible regret of a given game. To this end, several
results of independent interest are shown. (1) We identify a class of
regret-minimizing and admissible strategies that first assume that the
environment is collaborating, then assume it is adversarial---the precise
timing of the switch is key here. (2) Disregarding the computational cost of
numerical analysis, we provide an NP algorithm that checks that the regret
entailed by a given time-switching strategy exceeds a given value. (3) We show
that determining whether a strategy minimizes regret is decidable in PSPACE
Synthesis of Computable Regular Functions of Infinite Words
Regular functions from infinite words to infinite words can be equivalently
specified by MSO-transducers, streaming -string transducers as well as
deterministic two-way transducers with look-ahead. In their one-way
restriction, the latter transducers define the class of rational functions.
Even though regular functions are robustly characterised by several
finite-state devices, even the subclass of rational functions may contain
functions which are not computable (by a Turing machine with infinite input).
This paper proposes a decision procedure for the following synthesis problem:
given a regular function (equivalently specified by one of the
aforementioned transducer model), is computable and if it is, synthesize a
Turing machine computing it.
For regular functions, we show that computability is equivalent to
continuity, and therefore the problem boils down to deciding continuity. We
establish a generic characterisation of continuity for functions preserving
regular languages under inverse image (such as regular functions). We exploit
this characterisation to show the decidability of continuity (and hence
computability) of rational and regular functions. For rational functions, we
show that this can be done in (it was already known to be
in by Prieur). In a similar fashion, we also effectively
characterise uniform continuity of regular functions, and relate it to the
notion of uniform computability, which offers stronger efficiency guarantees
Real-Time Synthesis is Hard!
We study the reactive synthesis problem (RS) for specifications given in
Metric Interval Temporal Logic (MITL). RS is known to be undecidable in a very
general setting, but on infinite words only; and only the very restrictive BRRS
subcase is known to be decidable (see D'Souza et al. and Bouyer et al.). In
this paper, we precise the decidability border of MITL synthesis. We show RS is
undecidable on finite words too, and present a landscape of restrictions (both
on the logic and on the possible controllers) that are still undecidable. On
the positive side, we revisit BRRS and introduce an efficient on-the-fly
algorithm to solve it
SAT-Based Synthesis Methods for Safety Specs
Automatic synthesis of hardware components from declarative specifications is
an ambitious endeavor in computer aided design. Existing synthesis algorithms
are often implemented with Binary Decision Diagrams (BDDs), inheriting their
scalability limitations. Instead of BDDs, we propose several new methods to
synthesize finite-state systems from safety specifications using decision
procedures for the satisfiability of quantified and unquantified Boolean
formulas (SAT-, QBF- and EPR-solvers). The presented approaches are based on
computational learning, templates, or reduction to first-order logic. We also
present an efficient parallelization, and optimizations to utilize reachability
information and incremental solving. Finally, we compare all methods in an
extensive case study. Our new methods outperform BDDs and other existing work
on some classes of benchmarks, and our parallelization achieves a super-linear
speedup. This is an extended version of [5], featuring an additional appendix.Comment: Extended version of a paper at VMCAI'1
Alternating Tree Automata with Qualitative Semantics
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 article, 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 BĂŒchi 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-BĂŒchi 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
Variations on the Stochastic Shortest Path Problem
In this invited contribution, we revisit the stochastic shortest path
problem, and show how recent results allow one to improve over the classical
solutions: we present algorithms to synthesize strategies with multiple
guarantees on the distribution of the length of paths reaching a given target,
rather than simply minimizing its expected value. The concepts and algorithms
that we propose here are applications of more general results that have been
obtained recently for Markov decision processes and that are described in a
series of recent papers.Comment: Invited paper for VMCAI 201
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