Logic Colloquium ’99: Lecture Notes In Logic, 17 by Jan van Eicjk

By Jan van Eicjk

A compilation of papers offered on the 1999 eu summer season assembly of the organization for Symbolic good judgment, good judgment Colloquium '99 contains surveys and examine articles from many of the world's preeminent logicians. lengthy articles are according to tutorials given on the assembly and current obtainable expositions of present study in lively parts of good judgment, geometric version thought and descriptive set conception of crew activities. the opposite articles disguise present reseach subject matters in all components of mathematical good judgment, together with good judgment in machine technology, facts idea, set idea, version conception, computability idea, and philosophy.

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However, this extension would cause serious problems in our behavioral semantics. : a NAND with inputs tied together can be represented directly as an inverter, and a non· inverting buffer (ide nti ty gate) can be used to carry a feedback signal from an output to an input Circuit Structure 26 The third operation, hide, makes wires "internal.. to the circuit, so they can no longer be connected to other wires. Formally. C' = hide(D)(C) is defined when D � O. The effect is that [' = I and 0' = 0 - D.

5). E'{ II £'{ is equivalent 10 E, and can be converted to the form hide(Dl U Di)(E] II E4) form by applying Cg. This expression can be reduced to nonn al form by repealed applications of C l and C2 (to put the compositions in left associative form). If E = reoame(r)(E, ), there is a normal form Ej, .... s El such that Ej = hide(D)(E2). 5, Ej has an equivalent normal fonn Er = hide(D')(El> such that rCA D) n D' = 0. Define r' so that r' IA-D = r and r' ID is the identity function. 4 there is an normal form E" equivalent to E'.

In other w ords . a wire is a single equipotential region [69]. It is thus imporlantto be able lO model circuits that are not delay-insensitive. 6). MoreovCT. specifications can be checked automatiCally for delay insensitivity using the program of Chapter S. In the sections that follow. the second covers mathematical notation and background. relating to sequences and aulOmata. The third section defines prefix-closed trace struc­ tures and the operations of circuit algebra on them. The fourth section gives examples of trace structures for common circuits.

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