HYPER-MINIMIZATION FOR DETERMINISTIC TREE AUTOMATA

2013 ◽  
Vol 24 (06) ◽  
pp. 815-830 ◽  
Author(s):  
ARTUR JEŻ ◽  
ANDREAS MALETTI
Keyword(s):  
Finite Difference ◽  
Minimization Problem ◽  
Fast Algorithms ◽  
Input Tree ◽  
Tree Automaton ◽  
Tree Automata ◽  
Compression Technique ◽  
Bottom Up ◽  
Finite State ◽  
Transition Table

Hyper-minimization is a recent automaton compression technique that can reduce the size of an automaton beyond the limits imposed by classical minimization. The additional compression power is enabled by allowing a finite difference in the represented language. The necessary theory for hyper-minimization is developed for (bottom-up) deterministic tree automata. The hyper-minimization problem for deterministic tree automata is reduced to the hyper-minimization problem for deterministic finite-state string automata, for which fast algorithms exist. The fastest algorithm obtained in this way runs in time [Formula: see text], where m is the size of the transition table and n is the number of states of the input tree automaton.

A minimization algorithm for fuzzy top-down tree automata over lattices

2020 ◽  
pp. 1-10
Author(s):  
M. Ghorani ◽  
S. Garhwal
Keyword(s):  
Minimization Problem ◽  
Time Complexity ◽  
Tree Automaton ◽  
Tree Automata ◽  
Minimization Algorithm ◽  
Top Down ◽  
Minimal Form ◽  
Minimization Algorithms ◽  
The Given

In this paper, we study fuzzy top-down tree automata over lattices ( LTA s , for short). The purpose of this contribution is to investigate the minimization problem for LTA s . We first define the concept of statewise equivalence between two LTA s . Thereafter, we show the existence of the statewise minimal form for an LTA . To this end, we find a statewise irreducible LTA which is equivalent to a given LTA . Then, we provide an algorithm to find the statewise minimal LTA and by a theorem, we show that the output statewise minimal LTA is statewise equivalent to the given input. Moreover, we compute the time complexity of the given algorithm. The proposed algorithm can be applied to any given LTA and, unlike some minimization algorithms given in the literature, the input doesn’t need to be a complete, deterministic, or reduced lattice-valued tree automaton. Finally, we provide some examples to show the efficiency of the presented algorithm.

The Bottom-Up Position Tree Automaton and the Father Automaton

2020 ◽  
pp. 1-18
Author(s):  
Samira Attou ◽  
Ludovic Mignot ◽  
Djelloul Ziadi
Keyword(s):  
Tree Automaton ◽  
Tree Automata ◽  
Top Down ◽  
Bottom Up ◽  
Regular Tree ◽  
The One ◽  
Tree Expression

The conversion of a given regular tree expression into a tree automaton has been widely studied. However, classical interpretations are based upon a top-down interpretation of tree automata. In this paper, we propose new constructions based on Gluskov’s one and on the one by Ilie and Yu using a bottom-up interpretation. One of the main goals of this technique is to consider as a next step the links with deterministic recognizers, something which cannot be done with classical top-down approaches.

DECOMPOSITION OF WEIGHTED MULTIOPERATOR TREE AUTOMATA

2009 ◽  
Vol 20 (02) ◽  
pp. 221-245 ◽  
Author(s):  
TORSTEN STÜBER ◽  
HEIKO VOGLER ◽  
ZOLTÁN FÜLÖP
Keyword(s):  
Tree Automata ◽  
Commutative Monoid ◽  
Weighted Tree ◽  
Tree Language ◽  
Bottom Up ◽  
Tree Series ◽  
Finite State ◽  
Tree Languages ◽  
Tree Transformation

Weighted multioperator tree automata (for short: wmta) are finite-state bottom-up tree automata in which the transitions are weighted with an operation taken from some multioperator monoid. A wmta recognizes a tree series which is a mapping from the set of trees to some commutative monoid. We prove that every wmta recognizable tree series can be decomposed into a relabeling tree transformation, a recognizable tree language, and a tree series computed by a homomorphism wmta; vice versa, the composition of an arbitrary relabeling tree transformation, a recognizable tree language, and a tree series computed by a homomorphism wmta yields a wmta recognizable tree series. We use this characterization result for specific multioperator monoids and prove (1) a new decomposition of polynomial bottom-up tree series transducers over semirings and (2) a new characterization of tree series which are recognizable by weighted tree automata over semirings, in terms of projections of local tree languages.

scholarly journals Crisp-determinization of weighted tree automata over strong bimonoids

2021 ◽  
Vol vol. 23 no. 1 (Automata, Logic and Semantics) ◽  
Author(s):  
Zoltán Fülöp ◽  
Dávid Kószó ◽  
Heiko Vogler
Keyword(s):  
Finite Order ◽  
Positive Answer ◽  
Order Property ◽  
Sufficient Condition ◽  
Tree Automata ◽  
Weighted Tree ◽  
Bottom Up ◽  
Initial Algebra ◽  
Initial Algebra Semantics ◽  
Weighted Tree Automata

