Specification Formalisms: Temporal Logic and Automata on Infinite Words

1
Specification FormalismsTemporal Logic and
Automata on Infinite Words

• Literature Peled ch. 5
• Mads Dam

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Temporal Logic

• Logic of transition system executions
• Propositional/first-order logic state
  assertions
• Temporal assertions assertions on system
  executions
• Invariably (along this execution) x y z
• Sometime (along this execution) an
  acknowledgement packet is sent
• If T is infinitely often enabled (along this
  execution) then T is eventually executed
• Last packet received along channel a (along this
  execution) had the shape (b,c,d)
• No matter which execution is followed from now
  (this state), a reply will eventually (along that
  execution) be sent
• No matter what choice B made in the past, it
  would necessarily come to pass that ?

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Runs/Executions/Paths

• Fix transition system T (Q,R,Q0)
• Computation path (aka run, execution sequence)
• Infinite sequence
• ? q0q1q2...qi...
• such that for all i 0, qi R qi1
• Notation
• ?(k) qk ( kth state of ?)
• ?k qkqk1 ... ( kth suffix of ?, i.e. the
  path)

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LTL Linear Time Temporal Logic

• Logic of future time path properties
• ? Primitive state assertions
• Syntax
• ? ? ? ?Æ? ltgt? ? ? U ? O?
• ? ? holds now/at the current time instant
• ltgt? At some future time instant ? is true
• ? For all future time instants ? is true
• ? U ? ? is true until ? becomes true
• O? ? is true at the next time instant

5
Pictorially
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Semantics

• Satisfaction relation ? ² ?
• Assume interpretation function ? ? ? Q µ Q
• ?(?) Set of states for which ? holds
• ? ² ? iff ?(0) 2 ?(?)
• ? ² ? iff not ? ² ?
• ? ² ? Æ ? iff ? ² ? and ? ² ?
• ? ² ltgt? iff exists k2N. ?k ² ?
• ? ² ? iff for all k2N. ?k ² ?
• ? ² ? U ? iff exists k2N. ?k ² ? and for all i
  0 i lt k. ?i ² ?
• ? ² O? iff ?1 ² ?
• For transition system T (Q,R,Q0)
• T ² ? iff for all runs ? of T with ?(0)2 Q0, ? ² ?

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Some LTL Formulas

• ? Ç ? (? Æ ?)
• ? ! ? ? Ç ? (! is seriously overloaded!)
• ltgt? true U ?
• ? ltgt?
• ? V ? ? Ç (? U (?Æ?))
• (aka release in Peled)
• ltgt?
• ? holds from some point forever
• ltgt?
• ? holds infinitely often
• ltgt? ! ltgt?
• if ? holds infinitely often then so does ?

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Spring Example
release
release
q0
q1
q2
pull
extended
extended
malfunction

• Primitive state assertions extended, malfunction
• Sample paths
• q0 q1 q0 q1 q2 q2 q2 ...
• q0 q1 q2 q2 q2 ...
• q0 q1 q0 q1 q0 q1 ...

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Satisfaction by Single Path
release
release
? q0q1q0q1q2q2q2 ...
q0
q1
q2
pull
extended
extended
malfunction

• ? ² extended?
• ? ² Oextended?
• ? ² OOextended?
• ? ² ltgtextended?
• ? ² extended?
• ? ² ltgtextended?
• ? ² ltgtmalfunction?

? ² ltgtextended? ? ² extended U malfunction? ? ²
( extended) U extended? ? ² (ltgtextended) U
malfunction? ? ² (ltgt extended) U malfunction? ?
² ( extended ! Oextended)
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Satisfaction by Transition System
release
release
T
q0
q1
q2
pull
extended
extended
malfunction

• T ² extended?
• T ² Oextended?
• T ² OOextended?
• T ² ltgtextended?
• T ² extended?
• T ² ltgtextended?
• T ² ltgtmalfunction?

T ² ltgtextended? T ² extended U malfunction? T ²
( extended) U extended? T ² (ltgtextended) U
malfunction? T ² (ltgt extended) U malfunction? T
² ( extended ! Oextended)
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Example Mutex

• Assume there are 2 processes, Pl and Pr
• State assertions
• tryCSi Process i is trying to enter critical
  section
• E.g. tryCSl pcl l4
• inCSi Process i is inside its critical section
• E.g. inCSl pcl l5 Ç pcl l6
• Mutual exclusion
• ((inCSl Æ inCSr))
• Responsiveness
• (tryCSi ! ltgtinCSi)
• Process keeps trying until access is granted
• (tryCSi ! ((tryCSi U inCSi) Ç tryCSi))

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Example Fairness

• States Pairs (q,?)
• ? label of last transition taken, so
• q!? q
• (q,?) !? (q,?)
• ? Finite set of labels partitioned into subsets
  P
• P (finite) set of labels of some process
• State assertions
• enP Some transition labelled ? 2 P is enabled
• i.e. (q,?)2?(en?) iff 9 q.q!? q
• execP Label of last executed transition is in P
• i.e. (q,?)2?(execP) iff ?2 P
• Note enP Ç?2 Pen? and execP Ç?2 Pexec?

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Fairness Conditions

• Weak transition fairness
• Æ?2?ltgt(en? Æ exec?)
• Or equivalently
• Æ?2?(ltgten? ! ltgtexec?)
• Strong transition fairness
• Æ?2?(ltgten? ! ltgtexec?)
• Weak process fairness
• ÆPltgt(enP Æ execP)
• Strong process fairness
• ÆP (ltgtenP ! ltgtexecP)

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Branching Time Logic

• Sets of paths?

