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ece190_rec2

# ece190_rec2 - Propositional Logic Representations of...

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Propositional Logic: Representations of boolean functions and complete sets of operators Periklis A. Papakonstantinou ? University of Toronto We begin by introducing the concept of a boolean function. This concept has a very intuitive correspondence to the semantics of a propositional formula. Then, we discuss in what sense we represent boolean functions through propositional formulas. We conclude by getting into some of the details of the issue of complete sets of operators. Boolean functions We will say that a function is boolean if it maps true/false tuples to a true/false value. For several reasons it helps to think of “true” as “1” and of “false” as “0”. For example, digital circuits perform logical operations on 0 and 1 in exactly the same way we perform logical operations in propositional Logic. For instance, 1 1 = 1, 0 1 = 0. From now on we blur the distinction between 0 and “false” and 1 and “true”. In general, for a function f we write f : D R , to denote that D is the domain of f and R is the range of f . Clearly, f : d R just gives some but not all of the description of the function. For example, we write that f : R R for the function f which maps real numbers to real numbers as follows: f ( x ) = x 2 + 1. Functions that map R R have infinitely many points (the whole set R ) we must specify in order to describe them. In this handout we will restrict our attention to some special functions whose full description is finite (i.e. we can write it down on a piece of paper - as opposed to functions with infinite domains). Functions in our current interest have domains of constant size. For example, a function like f : { 0 , 1 } 2 → { 0 , 1 } has domain { 0 , 1 } 2 = { (0 , 0) , (0 , 1) , (1 , 0) , (1 , 1) } . Thus it suffices to specify the function on these four values to have its complete description. Here is the description of such a function f : x y f ( x, y ) 0 0 0 0 1 1 1 0 1 1 1 0 Remark 1. { 0 , 1 } 2 is the cartesian product { 0 , 1 } × { 0 , 1 } . For example, (0 , 1) ∈ { 0 , 1 } 2 , (1 , 1) ∈ { 0 , 1 } 2 etc. Similarly, for a constant k N we have that { 0 , 1 } k = { 0 , 1 } × { 0 , 1 } × . . . { 0 , 1 } | {z } k times . ? Please submit any typos, remarks, suggestions to [email protected] .

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2 Periklis Papakonstantinou (ECE 190, Fall 2006) Remark 2. The above function f corresponds to the addition in the binary system (note that 1 + 1 = 10 but the value of the function ignores the carry). Also, this function is sometimes called as XOR or as parity .
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