Dynamic Logic for While Programs: Cheat Sheet A.Platzer 1
1: Operators of Dynamic Logic for While Programs
DL Operator Meaning
e= ˜eequals true iff values of terms eand ˜eare equal
e≥˜egreater-or-equal true iff value of egreater-or-equal to value of ˜e
p(e1, . . . , ek) predicate true iff pholds for the value of (e1, . . . , ek)
¬Pnegation / not true if Pis false
P∧Qconjunction / and true if both Pand Qare true
P∨Qdisjunction / or true if Pis true or if Qis true
P→Qimplication / implies true if Pis false or Qis true
P↔Qbi-implication / equivalent true if Pand Qare both true or both false
∀x P universal quantifier / for all true if Pis true for all real values of variable x
∃x P existential quantifier / exists true if Pis true for some real value of variable x
[α]P[·] modality / box true if Pis true after all runs of HP α
hαiPh·i modality / diamond true if Pis true after at least one run of HP α
2: Statements and Effects of While Programs
Program Operation Effect
x:= eassignment assigns value of term eto variable x
?Qtest check truth of first-order formula Qat current state
α;βsequential composition βstarts after αfinishes
if(Q)αelse βif-then-else run αif Qis true at current state else run β
while(Q)αwhile loop repeats αas long as Qis true
α∗nondeterministic repetition repeats α n-times for any n∈N
3: Semantics of Dynamic Logic formula Pin interpretation I
ω|=e≥˜eiff ω[[e]] ≥ω[[˜e]]
ω|=p(e1, . . . , ek) iff (ω[[e1]], . . . , ω[[ek]]) ∈I(p) where predicate symbol pis interpreted as relation I(p)
ω|=¬Piff ω6|=Pthat is, it is not the case that ω|=P
ω|=P∧Qiff ω|=Pand ω|=Q
ω|=P→Qiff ω6|=Por ω|=Q
ω|=∃x P iff ν|=Pfor some state νwith ν=ωexcept for the value of x
ω|=∀x P iff ν|=Pfor all states νwith ν=ωexcept for the value of x
ω|=hαiPiff ν|=Pfor some state νsuch that (ω, ν)∈[[α]]
ω|= [α]Piff ν|=Pfor all states νsuch that (ω, ν)∈[[α]]
4: Semantics of While Program αis relation [[α]] ⊆ S × S between initial and final states
[[x:= e]] = {(ω, ν ) : ν=ωexcept ν[[x]] = ω[[e]]}
[[?Q]] = {(ω, ω) : ω|=Q}
[[if(Q)αelse β]] = {(ω, ν ) : (ω|=Qand (ω, ν )∈[[α]]) or (ω6|=Qand (ω , ν)∈[[β]])}
[[α;β]] = {(ω, ν ) : (ω, µ)∈[[α]],(µ , ν)∈[[β]]}
[[α∗]] = {(ω, ν) : there are n∈N, µ0=ω, µ1, µ2, . . . , µn=νsuch that (µi, µi+1)∈[[α]] for all 0≤i<n
[[while(Q)α]] = (ω, ν ) : µ0=ω, µ1, µ2, . . . , µn=νand µi|=Q, (µi, µi+1)∈[[α]] and µn6|=Qfor all 0≤i<n
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