Translating from logic to automata - PowerPoint PPT Presentation

1 / 30
About This Presentation
Title:

Translating from logic to automata

Description:

Translating from logic to automata (Book: Chapter 6) – PowerPoint PPT presentation

Number of Views:79
Avg rating:3.0/5.0
Slides: 31
Provided by: doro47
Category:

less

Transcript and Presenter's Notes

Title: Translating from logic to automata


1
Translating from logic to automata
(Book Chapter 6)
2
Why translating?
  • Want to write the specification in some logic.
  • Want model-checking tools to be able to check the
    specification automatically.

3
Generalized Büchi automata
  • Acceptance condition F is a setFf1 , f2 , ,
    fn where each fi is a set of states.
  • To accept, a run needs to pass infinitely often
    through a state from every set fi .

4
Translating into simple Büchi automaton
Version 0
b
c
q0
q2
q1
a
c
c
b
q0
q2
q1
c
a
Version 1
5
Translating into simple Büchi automaton
Version 0
c
q0
q2
q1
a
c
b
c
b
q0
q2
q1
c
a
Version 1
6
Translating into simple Büchi automaton
Version 0
c
q0
q2
q1
a
c
b
c
b
q0
q2
q1
c
a
Version 1
7
Preprocessing
  • Convert into normal form, where negation only
    applies to propositional variables.
  • ? becomes ltgt?.
  • ltgt? becomes ?.
  • What about (? U ?)?
  • Define operator V such that ( ? U ??) (?) R
    (?),
  • ( ? R ??) (?) U (?).

8
Semantics of pR q
p
p
p
p
p
p
p
p
p
q
q
q
q
q
q
q
q
q
p
p
p
p
p
q
q
q
q
q
9
  • Replace true by false, and false by true.
  • Replace (? \/ ?) by (?) /\ (?) and
    (? /\ ?) by (?) \/ (?)

10
Eliminate implications, ltgt,
  • Replace ? -gt ? by ( ?) \/ ?.
  • Replace ltgt? by (true U ?).
  • Replace ? by (false R ?).

11
Example
  • Translate ( ltgtP ) ? ( ltgtQ )
  • Eliminate implication ( ltgtP ) \/ ( ltgtQ )
  • Eliminate , ltgt( false R ( true U P ) ) \/ (
    false R ( true U Q ) )
  • Push negation inwards(true U (false U P ) )
    \/ ( false V ( true U Q ) )

12
The data structure
Name
13
The main idea
  • ? U ? ? \/ ( ? /\ O ( ? U ? ) )
  • ? V ? ? /\ ( ? \/ O ( ? R ? ) )
  • This separates the formulas to two partsone
    holds in the current state, and the otherin the
    next state.

14
How to translate?
  • Take one formula from New and add it to Old.
  • According to the formula, either
  • Split the current node into two, or
  • Evolve the node into a new version.

15
Splitting
Copy incoming edges, update other field.
16
Evolving
Copy incoming edges, update other field.
17
Possible cases
  • ? U ? , split
  • Add ? to New, add ? U ? to Next.
  • Add ? to New.
  • Because ?U ? ? \/ ( ? /\ O (?U ? )).
  • ? R ? , split
  • Add ???? to New.
  • Add ? to New, ? R ? to Next.
  • Because ? R ? ? /\ ( ? \/ O (? R ? )).

18
More cases
  • ? \/ ?, split
  • Add ? to New.
  • Add ? to New.
  • ? /\ ?, evolve
  • Add ???? to New.
  • O ?, evolve
  • Add ? to Next.

19
How to start?
init
Incoming
New
Old
aU(bUc)
Next
20
init
Incoming
aU(bUc)
init
init
21
Incoming
aU(bUc)
bUc
init
init
Incoming
Incoming
aU(bUc)
aU(bUc)
c
b
(bUc)
22
When to stop splitting?
  • When New is empty.
  • Then compare against a list of existing nodes
    Nodes
  • If such a with same Old, Next exists,just
    add the incoming edges of the new versionto the
    old one.
  • Otherwise, add the node to Nodes. Generate a
    successor with New set to Next of father.

23
init
Incoming
a,aU(bUc)
Creating a successor node.
aU(bUc)
Incoming
aU(bUc)
24
How to obtain the automaton?
X
  • There is an edge from node X to Y labeled with
    propositions P (negated or non negated), if X is
    in the incoming list of Y, and Y has propositions
    P in field Old.
  • Initial node is init.

a, b, c
Node Y
25
The resulted nodes.
26
Initial nodes
a, aU(bUc)
b, bUc, aU(bUc)
c, bUc, aU(bUc)
b, bUc
c, bUc
All nodes with incoming edge from init.
27
Include only atomic propositions
a
c
Init
b
c
b
28
Acceptance conditions
  • Use generalized Buchi automata, wherethere are
    several acceptance sets F1, F2, , Fn, and each
    accepted infinite sequence must include at least
    one state from each set infinitely often.
  • Each set corresponds to a subformula of form ?U?.
    Guarantees that it is never the case that ?U?
    holds forever, without ?.

29
Accepting w.r.t. bU c
30
Acceptance w.r.t. aU (bU c)
Write a Comment
User Comments (0)
About PowerShow.com