206,475 research outputs found
Counter Machines and Distributed Automata: A Story about Exchanging Space and Time
We prove the equivalence of two classes of counter machines and one class of
distributed automata. Our counter machines operate on finite words, which they
read from left to right while incrementing or decrementing a fixed number of
counters. The two classes differ in the extra features they offer: one allows
to copy counter values, whereas the other allows to compute copyless sums of
counters. Our distributed automata, on the other hand, operate on directed path
graphs that represent words. All nodes of a path synchronously execute the same
finite-state machine, whose state diagram must be acyclic except for
self-loops, and each node receives as input the state of its direct
predecessor. These devices form a subclass of linear-time one-way cellular
automata.Comment: 15 pages (+ 13 pages of appendices), 5 figures; To appear in the
proceedings of AUTOMATA 2018
Path Checking for MTL and TPTL over Data Words
Metric temporal logic (MTL) and timed propositional temporal logic (TPTL) are
quantitative extensions of linear temporal logic, which are prominent and
widely used in the verification of real-timed systems. It was recently shown
that the path checking problem for MTL, when evaluated over finite timed words,
is in the parallel complexity class NC. In this paper, we derive precise
complexity results for the path-checking problem for MTL and TPTL when
evaluated over infinite data words over the non-negative integers. Such words
may be seen as the behaviours of one-counter machines. For this setting, we
give a complete analysis of the complexity of the path-checking problem
depending on the number of register variables and the encoding of constraint
numbers (unary or binary). As the two main results, we prove that the
path-checking problem for MTL is P-complete, whereas the path-checking problem
for TPTL is PSPACE-complete. The results yield the precise complexity of model
checking deterministic one-counter machines against formulae of MTL and TPTL
ΠΠΎΡΡΡΠΎΠ΅Π½ΠΈΠ΅ ΠΏΡΠΈΠ±Π»ΠΈΠΆΠ΅Π½ΠΈΠΉ Π±ΠΈΡΠΈΠΌΡΠ»ΡΡΠΈΠΈ Π² ΠΎΠ΄Π½ΠΎΡΡΠ΅ΡΡΠΈΠΊΠΎΠ²ΡΡ ΡΠ΅ΡΡΡ
One-counter nets are finite-state machines operating on a variable (counter) which ranges over the natural numbers. Every transition can increase or decrease the value of the counter (the decrease is possible only if the result is non-negative, hence zero-testing is not allowed). The class of one-counter nets is equivalent to the class of Petri nets with one unbounded place, and to the class of pushdown automata where the stack alphabet contains one symbol. We present a specific method of approximation of the largest bisimulation of a one-counter net, based on the single-periodic arithmetics and a notion of stratified bisimulation.ΠΠ΄Π½ΠΎΡΡΠ΅ΡΡΠΈΠΊΠΎΠ²ΡΠ΅ ΡΠ΅ΡΠΈ ΠΏΡΠ΅Π΄ΡΡΠ°Π²Π»ΡΡΡ ΡΠΎΠ±ΠΎΠΉ ΠΊΠΎΠ½Π΅ΡΠ½ΡΠ΅ Π°Π²ΡΠΎΠΌΠ°ΡΡ Ρ Π΄ΠΎΠΏΠΎΠ»Π½ΠΈΡΠ΅Π»ΡΠ½ΡΠΌ ΡΠ΅Π»ΠΎΡΠΈΡΠ»Π΅Π½Π½ΡΠΌ Π½Π΅ΠΎΡΡΠΈΡΠ°ΡΠ΅Π»ΡΠ½ΡΠΌ ΡΡΠ΅ΡΡΠΈΠΊΠΎΠΌ. ΠΠ΅ΡΠ΅Ρ
ΠΎΠ΄ ΡΠΏΡΠ°Π²Π»ΡΡΡΠ΅Π³ΠΎ Π°Π²ΡΠΎΠΌΠ°ΡΠ° ΡΠ²Π΅Π»ΠΈΡΠΈΠ²Π°Π΅Ρ ΠΈΠ»ΠΈ ΡΠΌΠ΅Π½ΡΡΠ°Π΅Ρ Π·Π½Π°ΡΠ΅Π½ΠΈΠ΅ ΡΡΠ΅ΡΡΠΈΠΊΠ°, ΠΏΡΠΈ ΡΡΠΎΠΌ ΡΠΌΠ΅Π½ΡΡΠ΅Π½ΠΈΠ΅ Π²ΠΎΠ·ΠΌΠΎΠΆΠ½ΠΎ ΡΠΎΠ»ΡΠΊΠΎ Π² ΡΠΎΠΌ ΡΠ»ΡΡΠ°Π΅, ΠΊΠΎΠ³Π΄Π° ΡΠ΅Π·ΡΠ»ΡΡΠ°Ρ Π±ΡΠ΄Π΅Ρ Π½Π΅ΠΎΡΡΠΈΡΠ°ΡΠ΅Π»ΡΠ½ΡΠΌ; ΠΏΡΠΎΠ²Π΅ΡΠΊΠ° Π½Π° Π½ΠΎΠ»Ρ ΠΎΡΡΡΡΡΡΠ²ΡΠ΅Ρ. ΠΠ΄Π½ΠΎΡΡΠ΅ΡΡΠΈΠΊΠΎΠ²ΡΠ΅ ΡΠ΅ΡΠΈ ΡΠΊΠ²ΠΈΠ²Π°Π»Π΅Π½ΡΠ½Ρ ΠΏΠΎ Π²ΡΡΠ°Π·ΠΈΡΠ΅Π»ΡΠ½ΠΎΠΉ ΠΌΠΎΡΠ½ΠΎΡΡΠΈ ΡΠ΅ΡΡΠΌ ΠΠ΅ΡΡΠΈ Ρ Π½Π΅ Π±ΠΎΠ»Π΅Π΅ ΡΠ΅ΠΌ ΠΎΠ΄Π½ΠΎΠΉ Π½Π΅ΠΎΠ³ΡΠ°Π½ΠΈΡΠ΅Π½Π½ΠΎΠΉ ΠΏΠΎΠ·ΠΈΡΠΈΠ΅ΠΉ, Π° ΡΠ°ΠΊΠΆΠ΅ ΠΌΠ°Π³Π°Π·ΠΈΠ½Π½ΡΠΌ Π°Π²ΡΠΎΠΌΠ°ΡΠ°ΠΌ Ρ ΠΎΠ΄Π½ΠΎΡΠΈΠΌΠ²ΠΎΠ»ΡΠ½ΡΠΌ ΡΡΠ΅ΠΊΠΎΠ²ΡΠΌ Π°Π»ΡΠ°Π²ΠΈΡΠΎΠΌ.
