Reversal of a Language By: Travis Pullen Paper • • Reversal of regular languages and state complexity By: Juraj Sebej History • • Goes back to the early 40’s and 50’s • FA’s • NFA’s • DFA’s Research has intensified in recent years State Complexity • The state complexity of a regular language is the number of states in its minimal dfa. A regular language with deterministic state complexity n is called an n-state dfa language. DFA • • Deterministic Finite Automata Finite state machine • accepts/rejects • produces a unique computation • must have an input symbol to transition from state to state NFA • • • Non-Deterministic Finite Automata Were introduced by Micheal O. Rabin and Dana Scott in 1958 An NFA can be converted into an equivalent DFA • Does not have to follow the same rules as a DFA Brief Review • Converting an NFA to a DFA 0 ^ -0 1 0 1 0,1 2 1 State 0 1 Z1: 0 0,1 : Z2 empty Z2: 0,1 0,1,2 : Z3 2 : Z4 Z3 : 0,1,2 0,1,2 : Z3 0,2 : Z5 Z4 : 2 empty 0,2 : Z5 Z5 : 0,2 1,0 : Z2 0,2 : Z5 Converted Empty 0 1 Z1 0 Z2 1 Z4 0 0 Z5 1 Z6 1 Normal FA (DFA) • • We normally read machines from left to right For a reversal we are going to be reading the machines from right to left Example • DFA 1 — 0 0 0 0,1 1 + 1 Closed? • • Now we know that languages are closed under union, intersection, and complement Is it closed under the reversal? Closed Reversal • If a language can be recognized by a DFA that reads strings from right to left, then there is a “normal” DFA that accepts the same language. Reversal of a Language • • The reversal wR of a string w is defined as follows: εR = ε and if w = a1a1 ···an with ai∈Σ, then wR=anan−1···a2a1. The reversal of a language L is the language LR={wR|w∈L}. The reversal of a dfa A = (Q,Σ,δ,s,F) is the nfa AR obtained from A by reversing all transitions and by swapping the role of starting and accept(-ing) states, that is AR = Q, Σ, δR, F, {s}, where δR(q,a)={p∈Q: δ(p,a)=q}. How to get the Reversal • • Assume that L is a regular language • Let M be the DFA that recognizes the language L So we need a DFA, MR that will accepts LR Example 1 — 0 0 0 0,1 1 + 1 Example 0 1 + 0 1 0 0,1 1 — Converted Reversal • Called the subset automaton State 0 1 Z1 1 : Z1 1,2 : Z2 Z2 1,2,3 : Z3 1,2 : Z2 Z3 1,2,3,4 : Z4 1,2 : Z2 Z4 1,2,3,4 : Z4 1,2,3,4 : Z4 0 Z1 1 1 0,1 Z2 0 Z4 1 Z3 0 Sources • • • • http://upload.wikimedia.org/wikipedia/commons/thumb/0 /0e/NFAexample.svg/250px-NFAexample.svg.png http://upload.wikimedia.org/wikipedia/commons/thumb/9 /94/DFA_example_multiplies_of_3.svg/250pxDFA_example_multiplies_of_3.svg.png Cohen, Daniel I. A. Introduction to Computer Theory. 2nd ed. New York: Wiley, 1986. Print. Sebej, Juraj. "Reversal of Regular Languages and State Complexity." Http://ceur-ws.org/Vol-683/paper8.pdf. Theoretical Computer Science, 31 Aug. 2012. Web. 6 May 2015. Questions?