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Stateless determinisitc R-automata with constant window size – with window size 1. • CD-systems of them – LATA 2010: FINITE AUTOMATA with Translucent ...

Globally Deterministic CDSystems of Stateless R(1)Automata Benedek Nagy Department of Computer Science Faculty of Informatics University of Debrecen, Hungary

LATA 2011

Friedrich Otto Fachbereich Elektrotechnik/Informatik Kassel University, Germany

Tarragona, Spain

(Deterministic Finite Automata with translucent letters)

Outline • Stateless determinisitc R-automata with constant window size – with window size 1 • CD-systems of them – LATA 2010: FINITE AUTOMATA with Translucent Letters • DETERMINISTIC VARIANTS: – Strictly det. CD-systems – Globally det. CD-systems

Stateless Restarting (R-)automata • Restarting automata – linguistic motivation Principle of analysis by reduction • ( ,¢,$,k, ), where is a finite alphabet, ¢, $ markers left, right border of the workspace k ≥ 1 is the size of the read/write window, is the transition relation there are 3 types of transitions • Deterministic machine – at most 1 transition

Transitions there are 3 types of transitions: • move-right steps (MVR), which shift the window one step to the right, • combined rewrite/restart steps, which delete one or more symbols from the content u of the window, thereby shortening the tape, and place the window over the left end of the tape, and • accept steps (Accept), which cause the automaton to halt and accept.

Det R-automata with k=1 • The alphabet can be partitioned:

Accepted language • Without rewriting/restarting step (tail computation)

• Allowing rewriting/restarting steps (cycles):

CD systems • Cooperating Distributed system (simplecomplex) • More than 1 device, work one after the other several modes are known • For stateless deterministic R-automata we use =1 mode (the simplest combination) • We need: initial component(s) and define the successor(s) for each component

CD-Systems of Stateless Deterministic R(1)-Automata • with • components Stateless det. R(1)-automata • Sucessor relations • Initial indices are:

and

How it works:

FINITE AUTOMATA with TRANSLUCENT LETTERS • Let the tape be divided to | | slices • At a ‘state’ (i.e. component) the machine can see the first desired letter, if only ‘translucent’ letters are before

• Alternative, equivalent model.

Alternative Introduction • The finite-state acceptor is fundamental. Its deterministic version (DFA) and its nondeterministic version (NFA) both accept exactly the regular languages. • Applications: compiler construction, text editors, hardware design, etc. • Expressiveness is limited. Other models: PDA , LBA, TM. Price of larger expressive power is: algorithmic questions become more complex or even undecidable.

… and motivation • Hence, when dealing with applications, for example in natural language processing or concurrency control, it is of importance to find models of automata that reconcile two contrasting goals: sufficient expressiveness and a moderate degree of complexity.

Finite state automata • • • • • •

A(Q,T,I,F, ) States Input alphabet (set of) initial state (set of) final states Transition function NFA: Q x T  2Q DFA: Q x T  Q • (then | I | = 1 also)

FA with translucent letters • For each state q, the letters from the set (q) are translucent for q, that is, in state q the automaton A does not see these letters. • A is called deterministic, abbreviated as DFAwtl, if | I | = 1 and if | (q,a) | = 1 for all states q and all letters a

Deterministic vs. non-det. • DFA, NFA : Regular • NFA with translucent letters vs. Deterministic versions

Strictly deterministic CD-systems • By definition : DET. CD-Systems of Stateless Deterministic R(1)-Automata

• Completely deterministic and very restricted (no guessing at all, the next component is independent of the read letter)

Strictly deterministic CD-systems • The finite language L= {aaa, bb} is not accepted by them. • EXAMPLE

Strictly deterministic CD-systems • The accepted language class is incomarable to the class of – finite – regular – context-free

languages.

Globally Deterministic CD-R(1)-Systems It is global successor function:

(stl-det-global-CD-R(1)) • The next component depends on the read letter (as at DFA)… • By simulation of a DFA, (having empty set as translucent letters in each state) every Regular language is in our family. • This is the class of DFAwtl

Normal from (NF) machine • For every components (1.) it is reachable from the initial component can accept only on $. (2.) (3.)

One can construct equivalent NF machine..

Hierarchy It is a rational trace language, but not DFAwtl (there is no global-det CD-R(1) accept it) are both incomarable with the set of RAT (rational trace languages) strictDET

globalDET DFAwtl

localDET NFAwtl

Closure properties of globalDET • The class L(DFAwtl) is not closed under union, intersection with Reg, commutative closure, product. • It is closed under complementation.

Decision problems • The membership problem, the emptiness problem, the universe problem, and the finiteness problem are effectively decidable for stl-det-global-CD-R(1)-systems. • Undecidable:

Conclusions • Deterministic variants of NFAwtl, i.e., CD-R(1) systems are investigated • strict DET : very limited (not all finite), but not CF language can be accepted… • global DET : all Reg (extension of DFA) • closure properties and decision problems were presented

Further literature on the topic • CD-Systems of Stateless Deterministic R(1)-Automata Accept all Rational Trace Languages, LATA 2010, LNCS 6031 (2010), 463-474. • On CD-systems of stateless deterministic R-automata with window size one, Kasseler Informatikschriften 2010, 2, Kassel University, Germany; Journal of Computer and System Sciences - JCSS, accepted • Finite-State Acceptors with Translucent Letters, ICAART 2011 - 3rd Int. Conf. on Agents and Artificial Intelligence, BILC 2011 - 1st Int. Workshop on AI Methods for Interdisciplinary Research in Language and Biology, 3-13. • On Globally Deterministic CD-Systems of Stateless R-Automata with Window Size One, Kasseler Informatikschriften 2011, 1, Kassel University • An automata-theoretical characterization of context-free trace languages, SOFSEM 2011, LNCS 6543 (2011), 406–417. • CD-Systems of Stateless Deterministic R(1)-Automata Governed by an External Pushdown Store, Kasseler Informatikschriften 2010, 4, Kassel University, Germany • Deterministic pushdown-CD-systems of stateless deterministic R(1)automata, AFL 2011, accepted

Support The presented results are results of a bilateral cooperation between the two author’s research groups. Supported by the Balassi Intézet Magyar Ösztöndíj Bizottsága (MÖB) and the Deutsche Akademischer Austauschdienst (DAAD). The work is also supported by the TÁMOP 4.2.1./B-09/1/KONV-2010-0007 project. The project is implemented through the New Hungary Development Plan, co-financed by the European Social Fund and the European Regional Development Fund.

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