Linear Temporal Logic

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Linear Temporal Logic
Roxana Ragneala, roxana.ragneala_at_gmail.com
Seminar The Time Machine Supervisor Klaus Dräger
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History of Temporal Logic
Philosophers
Arthur Prior
Amir Pnueli
Specification of concurrent systems
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Framework

• Temporal Logic is a class of Modal Logic
• Allows qualitatively describing and reasoning
  about changes of the truth values over time
• Usually implicit time representation
• Provides variety of temporal operators
  (sometimes, always)
• Different views of time (branching vs. linear,
  discrete vs. continuous, past vs. future, etc.)

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Branching vs. Linear Time
Linear - only one possible future in a moment
Branching - may split to different courses
depending on possible futures
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LTL

• In LTL time is
• implicit,
• discrete,
• infinite in the future and
• has an initial moment with no predecessors
• A model of LTL formula is an infinite sequence of
  states p s0, s1, s2,

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LTL

• Elements
• Atomic propositions AP
• Boolean operators ? ? ?
• Temporal operators G F X U R
• Syntax
• F P
• F ? ? ? ? ? ? F ? F
• G F F F X F ? U ? ? R ?

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Semantic Intuition

• G f always f
• F f eventually f
• X f next state
• f U r until
• f R r releases

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Examples
p,q p,q p,q p,q p,q p,q
p ? q X(p ? q)
True
G p
False
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More Examples
G(p -gtXq) G(p -gt X(q U r)) GF p p R q
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Semantic

• Semantic is given with respect to path
• p s0 s1 s2
• Suffix of the path starting at si
• pi si si1 si2
• A system satisfies an LTL formula f if each path
  through the system satisfies f.

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Semantic

• p ? iff ? ? L(s0)
• p ?? iff not p ?
• p ? ? ? iff p ? and p ?
• p ? ? ? iff p ? or p ?
• p X ? iff p1 ?
• p F ? iff exists i ? 0 pi ?
• p G ? iff for all i ? 0 pi ?
• p ? U ? iff exists i ? 0 pi ?
• and for all 0 ? j lt i pj ?

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From LTL to Automata
Automaton for Fp
,p
p
s0
s1
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LTL to Büchi Automata

• First, we bring the LTL formulas in a normal form
• Rules

• p ? q ?p ? q
• p ? q (?p ? q) ? (?q ? p)
• ?(p ? q) ?p ? ?q
• ?(p ? q) ?p ? ?q
• ??p p

• ?(p U q) ?p R ?q
• ?(p R q) ?p U ?q
• F p true U p
• G p false R p
• ? X p X ?p

GF p ? F r 8,9 (false R (Fp)) ? (true U p) 8
(false R (true U p)) ?(true U p) 1
?(false R (true U p)) ? (true U p) 7 (true U
?(true U p)) ? (true U p) 6
(true U (false R ?p)) ? (true U p)
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Büchi Automata

• Automaton A (S,S,d,I,F)
• S finite alphabet
• S set of states
• d transition relation
• I set of initial states
• F set of acceptance states
• A run p of A on ? word a
• p q0,q1,q2,, such that q0 ? I and (qi,ai,qi1)
  ? d
• The run p is accepting if
• Inf(p)nF ?

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LTL to Büchi Automata A?

• S sets of subformulas of ?
• e.g ?p1U?p2, a state is given by s p1,?p2,
  p1U?p2
• Consider a word ss0,s1,s2 such that s ?
  where,
• e.g.,? ?1 U ?2
• Mark each position i with the subset of formulas
  si of ?
• that hold true there (s0, s1, - s0,s1,)
• Clearly, ? ?s0. But then, by consistency either
• ?1 ?s0 and ? ?s1 or
• ?2 ?s0

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LTL to Büchi Automata A?
sub(?) sets of subformulas of ? A?(Q, S, R, L,
Init, F) Qs sub(?) s.t. s is locally
consistent For s to be locally consistent we
should e.g. have

• false s
• if ?1 ? ?2 ? s then ?1 ? s and ?2 ? s
• if ?1 ? ?2 ? s then ?1 ? s or ?2 ? s
• if pi ? s then ?pi s and if ?pi ? s then pi
  s
• if ?1 U ?2 ? s then ?1 ? s or ?2 ? s

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LTL to Büchi Automata A?
L Q? S We want a word ss0,s1,s2 to be in
L(A?) iff there is a run ps0,s1,such that
i?N, we have that si satisfies L(si)
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Temporal Operators

• p U q (q ? (p ? X(p U q)))
• Note q has to be true at some point!
• p R q (q ? (p ? X(p R q)))

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LTL to Büchi Automata A?
LQ? S We want a word ss0,s1,s2 to be in L(A?)
iff there is a a run ps0,s1,such that
i?N, we have that si satisfies L(si) R
Q x Q where (s,s) ? R iff

• if ?1 U ?2 ? s then ?2 ? s or (?1 ? s and (?1 U
  ?2) ? s)
• if ?1 R ?2 ? s then ?2 ? s and (?1 ? s or (?1 R
  ?2) ? s)
• If X? ? s then ? ? s

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LTL to Büchi Automata A?
Init s ? Q ? ? s F for each ?1U?2 ?
sub(?) there is a set Fi ? F such that Fis ?
Q if ?1U?2 ?s then ?2 ?s Lemma L(?)L(A?)
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Example Fp
true U p p
true U p
p

• F p true U p
• Init s ? sub(true U p) (true U p) ? s

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Example Fp
true U p p
true U p
p

• true U p p ? X (true U p)

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Example Fp
true U p p
true U p
p

• true U p p ? X (true U p)

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Example Fp
true U p p
true U p
p

• true U p p ? X (true U p)

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Example Fp
true U p p
true U p
p

• true U p p ? X (true U p)

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Example Fp
true U p p
true U p
p

• F Ftrue U ps ? sub(true U p) if (true U
  p) ?s then p ?s

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Branching Time

• Path quantifiers
• A for all paths
• E for some paths
• Examples
• CTL AGp, EFp, AGEXp, A(GFp), E(GFp)
• CTL AGp, EFp, AGEXp, EGEFP

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Comparison

• Different views of time branching with linear
• Incomparable expressive power
• FGp is not expressible in CTL
• AGEFp is not expressible in LTL
• Performance
• CTL run in time O(Pxf)
• LTL run in time O(Px2O(f)) and space
  O((flog(P))2)
• CTL characterizes bisimulation
• CTL is more used in industry

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