Boolean Theorems

These Boolean theorems (sometimes called switching theorems) are used in problems involving binary states. The two states may be considered as functional propositions, true or false (hence the alternate name propositional calculus ). But in physical devices, such as switches, controls, or computers, the two states may be on or off, short circuit or open circuit, high voltage or low voltage, or presence or absence of a hole in a card or tape, and the digits 1 and 0 are arbitrarily used. 

The more general implications of De Morgan s theorem can be considered after defining two new terms, a dual and a literal. To obtain the dual of a Boolean expression, one must interchange all occurrences of a -t- and a and of a 1 and a 0. A literal is defined as any single variable within the dual expression. For example, in the expression... 

The simple idea behind the post-synthesis verification procedure is a3 follows. A concrete specification and a concrete synthesis result are defined by introducing the enumeration types for OPS,. .., BUSS and by introducing the definitions of the hardware model functions flow,. .., ball. The properties are first order terms with quantifications over finite sets represented by enumeration types and over natural numbers representing control steps. We have proved a theorem that allows to replace each quantification over natural numbers by a quantification over the finite set of operations. So all quantifications reach over finite sets. Each 3-quantification is replaced by a disjunction and each V-quantification is replaced by a conjunction. The remaining terms are boolean terms and applications of hardware model functions to constants of enumeration types or constant natural numbers. Rewriting with the definitions of the model functions reduce the applications to constants. What remains are comparisons of constants which are replaced by truth values. Final rewriting completes the proof. 

The matrix condition IOC, stated by Theorem 2.2 or by Remark 2.6, is called the intrinsic order criterion, because it is independent of the basic probabilities p, and it only depends on the relative positions of the Os and Is in the binary n-tuples u, v. Theorem 2.2 naturally leads to the following partial order relation on the set 0,1 " (3). The so-called intrinsic order will be denoted by , and we shall write u > V (u < v)to indicate that u is intrinsically greater (less) than or equal to v. The partially ordered set (from now on, poset, for short) ( 0,1 ", on n Boolean variables will be denoted by / . 

Theorem 3. To every function computed by a uniform family of boolean circuits there is a uniform family of mbC-circuits that computes such function. 

The previous theorem can be generalized to several other paraconsistent logics, that is, to those where a conservatively translation function from CPL can be defined taking into account that such translation function must be effectively calculated. In the other direction, the existence of uniform families of boolean circuits to every uniform family of L-circuits (for any logic L provided with PRC) is guaranteed by the classical computability of roots for polynomials over finite fields. Then, the L-circuits model does not invalidate Church-Turing s thesis. 

A theorem is recorded ais a (hopefully true) Boolean expression. 

By applying the theorems set out in the previous sub-section, we can formally derive the constructive definition of readtable from the above implicit form. The derivation is presented as a single Boolean expression, composed with the implication operator ==>. Of course, Haskell cannot ensure the truth of the derivation, but does perform valuable syntax and type checking. 

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