A Completeness Theorem in Modal Logic (paper) by Kripke Saul

By Kripke Saul

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1): • the constraints of the form xi ∼ c with xi ∈ X , ∼ ∈ {<, ≤, ≥, >} and c ∈ IN are expressed by xi − x0 ∼ c, • these of the form xi = c or xi − xj = c – by the conjunctions xi − x0 ≤ c ∧ x0 − xi ≤ −c and xi − xj ≤ c ∧ xj − xi ≤ −c, respectively, • true is expressed by x0 − x0 < ∞, and • all the constraints of the form xi − xj > c with xi , xj ∈ X + and c ∈ IN are replaced by xj − xi < −c, whereas these of the form xi − xj ≥ c – by xj − xi ≤ −c. The constraints of the form xi − xj < −∞ are introduced to express constraints which are never satisﬁed.

For simplicity of the description below, let ((m(p1 ), . . , m(p8 )), (clock T (t1 ), . . , clockT (t6 ))) denote the concrete state (m, clock T ) ∈ Σ T . 5, 0)) →T c . . A state σ R ∈ Σ R is reachable if there exists a (σ R )0 -run ρR and i ∈ IN such that σ R = σiR + δ for some 0 ≤ δ ≤ δi , where σiR + δi is an element of ρR . The set of all the reachable states of N is denoted by ReachR N . A marking m . The set of all the reachable is reachable if there is a state (m, ·) ∈ ReachR N markings of N is denoted by RMN (it is easy to see that this set does not depend on the way the concrete states of the net are deﬁned).

By a partition of a set B we mean a family of its disjoint subsets B such that B = B. B ∈B 32 • • • • 2 Timed Automata X Z := {v ∈ IRn0+ | (∃v ∈ Z) v ≤ v}, Z ⇑ Z = {v ∈ Z | (∃v ∈ Z ) v ≤ v ∧ (∀v ≤ v ≤ v ) v ∈ Z ∪ Z }, Z[X := 0] = {v[X := 0] | v ∈ Z}, X | v[X := 0] ∈ Z}. , Z[X := 0] and [X := 0]Z) and the standard intersection preserve zones [8, 159]. A description of the implementation of Z\Z , following [8], is given also in Sect. 3. Some examples of the operations are presented in Fig. 2. x2 x2 5 x2 Z \ Z = {Z1 , Z2 } 7 6 Z \ Z = {Z1 , Z2 , Z3 } 6 6 Z3 Z Z Z1 3 3 1 1 3 4 6 x1 x2 Z1 3 Z2 1 3 4 5 6 x1 x2 6 Z2 3 4 5 6 x1 6 x1 x2 6 6 Z Z ∩Z 3 Z 3 1 x1 3 4 x1 3 2 x2 x2 x2 6 6 6 Z⇑Z 3 3 Z ⇑Z Z[x1 := 0] 1 3 4 x1 x2 4 x1 x1 x2 x2 7 Z [x1 := 0] 5 [x1 := 0]Z = ∅ [x1 := 0]Z 3 3 x1 x1 Fig.