Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. If you continue browsing the site, you agree to the use of cookies on this website. See our User Agreement and Privacy Policy.

Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. If you continue browsing the site, you agree to the use of cookies on this website. See our Privacy Policy and User Agreement for details.

Like this presentation? Why not share!

938 views

821 views

821 views

Published on

Discussed the application and language of ω-automaton

No Downloads

Total views

938

On SlideShare

0

From Embeds

0

Number of Embeds

1

Shares

0

Downloads

2

Comments

0

Likes

2

No embeds

No notes for slide

- 1. ω-automaton<br />
- 2. What is ω-automaton<br />Aω-automaton is a deterministic or nondeterministic automaton that runs on infinite, rather than finite, strings as input. <br />Since ω-automata do not stop, they have a variety of acceptance conditions rather than simply a set of accepting states.<br />
- 3. Classes<br />Buchi automata, Rabin automata, Streett automata, parity automataand Muller automata, each deterministic or non-deterministic<br />These classes of ω-automata differ only in terms of acceptance condition.<br />
- 4. Deterministic ω-automata<br />A = (Q,Σ,δ,q0,F)<br />Q is a finite set. The elements of Q are called the states of A.<br />Σ is a finite set called the alphabet of A.<br />δ: Q × Σ -> Q is a function, called the transition function of A. <br />q0is an element of Q, called the initial state.<br />Fis the acceptance condition, formally a subset of Qω.<br />
- 5. DBA(cont.)<br />An input for A is an infinite string over the alphabet Σ, i.e. it is an infinite sequence α = (a1,a2,a3,...).<br />The run of A on such an input is an infinite sequence β = (r0,r1,r2,...) of states, defined as follows:<br />r0 = q0<br /> r1= δ(r0,a1)<br /> r2= δ(r1,a2)<br /> ... rn = δ(rn-1,an)<br />
- 6. DBA(cont.)<br />The main purpose of an ω-automaton is to define a subset of the set of all inputs: The set of accepted inputs. <br />Whereas in the case of an ordinary finite automaton every run ends with a state rn and the input is accepted if and only if rn is an accepting state,<br />the definition of the set of accepted inputs is more complicated for ω-automata. Here we must look at the entire run β.<br />The input is accepted if the corresponding run is in F. The set of accepted input ω-words is called the recognized ω-language by the automaton, which is denoted as L(A).<br />
- 7. Acceptance Condition<br />A Büchi automatonis an ω-automaton A that uses the following acceptance condition, for some subset F of Q:<br />A accepts exactly those runs βfor which Inf(β) ∩ F is not empty, i.e. there is an accepting state that occurs infinitely often in β. <br />Since F is finite, this is equivalent to the condition that βnis accepting for infinitely many natural numbers n.<br />
- 8. DBA(example)<br />
- 9. Non-Deterministic ω-automata<br />A = (Q,Σ,Δ,Q0,F)<br />Q is a finite set. The elements of Q are called the states of A.<br />Σ is a finite set called the alphabet of A.<br />Δ is a subset of Q × Σ × Q and is called the transition relation of A.<br />Q0 is a subset of Q, called the initial set of states.<br />Fis the acceptance condition, a subset of Qω.<br />
- 10. NBA (cont.)<br />Unlike a deterministic ω-automaton which has a transition function δ, the non-deterministic version has a transition relation Δ. Note that Δ can be regarded as a function : Q × Σ -> P(Q) from Q × Σ to the power setP(Q). <br />Thus, given a state qn and a symbol an, the next state qn+1 is not necessarily determined uniquely, rather there is a set of possible next states.<br />A run of A on the input α = (a1,a2,a3,...) is any infinite sequence ρ = (r0,r1,r2,...) of states that satisfies the following conditions:<br /> r0is an element of Q0.<br /> r1is an element of Δ(r0,a1).<br /> r2is an element of Δ(r1,a2).<br /> ... rn is an element of Δ(rn-1,an).<br />
- 11. NBA (cont.)<br />A nondeterministic ω-automaton may admit many different runs on any given input, or none at all. <br />The input is accepted if at least one of the possible runs is accepting. Whether a run is accepting depends only on F, as for deterministic ω-automata.<br />Every deterministic ω-automaton can be regarded as a nondeterministic ω-automaton by taking Δ to be the graph of δ.<br />The definitions of runs and acceptance for deterministic ω-automata are then special cases of the nondeterministic cases.<br />
- 12. NBA(example)<br />
- 13. Power of ω-automata<br />An ω-language over Σ is said to be recognized by an ω-automaton A (with the same alphabet) if it is the set of all ω-words accepted by A. <br />The expressive power of a class of ω-automata is measured by the class of all ω-languages which can be recognized by some automaton in the class.<br />
- 14. Applications<br />ω-automata are useful for specifying behavior of systems that are not expected to terminate, such as hardware, operating systems and control systems.<br />For such systems, you may want to specify a property such as "for every request, an acknowledge eventually follows", or its negation "there is a request which is not followed by an acknowledge".<br />Thus the property of infinite words: one cannot say of a finite sequence that it satisfies this property.<br />
- 15. Any Question<br />

No public clipboards found for this slide

×
### Save the most important slides with Clipping

Clipping is a handy way to collect and organize the most important slides from a presentation. You can keep your great finds in clipboards organized around topics.

Be the first to comment