The Language Theory of Bounded Context-Switching

1
The Language Theory of Bounded Context-Switching

• Gennaro Parlato (U. of Illinois, U.S.A.)
• Joint work with
• Salvatore La Torre (U. of Salerno, Italy)
• P. Madhusudan (U. of Illinois,
  U.S.A.)

2
What is this talk about?

• Our work is motivated by the verification of
  concurrent programs with recursive procedures
  communicating through shared variables
• Reachability/Emptiness is undecidable for such
  programs
• Restricted reachability has gained a lot of
  attention in the literature for programs with
  variable ranging over finite domain
• Reachability within a fixed number of
  context-switches is decidable
• Many errors manifest themselves within few
  context-switches
• We undertake a language- and automata-theoretic
  study of concurrent programs under bounded
  context-switches
• Multi-stack Pushdown Automata are a natural model
  for such programs

3
Multi-stack Visibly Pushdown Automata (MVPAs)

• VISIBLE ALPHABET
• Each stack has its own alphabet
• - Pushi, Popi, Inti (disjoint sets)
• ? i (Pushi U Popi U Inti) is the alphabet of
  stack i
• Symbols describe the behavior of the automaton
• ? i and ? j are pairwise disjoint for i?j
• ? ? 1 U? 2 U U ? n is the input alphabet

Finite Control

• Finite number of states
• - including an initial state set of final
  states
• Moves only the symbol a
• - Internal move for stack i if a?Inti
• - Push move onto stack i if a?Pushi
• Pop move from stack i if a?Popi
• The symbol of ? determines the kind of move to
  make

stack-1
stack-2
stack-n
4
Bounded round executions

• We consider only executions going through k
  rounds
• A round is a word in the language
• Round ? 1 . ? 2 . ? n
• Roundk is the set of all k-round words
• A k-round word can be seen as a matrix
• A row is a round
• The concatenation of the word on column i is the
  stack i projection of the input word w
• round 1 w11 w12
  w13 w1n
• round 2 w21 w22
  w23 w2n
• 
  
• round k wk1 wk2
  wk3 wkn
• w w11 w12 w13 w1n w21w22 w23 w2n .
  wk1 wk2 wk3 wkn

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A k-round execution can be seen as
w1n
round 1
w11
w12
q11
w2n
round 2
w21
w22
q21
w3n
round 3
w31
w32
q31
wkn
round k
wk1
wk2
qk1
stack 1
stack 2
stack n
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Example a language recognized by a 2-round
MVPA

• L ai xj bi yj i,j gt 0
• Push onto stack 1 reading a
• Push _at_ onto stack 2 reading x
• Pop form stack 1 reading b
• Pop _at_ from stack 2 reading y
• (The first symbol pushed onto each stack is
  encoded differently to check later whether the
  stack is empty)
• L can be accepted by a 2-round MVPA
• L is not context-free language
• All recognized languages are context-sensitive
• (La Torre,
  Madhusudan, Parlato, LICS07)

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Bounded-round MVPLs Class of languages accepted
by bounded-round MVPAs

• A sub-class of context-sensitive languages
• Visibily implies closure under
• Union
• Intersection
• Nondeterministic and deterministic versions are
  equivalent
• Closed under complement (through
  Determinizability)
• Decidable Emptiness and Membership problems
• Universality and inclusion problems are decidable
• closure under Boolean operations
• decidability of the emptiness problem

8
Related Work

• Visibly pushdown automata
• 
  (Alur, Madhusudan, STOC04)
• Bounded-phase multi-stack visibly push-down
  automata
• (not determinizable)
• (La
  Torre, Madhusudan, Parlato, LICS07)
• Visibly ordered pushdown automata
• (NOT determinizable, wrong determinizability
  proof is given)
• 
  (Carotenuto, Murano, Peron, DLT07)

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• Determinization
• (main technical
  result)

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Deterministic bounded-round VMPAs

• A VMPA is deterministic if
• for any state, and
• for any input symbol
• at most one move is allowed
• Bounded-round VMPAs are determinizable
• If A is a k-round MVPA, then
• there exists a k-round MVPA AD such that
  L(A)L(AD)
• Boundedness of the number of rounds is crucial
  for our proof
• The class of MVPAs is not closed under
  determinization
• 
  (La Torre,
  Madhusudan, Parlato, LICS07)
• Determinization construction

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A run can be seen as
w1n
round 1
w11
w12
q11
w2n
round 2
w21
w22
q21
w3n
round 3
w31
w32
q31






wkn
round k
wk1
wk2
qk1
stack 1
stack 2
stack n
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Interfaces

• An interface of stack i Is definded
• w.r.t. a word of stack i
• w1i w2i w3i wki
• ( represents a context-switch)
• It corresponds to the pair (INi, OUTi) where
• Ini ( q1j, q2j, q3j, qkj )
• OUTi ( q1j, q2j, q3j, qkj )
• Interfaces of stack i can be computed by a
• non deterministic VPA Ai
• STATES (q, Interface, round)
• q is any A state
• Interface is an encoding of the interface
• computed until now
• round tracks the round under simulation
• MOVES

round 1
w1j
round 2
w2j
round 3
w3j




round k
wkj
INi
OUTi
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Why consider interfaces
w1n
round 1
w11
w12
q11
w2n
round 2
w21
w22
q21
w3n
round 3
w31
w32
q31






wkn
round k
wk1
wk2
qk1
stack 1
stack 2
stack n

• Theorem (accepting condition)
• A k-round word w is accepted by A iff there is an
  interface Inti, one
• for each stack i on the word wi, such that
• Int1 composes Int2 composes composes Intn
• Intn wraps Int1
• q11 is the initial state qkn is a final state

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Construction of a deterministic MVPA AD

• Ai can be determinized (AiD) (Alur,
  Madhusudan, STOC04)
• Every AiD state encodes the set of all possible
  interfaces on any k-round word wi (wi is the
  subword of w composed only by symbols of stack i)
• The deterministic automaton AD simulates in
  parallel all the AiD independently, switching
  from one to another reading the input word
• It does not care if interfaces compose/wrap
• Composition and wrapping is checked only at the
  end of the computation for acceptance (see
  accepting condition)

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Idea of the simulation
w1n
round 1
w11
w12
q11
w2n
round 2
w21
w22
q21
w3n
round 3
w31
w32
q31






wkn
round k
wk1
wk2
qk1
stack 1
stack 2
stack n

• Reading a symbol of wij, except the first one,
  simulate AjD
• Reading the first symbol of wij, simulate in
  parallel ( is not in ?)
• Aj-1D on the symbol , and
• AjD on the first symbol of wij
• AD is
  deterministic

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going back to the construction of AD

• Ai can be determinized (AiD) (Alur,
  Madhusudan, STOC04)
• Every AiD state encodes the set of all possible
  interfaces on the k-round word wi
• The deterministic automaton AD simulates in
  parallel all the AiD switching from one AiD to
  another reading the input word
• After reading w every AiD has computed the set of
  all possible interfaces on wi
• Final states AD accepts a word w iff there is a
  set of interfaces, one for each stack, that
  satisfy the accepting condition

17

• Conclusion

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Conclusion

• We have defined
• a robust sub-class of
  context-sensitive languages
• Determinazable
• (we conjecture that finding a larger class in
  terms of patterns is unlikely)
• Closed under all Boolean operations
• Decidable emptiness, universality and inclusion
  problems
• MSO characterization, Parikh theorem
• (La
  Torre, Madhusudan, Parlato, LICS07)
• Same results if we consider bounded
  context-switch words instead of bounded round
  words