Π ΡΠ°Π±ΠΎΡΠ΅ ΠΏΡΠ΅Π΄ΡΡΠ°Π²Π»Π΅Π½ ΠΌΠ΅ΡΠΎΠ΄ ΠΏΡΠΈΠ±Π»ΠΈΠΆΠ΅Π½ΠΈΡ Π½Π°ΠΈΠ±ΠΎΠ»ΡΡΠ΅ΠΉ Π±ΠΈΡΠΈΠΌΡΠ»ΡΡΠΈΠΈ Π² ΠΎΠ΄Π½ΠΎΡΡΠ΅ΡΡΠΈΠΊΠΎΠ²ΠΎΠΉ ΡΠ΅ΡΠΈ, ΠΎΡΠ½ΠΎΠ²Π°Π½Π½ΡΠΉ Π½Π° ΠΈΡΠΏΠΎΠ»ΡΠ·ΠΎΠ²Π°Π½ΠΈΠΈ ΠΎΠ΄Π½ΠΎΠΏΠ΅ΡΠΈΠΎΠ΄ΠΈΡΠ΅ΡΠΊΠΎΠΉ ΡΠΈΠΌΠ²ΠΎΠ»ΡΠ½ΠΎΠΉ Π°ΡΠΈΡΠΌΠ΅ΡΠΈΠΊΠΈ ΠΈ ΠΏΠΎΠ½ΡΡΠΈΡ ΡΠ°ΡΡΠ»ΠΎΠ΅Π½Π½ΠΎΠΉ Π±ΠΈΡΠΈΠΌΡΠ»ΡΡΠΈΠΈ
Remarks on Parikh-recognizable omega-languages
Several variants of Parikh automata on infinite words were recently
introduced by Guha et al. [FSTTCS, 2022]. We show that one of these variants
coincides with blind counter machine as introduced by Fernau and Stiebe
[Fundamenta Informaticae, 2008]. Fernau and Stiebe showed that every
-language recognized by a blind counter machine is of the form
for Parikh recognizable languages , but
blind counter machines fall short of characterizing this class of
-languages. They posed as an open problem to find a suitable
automata-based characterization. We introduce several additional variants of
Parikh automata on infinite words that yield automata characterizations of
classes of -language of the form for all
combinations of languages being regular or Parikh-recognizable. When
both and are regular, this coincides with B\"uchi's classical
theorem. We study the effect of -transitions in all variants of
Parikh automata and show that almost all of them admit
-elimination. Finally we study the classical decision problems
with applications to model checking.Comment: arXiv admin note: text overlap with arXiv:2302.04087,
arXiv:2301.0896
Highly Undecidable Problems For Infinite Computations
We show that many classical decision problems about 1-counter
omega-languages, context free omega-languages, or infinitary rational
relations, are -complete, hence located at the second level of the
analytical hierarchy, and "highly undecidable". In particular, the universality
problem, the inclusion problem, the equivalence problem, the determinizability
problem, the complementability problem, and the unambiguity problem are all
-complete for context-free omega-languages or for infinitary rational
relations. Topological and arithmetical properties of 1-counter
omega-languages, context free omega-languages, or infinitary rational
relations, are also highly undecidable. These very surprising results provide
the first examples of highly undecidable problems about the behaviour of very
simple finite machines like 1-counter automata or 2-tape automata.Comment: to appear in RAIRO-Theoretical Informatics and Application
Generating Permutations with Restricted Containers
We investigate a generalization of stacks that we call
-machines. We show how this viewpoint rapidly leads to functional
equations for the classes of permutations that -machines generate,
and how these systems of functional equations can frequently be solved by
either the kernel method or, much more easily, by guessing and checking.
General results about the rationality, algebraicity, and the existence of
Wilfian formulas for some classes generated by -machines are
given. We also draw attention to some relatively small permutation classes
which, although we can generate thousands of terms of their enumerations, seem
to not have D-finite generating functions
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