Complementing B

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Complementing Büchi Automata Safras construction

• Debmalya Panigrahi

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Agenda

• Determinization technique
• A naïve attempt Subset construction
• Towards the solution
• The construction
• Correctness
• Soundness
• Completeness
• Complexity
• State space optimality

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Complementation

• What do we do for classical automata?
• What is the problem with such a technique for
  Büchi automata (BA)?

DFA for L
FA for L
DFA for Lc
NBA strictly more expressive than DBA
DBA not closed under complementation
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Determinization Subset construction A naïve
attempt

• Why doesnt subset construction work in the
  infinite case?
• L all strings with finite number of as
• Multiple infinite runs with finite number of
  distinct positions visiting final states

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Determinization Towards the solution

• Breakpoint Recursive definition
• Each reachable state has at least one run that
  has seen an accepting state since last breakpoint
• Initial set of states is a breakpoint
• Mark accepting states and all states whose
  predecessor is marked- delete all markings at
  breakpoint
• Accept if there are infinite number of
  breakpoints (green flashes!)
• One bad run spoils the party!!
• L all strings with finite number of as
• Need to separate good runs (marked infinite
  times) from bad runs (marked finite number of
  times) separate marked from unmarked states in
  subset

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Determinization The solution

• Do not wait for all states to be marked (might
  have to wait infinitely long!)
• After finite time, take set of states already
  marked and make them child of node
• What can happen eventually?
• Case 1 Root eventually becomes green (good
  case!)
• Remove all descendants and start again
• Case 2 Root has bad state
• Bad states do not propagate down the tree
• Accept if ANY node flashes infinitely often!
• Depth of tree bounded by construction
• To bound width, ensure siblings are disjoint

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The basic scheme
Determinization
Standard construction
DRA for L
BA for L
DSA for Lc
BA for Lc
Nothing to do!
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The formal construction States

• Each state of the DRA is a labelled ordered
  rooted tree
• Each node has a
• Label a subset of the states of the BA
• Colour green or white
• Nodes also have an ordering
• Sibling order
• Extended to an ordering of non-ancestral nodes

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The formal construction States

• Constraints on the construction of the tree
• Union of labels of children is a proper subset of
  label of parent (depth bound!)
• Non-ancestral nodes have disjoint labels (width
  bound!)
• Leads to each state being specific to a node
• How many nodes does the tree have?

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The formal construction Start state

• Single node
• Contains all start states of the BA
• Coloured white

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The formal construction Transitions

• A sequence of actions on reading the symbol a
• Colour of all nodes reset to white
• Copy all final (BA) states in a tree node as
  youngest son for each tree node
• Change each tree node to all successor states (of
  current states of the BA on the symbol a)
• Remove a (BA) state from a node if it already
  exists on a node to the left
• Remove all nodes with empty labels (terminating
  run!)
• If a node has a label equal to the union of
  labels of all its children, remove all the
  children and colour the node green

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The formal construction Acceptance condition

• A run is accepting if some node turns green
  infinitely often
• What do we mean by some node?
• In general, nodes can both be deleted and added
• Policies for managing names
• An infinite vocabulary leads to a restricted
  Rabin acceptance condition cannot give bound on
  number of trees in a run
• A finite vocabulary with 2n (sufficient!) names
  leads to a full Rabin acceptance condition
• A word is accepted if the (deterministic!) run
  for the word is an accepting run

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Correctness

• Soundness If a word is accepted by the DRA, it
  is accepted by the BA
• Completeness If a word is accepted by the BA, it
  is accepted by the DRA

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Complexity

• Total number of labelled ordered trees is
  2O(nlogn)
• Number of ordered trees without labels is
  2O(nlogn)
• Each ordered tree can be labelled in 2O(nlogn)
  ways
• This is provably optimal
• Complemented BA also has 2O(nlogn) states

DRA for L
BA for L
DSA for Lc
BA for Lc
O(2O(nlogn))
O(2O(nlogn))
O(1)
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References

• On the Complexity of ?-Automata Shmuel Safra

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THANK YOU