Finite Automata: Understanding States and Transitions, Slides of Theory of Automata

Download Finite Automata: Understanding States and Transitions and more Slides Theory of Automata in PDF only on Docsity! 1 Finite Automata Motivation An Example Docsity.com 2 Informal Explanation Finite automata are finite collections of states with transition rules that take you from one state to another. Original application was sequential switching circuits, where the “state” was the settings of internal bits. Today, several kinds of software can be modeled by FA. Docsity.com 5 Automata to Code  In C/C++, make a piece of code for each state. This code: 1. Reads the next input. 2. Decides on the next state. 3. Jumps to the beginning of the code for that state. Docsity.com 6 Example: Automata to Code 2: /* i seen */ c = getNextInput(); if (c == ’n’) goto 3; else if (c == ’i’) goto 2; else goto 1; 3: /* ”in” seen */ . . . Docsity.com 7 Automata to Code – Thoughts How would you do this in Java, which has no goto? You don’t really write code like this. Rather, a code generator takes a “regular expression” describing the pattern(s) you are looking for.  Example: .*ing works in grep. Docsity.com 10 Strange Planet – (2) Observation: the number of individuals never changes. The planet fails if at some point all individuals are of the same species.  Then, no more breeding can take place. State = sequence of three integers – the numbers of individuals of species a, b, and c. Docsity.com 11 Strange Planet – Questions In a given state, must the planet eventually fail? In a given state, is it possible for the planet to fail, if the wrong breeding choices are made? Docsity.com 12 Questions – (2) These questions mirror real ones about protocols.  “Can the planet fail?” is like asking whether a protocol can enter some undesired or error state.  “Must the planet fail” is like asking whether a protocol is guaranteed to terminate. • Here, “failure” is really the good condition of termination. Docsity.com 15 Strange Planet with 3 Individuals 300 003030 111 a c b Notice: four states are “must-fail” states. The others are “can’t-fail” states. 102210 a c 201021 bb 012120 a c State 111 has several transitions. Docsity.com 16 Strange Planet with 4 Individuals Notice: states 400, etc. are must-fail states. All other states are “might-fail” states. 400 022 130103 211 a c b b c a 040 202 013310 121 b a c c a b 004 220 301031 112 c b a a b c Docsity.com 17 Taking Advantage of Symmetry The ability to fail depends only on the set of numbers of the three species, not on which species has which number. Let’s represent states by the list of counts, sorted by largest-first. Only one transition symbol, x. Docsity.com 20 6 Individuals 321 600 411 330 222 Notice: 600 is a must-fail state; 510, 420, and 321 are can’t-fail states; 411, 330, and 222 are “might-fail” states. 420 510 Docsity.com 21 7 Individuals 331 700 430 421 322 Notice: 700 is a must-fail state; All others are might-fail states. 511 520 610 Docsity.com 22 Questions for Thought 1. Without symmetry, how many states are there with n individuals? 2. What if we use symmetry? 3. For n individuals, how do you tell whether a state is “must-fail,” “might- fail,” or “can’t-fail”? Docsity.com