• Or computation tree?

. . . .
. . . .
. . . .
. . . .
. . . .
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Computation Tree Logic - CTL

• Syntax
• ? ? ? ?Æ? AF? AG? A(? U ?) AX?
• Formulas hold of states, not paths
• A Path quantifier, along all paths from this
  state
• F ltgt, G , X O
• So
• AF? Along all paths, at some future time instant
  ? is true
• AG? Along all paths, for all future time
  instants ? is true
• A(? U ?) Along all paths, ? is true until ?
  becomes true
• AX? ? is true for all next states
• Note CTL is closed under negation so also
  express dual modalities EF, EG, EU, EX (E is
  existential path quantifier)

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CTL, Semantics

• Interpretation function ? ? ? Q µ Q the same
• q ² ? iff q 2 ?(?)
• q ² ? iff not q ² ?
• q ² ? Æ ? iff q ² ? and q ² ?
• q ² AF? iff for all ? such that ?(0)q exists k2N
  such that ?(k) ² ?
• q ² AG? iff for all ? such that ?(0)q, for all
  k2N, ?(k) ² ?
• q ² A(? U ?) iff for all ? such that ?(0)q,
  exists k2N. ?(k) ² ? and for all i 0 i lt k.
  ?(i) ² ?
• q ² AX? iff for all ? such that ?(0) q, ?(1) ²
  ?
• (iff for all q such that q ! q, q² ?)
• For transition system T (Q,R,Q0)
• T ² ? iff for all q02 Q0, q0 ² ?

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CTL LTL Brief Comparison

• LTL in branching time framework
• ? ? A? ( ? to hold for all paths)
• CTL LTL EF? not expressible in LTL
• LTL CTL ltgt? not expressible in CTL
• CTL Extension of CTL with free alternation A,
  F, G, U, X
• Advantages and disadvantages
• LTL often more natural
• Satisfiability LTL PSPACE complete, CTL
  DEXPTIME complete
• Model checking LTL PSPACE complete, CTL In P

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Automata Over Finite Words

• Finite state automaton A (Q,?,?,I,F)
• Q Finite set of states
• ? Finite alphabet
• ? µ Q ? Q Transition relation
• Write q!a q for ?(q,a,q) as before
• I µ Q Start states
• F µ Q Accepting states
• Word a1a2...an is accepted, if there is sequence
• q0 !a1 q1 !a2 ... !an qn
• such that q02 I and qn2 F

a
b
b
a
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Automata Over Infinite Words

• Intuition Letters a2? might represent states, or
  state properties
• A computation path is an infinite word over
  object states
• Infinite word w
• Function w N ! ?
• Equivalently Infinite sequence w a0a1a2 ... an
  ...
• Buchi automaton Finite state automaton, but on
  infinite words
• Word w is accepted if accepting state visited
  infinitely often

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Example
a
b
b
a

• Which infinite words are accepted?
• ababab ... ( ab?) ?
• aaaaaa... ( a?) ?
• bbbbbb... ( b?) ?
• aaabbbbb... ( aaab?) ?
• ababbabbbabbbba... ?

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Nondeterminism

• What is the language accepted by this automaton?
• What is the corresponding LTL property if b
  inCS and a b?

a
a
a,b
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Another Example

• Letters represent propositions
• Example ltgtinCS, ainCS, b inCS

a
b
b
a
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Yet More Examples

• a inCS1 Æ inCS2
• b a
• c true
• Property a
• Property (d ! ltgte)
• Idea
• q0 Have seen d Ç e
• q1 Saw d, now wait for e

a
c
b
Or just
b
d ! e
e
dÆ e
q0
q1
e
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Even More...

• Property (a ! (bUc))
• Idea
• q0 Body of immediately ok
• q1 Awaiting c
• Property ?(a ! (bUc)) ltgt(a Æ ?(bUc))
• Idea
• ?(bUc) b becomes false some time without c
  having become true first
• q0 Waiting ...
• q1 Have seen a with b and ?c
• q2 Committing ...

a Ç c
b Æ c
a Æ b Æ c
q0
q1
c
true
b Æ c
a Æ b Æ c
q0
q1
q2
aÆ b Æ c
b Æ c
true
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Deterministic Buchi Automata

• Consider ? ltgta where ? a,b
• Suppose A recognizes ?
• A deterministic
• A reaches accepting state on some input an1
• And on an1ban2
• And on an1ban2ban3
• And on an1ban2ban3b ... b ... b ...
• So Nondeterministic Buchi automata strictly more
  expressive than deterministic ones
• And Deterministic B. A. not closed under
  complement

a
a
a,b
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Alternative Formalisms

• Next lecture LTL ? Buchi automata
• Buchi automate strictly richer than LTL
• B. A. recognisable languages remarkably stable
• Monadic second order logic of successor
• 9 X(02 X Æ 8 y8 z(succ(y,z) ! (y2X z2X))
• Æ 8 y(y2X ! a(y)))
• (all even symbols are as)
• LTL with propositional quantification
• 9 X((X Æ (X OX) Æ (x ! a))
• ?-regular expressions
• a((a b)a)?
• Linear-time ?-calculus
• ? X.a Æ OOX