What Is Mathematical Logic? by C. J. Ash, J. N. Crossley


By C. J. Ash, J. N. Crossley

Publish yr note: initially released in 1972

This advent to the most rules and result of mathematical common sense is a significant remedy aimed at non-logicians. beginning with a old survey of common sense in precedent days, it strains the 17th-century improvement of calculus and discusses glossy theories, together with set concept, the continuum speculation, and different principles.

From 1972 edition.

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The following abbreviations are used in the definitions of both TA and TPNs. Let IR denote the set of real numbers, Q – the set of rational numbers, ZZ – the set of integers, and IN – the set of naturals (including 0). For each S ∈ {IR, Q, ZZ} by S0+ (S+ ) we denote a subset of S consisting of all its nonnegative (respectively positive) elements. Moreover, by IN+ we mean the set of positive natural numbers. When we deal with elements of IR0+ ∪ {∞}, by the notations “≤ b” and “[a, b]” we mean “< b” and “[a, b)” if a ∈ IR0+ and b = ∞.

PnP } is a finite set of places, T = {t1 , . . , tnT } is a finite set of transitions, where P ∩ T = ∅, F : (P × T ) ∪ (T × P ) −→ IN is the flow function, and m0 : P −→ IN is the initial marking of P. Intuitively, Petri nets are directed weighted graphs of two types of nodes: places (representing conditions) and transitions (representing events), whose arcs correspond to these elements in the domain of the flow function, for which the value of this function is positive. The arcs are assigned positive weights according to the values of F .

A concrete state σ N of N is defined as an ordered pair (m, clock N ), where • m is a marking, and • clock N : I −→ IR0+ is a function which for each index i ∈ I gives the time elapsed since the marked place p ∈ Pi of the process Ni of N became marked most recently. For δ ∈ IR0+ , by clock N + δ we denote the function given by (clock N + δ)(i) = clock N (i)+δ for all i ∈ I. Moreover, let (m, clock N )+δ denote (m, clock N +δ). The (dense) concrete state space of N is now a transition system CcN (N ) = (Σ N , (σ N )0 , →N c ), where • Σ N is the set of all the concrete states of N , • (σ N )0 = (m0 , clock0N ) with clock0N (i) = 0 for each i ∈ I is the initial state, and • a timed consecution relation →N c ⊆ Σ N × (T ∪ IR0+ ) × Σ N is defined by action- and time successors as follows: δ – for δ ∈ IR0+ , (m, clock N ) →N c (m, clock N + δ) iff · for each t ∈ en(m) there exists i ∈ I with •t ∩ Pi = ∅ such that (clock N + δ)(i) ≤ Lf t(t) (time successor), t – for t ∈ T , (m, clock N ) →N c (m1 , clock1N ) iff · t ∈ en(m), · for each i ∈ I with •t ∩ Pi = ∅ we have clock N (i) ≥ Ef t(t), · there is i ∈ I with •t ∩ Pi = ∅ such that clock N (i) ≤ Lf t(t), · m1 = m[t , and · for all i ∈ I we have clock1N (i) = 0 if •t ∩ Pi = ∅ and clock1N (i) = clock N (i) otherwise (action successor).

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