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.

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

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.

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.

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.

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.

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

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?