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.)

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

Multi-stack Visibly Pushdown Automata (MVPAs)

…

stack-1 stack-2 stack-n

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 aInti

- Push move onto stack i if aPushi

- Pop move from stack i if aPopi

The symbol of ∑ determines the kind of move to make

VISIBLE ALPHABET

Each stack has its own alphabet - Pushi, Popi, Inti (disjoint sets)

∑ i = (Pushi U Popi U Inti) is the alphabet of stack iSymbols describe the behavior of the automaton

∑ i and ∑ j are pairwise disjoint for ij

∑ = ∑ 1 U∑ 2 U … U ∑ n is the input alphabet

Bounded round executions We consider only executions going through k rounds

A round is a word in the languageRound = ∑ 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

A k-round execution can be seen as

w11

w21

w31

wk1

w12

w22

w32

wk2

w1n

w2n

w3n

wkn

round 1

round 2

round 3

round k

stack 1 stack 2 stack n

q11

q21

q31

qk1

q’12

q’22

q’32

q’k2

q’11

q’21

q’31

q’k1

q12

q22

q32

qk2

q1n

q2n

q3n

qkn

q’1n

q’2n

q’3n

q’kn

Example: a language recognized by a 2-round MVPA

L= { ai xj bi yj | i,j > 0 } Push $ onto stack 1 reading a Push @ onto stack 2 reading x Pop $ form stack 1 reading b Pop @ 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, LICS’07)

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

Related Work Visibly pushdown automata (Alur, Madhusudan, STOC’04)

Bounded-phase multi-stack visibly push-down automata

(not determinizable) (La Torre, Madhusudan, Parlato, LICS’07)

Visibly ordered pushdown automata

(NOT determinizable, wrong determinizability proof is given) (Carotenuto, Murano, Peron, DLT’07)

Determinization (main technical result)

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, LICS’07)

Determinization construction …

A run can be seen as …

w11

w21

w31

wk1

w12

w22

w32

wk2

w1n

w2n

w3n

wkn

round 1

round 2

round 3

round k

stack 1 stack 2 stack n

q11

q21

q31

qk1

q’12

q’22

q’32

q’k2

q’11

q’21

q’31

q’k1

q12

q22

q32

qk2

q1n

q2n

q3n

qkn

q’1n

q’2n

q’3n

q’kn

… … …

… …

…

Interfaces

w1j

w2j

w3j

wkj

round 1

round 2

round 3

round k

INi

q’1i

q’2ii

q’3i

q’ki

q1i

q2i

q3i

qki

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 = ( q’1j, q’2j, q’3j, … q’kj )

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:- All A’s moves on stack i symbols - internal moves on the fresh symbol # -update q with any q’ (guess the state at next round)

-update Interface storing q and q’ -increment round

…… …

…

OUTi

Why consider interfaces

w11

w21

w31

wk1

w12

w22

w32

wk2

w1n

w2n

w3n

wkn

round 1

round 2

round 3

round k

stack 1 stack 2 stack n

q11

q21

q31

qk1

q’12

q’22

q’32

q’k2

q’11

q’21

q’31

q’k1

q12

q22

q32

qk2

q1n

q2n

q3n

qkn

q’1n

q’2n

q’3n

q’kn

… … …

… …

…

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 & q’kn is a final state

Construction of a deterministic MVPA AD

Ai can be determinized (AiD) (Alur, Madhusudan, STOC’04)

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)

Idea of the simulation

w11

w21

w31

wk1

w12

w22

w32

wk2

w1n

w2n

w3n

wkn

round 1

round 2

round 3

round k

stack 1 stack 2 stack n

q11

q21

q31

qk1

q’12

q’22

q’32

q’k2

q’11

q’21

q’31

q’k1

q12

q22

q32

qk2

q1n

q2n

q3n

qkn

q’1n

q’2n

q’3n

q’kn

… … …

… …

…

Reading a symbol of wij, except the first one, simulate AjD

Reading the first symbol of wij, simulate in parallel (# is not in ∑) Aj-1

D on the symbol #, and Aj

D on the first symbol of wij

AD is deterministic

… going back to the construction of AD

Ai can be determinized (AiD) (Alur, Madhusudan, STOC’04)

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

Conclusion

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, LICS’07)

Same results if we consider bounded context-switch words instead of bounded round words