We consider weighted tree automata (wta) over strong bimonoids and their initial algebra semantics and their run semantics. There are wta for which these semantics are different; however, for bottom-up deterministic wta and for wta over semirings, the difference vanishes. A wta is crisp-deterministic if it is bottom-up deterministic and each transition is weighted by one of the unit elements of the strong bimonoid. We prove that the class of weighted tree languages recognized by crisp-deterministic wta is the same as the class of recognizable step mappings. Moreover, we investigate the following two crisp-determinization problems: for a given wta ${\cal A}$, (a) does there exist a crisp-deterministic wta which computes the initial algebra semantics of ${\cal A}$ and (b) does there exist a crisp-deterministic wta which computes the run semantics of ${\cal A}$? We show that the finiteness of the Nerode algebra ${\cal N}({\cal A})$ of ${\cal A}$ implies a positive answer for (a), and that the finite order property of ${\cal A}$ implies a positive answer for (b). We show a sufficient condition which guarantees the finiteness of ${\cal N}({\cal A})$ and a sufficient condition which guarantees the finite order property of ${\cal A}$. Also, we provide an algorithm for the construction of the crisp-deterministic wta according to (a) if ${\cal N}({\cal A})$ is finite, and similarly for (b) if ${\cal A}$ has finite order property. We prove that it is undecidable whether an arbitrary wta ${\cal A}$ is crisp-determinizable. We also prove that both, the finiteness of ${\cal N}({\cal A})$ and the finite order property of ${\cal A}$ are undecidable.

scholarly journals Encoding of Terms in EMB-Based Mealy FSMs

2020 ◽  
Vol 10 (8) ◽  
pp. 2762 ◽  
Author(s):  
Alexander Barkalov ◽  
Larysa Titarenko ◽  
Małgorzata Mazurkiewicz ◽  
Kazimierz Krzywicki
Keyword(s):  
State Transition ◽  
Logic Circuits ◽  
Sequential Algorithm ◽  
State Variables ◽  
Embedded Memory ◽  
Chip Area ◽  
Memory Block ◽  
Finite State ◽  
Transition Table ◽  
Number Of Classes

A method is proposed targeting implementation of FPGA-based Mealy finite state machines. The main goal of the method is a reduction for the number of look-up table (LUT) elements and their levels in FSM logic circuits. To do it, it is necessary to eliminate the direct dependence of input memory functions and FSM output functions on FSM inputs and state variables. The method is based on encoding of the terms corresponding to rows of direct structure tables. In such an approach, only terms depend on FSM inputs and state variables. Other functions depend on variables representing terms. The method belongs to the group of the methods of structural decomposition. The set of terms is divided by classes such that each class corresponds to a single-level LUT-based circuit. An embedded memory block (EMB) generates codes of both classes and terms as elements of these classes. The mutual using LUTs and EMB allows diminishing chip area occupied by FSM circuit (as compared to its LUT-based counterpart). The simple sequential algorithm is proposed for finding the partition of the set of terms by a determined number of classes. The method is based on representation of an FSM by a state transition table. However, it can be used for any known form of FSM specification. The example of synthesis is shown. The efficiency of the proposed method was investigated using a library of standard benchmarks. We compared the proposed with some other known design methods. The investigations show that the proposed method gives better results than other discussed methods. It allows the obtaining of FSM circuits with three levels of logic and regular interconnections.

BISIMULATION MINIMIZATION OF TREE AUTOMATA

2007 ◽  
Vol 18 (04) ◽  
pp. 699-713 ◽  
Author(s):  
PAROSH AZIZ ABDULLA ◽  
JOHANNA HÖGBERG ◽  
LISA KAATI
Keyword(s):  
Tree Automata ◽  
Total Size ◽  
Maximum Rank ◽  
Input Alphabet ◽  
Transition Table

We extend an algorithm by Paige and Tarjan that solves the coarsest stable refinement problem to the domain of trees. The algorithm is used to minimize nondeterministic tree automata (NTA) with respect to bisimulation. We show that our algorithm has an overall complexity of [Formula: see text], where [Formula: see text] is the maximum rank of any symbol in the input alphabet, m is the total size of the transition table, and n is the number of states.

scholarly journals A Congruence-Based Perspective on Finite Tree Automata

2022 ◽  
Vol 184 (1) ◽  
pp. 1-47
Author(s):  
Pierre Ganty ◽  
Elena Gutiérrez ◽  
Pedro Valero
Keyword(s):  
Expressive Power ◽  
Tree Automata ◽  
Minimization Algorithm ◽  
Top Down ◽  
Finite Tree ◽  
Bottom Up ◽  
Original Result

We provide new insights on the determinization and minimization of tree automata using congruences on trees. From this perspective, we study a Brzozowski’s style minimization algorithm for tree automata. First, we prove correct this method relying on the following fact: when the automata-based and the language-based congruences coincide, determinizing the automaton yields the minimal one. Such automata-based congruences, in the case of word automata, are defined using pre and post operators. Now we extend these operators to tree automata, a task that is particularly challenging due to the reduced expressive power of deterministic top-down (or equivalently co-deterministic bottom-up) automata. We leverage further our framework to offer an extension of the original result by Brzozowski for word automata.

scholarly journals Bottom-Up derivatives of tree expressions

2021 ◽  
Vol 55 ◽  
pp. 4
Author(s):  
Samira Attou ◽  
Ludovic Mignot ◽  
Djelloul Ziadi
Keyword(s):  
Fixed Point ◽  
Tree Automaton ◽  
Bottom Up ◽  
Regular Tree ◽  
Partial Derivatives ◽  
New Family ◽  
Derivatives Of

In this paper, we extend the notion of (word) derivatives and partial derivatives due to (respectively) Brzozowski and Antimirov to tree derivatives using already known inductive formulae of quotients. We define a new family of extended regular tree expressions (using negation or intersection operators), and we show how to compute a Brzozowski-like inductive tree automaton; the fixed point of this construction, when it exists, is the derivative tree automaton. Such a deterministic tree automaton can be used to solve the membership test efficiently: the whole structure is not necessarily computed, and the derivative computations can be performed in parallel. We also show how to solve the membership test using our (Bottom-Up) partial derivatives, without computing an automaton.

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