Download presentation
Presentation is loading. Please wait.
1
Infinite Automata -automata is an automaton that accepts infinite strings A Buchi automaton is similar to a finite automaton: S is a finite set of states, the input alphabet, is the transition relation, r the initial state, and specifies the acceptance condition. An infinite string is accepted if there is a “run” of states on w, s(w), such that s(w) intersects Q infinitely often.
2
Propositional Linear Temporal Logic (LTL) 1. If p is a propositional formula, then p is an LTL formula 2. If p and q are LTL formulae, the so are (until), (next N), (eventually – finally F) Theorem: LTL Buchi Theorem: Deterministic-Buchi ND-Buchi (always – globally G) Example: request-acknowledge pattern: G (req => F ack) Safety: nothing bad ever happens Liveness: Something good eventually happens
3
Other infinite automata There are many other types of -automata (Det. = ND) Muller (infinitely occuring states is contained in Q) Rabin ({(R 1,G 1 ),…,(R n,G n )}) Street ({(R 1,G 1 ),…,(R n,G n )}) They differ from ND-Buchi only in how their acceptance condition is states. Some are related by having complementary acceptance conditions. Buchi Muller Rabin Street These automata differ in the compactness of their representation of any particular language. The set of all languages of ND-Buchi are called the -regular languages.
4
Multi-Valued Relations (single output): where are finite sets of values. The variables are multi-valued variables which can take on any value in Typically, we take (but could be symbolic) If for all minterms then R is deterministic. It is well-defined if for all minterms. If R is a binary-output MV-relation. Can represent R as a set of binary-output relations: where.
5
Minimizing binary-output MV function We seek a SOP MV expression with the minimum number of product terms. An MV-SOP is of the form or in general a sum (OR) of products of literals. A literal is where If then ( x 2 {0,1,2} =1 if P 2 ={0,1,2} ) and can be dropped from the expression. In general, a binary-output relation is simply an incompletely specified binary function on n multi-valued variables. A minimum SOP can be found by using ESPRESSO-MV.
6
Minimizing a Multi-Valued Output MV relation These can be represented with m binary output functions. We seek a SOP expression for each output where the total number of product terms is minimum, i.e. where is minimum. (P 0 ={0,1,…,m-1}) We will see how this can be done using a variation of Quine-McCluskey.
7
Multi-output MV relations A relation is called a multi-output MV relation if R is binary and the are treated as outputs. This relation is between a vector of inputs and a vector of outputs: It is well-defined if for all, there exists at least one such that It is output-symmetric if and only if such that This is equivalent to the following. Let Then
8
Questions and Problems: How do we minimize such a relation? What does minimum representation mean? Find the largest output-symmetric relation contained in a given one! Find the smallest output-symmetric relation containing a given one! Are the above two problems well posed?
Similar presentations
© 2025 SlidePlayer.com. Inc.
All rights reserved.