Scope-Bounded Pushdown Languages

Salvatore La Torre Università degli Studi di Salerno

joint work with

Margherita Napoli Università degli Studi di Salerno

Gennaro Parlato University of Southampton

Multi-stack Pushdown Automata (MPA)• n stacks sharing a finite control

– states: s, , , ...........,

– transitions :• push one symbol onto stack i• pop one symbol from stack i• internal move: stacks stay unchanged, only

control location is altered

• input is from a one-way read-only tape

• model of concurrency– captures the control flow of concurrent programs

with shared memory and recursive procedure calls

1 2 n

Visible alphabets

• Alphabet is partitioned into: – calls (cause a push operation)– returns (cause a pop operation)– internals (stacks are not used)

• For n > 1 stacks, alphabet is also partitioned according to stacks– the stack to operate is uniquely identified by

the input symbol (it is visible in the input)

What visibility gains for MPA?• Stack usage is synchronized with the input

– parallel simulation of multiple runs, cross product construction, subset-like constructions

• 1-stack (VPL):– intersection and determinization– universality, inclusion and equality

• n-stacks (MVPL):– just intersection– emptiness is undecidable: the runs of MPA are visible!– checking for emptiness of MVPL equals to decide

reachability for MPA

Theme of the talk

• The formal language theory of visibly n-stack languages of k-scoped words (SMVPL)

Visible alphabet (retns) --st1: a, a’ st2: b, b’ internal: e

a e b a a’ a’ b a b’ e b’ a’

scope of matching relation over S: max number of S-contexts between matching call/retn– scope is 2 for GREEN and 3 for RED

matching relation (matching call/retn)

Conte

xt sp

litting

word is 3-scoped

A few observations....• Interest for restrictions of MPA mainly comes from

verification– bugs of concurrent programs are likely to occur within

few context-switches [Musuvathi-Qadeer, PLDI ‘07]– efficient sequentializations of multithreaded programs

[Lal-Reps,CAV’08]

• Robust automata theories are useful tools for other domains– Automata-theoretic approach to verification (model-

checking)– Pattern matching problems – …

• K-scoped visibly languages indeed form a robust class…

k-scoped MVPA

• Closure under Boolean operations

• Det./nondet. models are equivalent

• Decidable emptiness [La Torre-Napoli, CONCUR’11], inclusion, equality, and universality

• Logical characterization (MSO with matching relations)

• Parikh theorem

• Sequentializable: computations can be simulated with one stack (rearranging order of inputs)

• Decidable temporal logic model-checking [La Torre-Napoli,TCS’12] [Atig-Bouajjani-Kumar-Saivasan, ATVA’12]

More related work • Visibly pushdown languages [Alur-Madhusudan J.

ACM'09] [Melhorn ICALP'80]

Restricted MPAs:• Emptiness/reachability/closure properties

[Carotenuto et al. DLT’07] [Atig et al. DLT’08] [Seth,CAV’10] [LaTorre et al. LATIN'10] [LaTorre et al. MFCS'14]

• Model-checking [Atig, FSTTCS’10] [Bollig et al. MFCS’11] [Bollig et al. LICS’13] [Bansal-Demri, CSR’13]

• MSO of multiply nested words [Madhusudan-Parlato POPL'11] [Cyriac et al. CONCUR'12]

• ............

Rest of the talk

• Determinization construction

• Brief comparison with the known MPA classes of languages

• Conclusions

MVPL are nondeterministic• L = {(ab)i cjdi-j xjyi-j | i,j>0}

is inherently nondeterministic for MPA [La Torre-Madhusudan-Parlato, LICS’07]

– j is arbitrary and needs to be the same for both stacks

– a guess is needed when pushing both stacks

• L is not SMVPL – For any j, (ab)k cjdk-j xjyk-j is (k+1)-scoped

a b a b a b ……… a b c………… d x………… y

Determinization of SMPA

• Summaries of computations for SMPA – Linear interfaces– Switching masks

• PDA computing linear interfaces– linear interface automaton (LIA)

• Simulation of SMPA by deterministic composition of deterministic LIAs

(using switching masks)

View of runs by stacks

stack 1 stack 2 stack 3

g1q1

q2 r1q2 q3

g2

q3 q4

b1q4 q5

r2

q5

q6

Input word (contexts) w = g1 r1 g2 b1 r2 b2 r3 b3 r4 g3

Run (without stacks)

q1 q2 q3 q4 q5 q6 q7 q8 q9 q10 q11

g3q10 q11r3

q7

q8

r4

q9

q10

b2q6 q7

b3q8 q9

g1 g2 g3r1 r2 r3 r4b1 b2 b3

b1

b2

b3

r1

r2

r3

r4

g3

g2

g1

Linear Interface (LI)

stack 1 stack 2 stack 3

q1 q2 q2 q3

q3q4

q4q5

q5q6

q10q11 q7 q8

q9

q6 q7

q8q9

q10

• k-LI for a stack S just summarizes starting and ending control states for k consecutive contexts of S in a run (starting from stack S empty)

3-LI for stack 1 2-LI for stack 2

b1

b2

b3

r1

r2

r3

r4

g3

g2

g1

Switching Mask (SM)

stack 1 stack 2 stack 3

q1 q2 q2 q3

q3q4

q4q5

q5q6

q10q11 q7 q8

q9

q6 q7

q8q9

q10

• a tuple of LI, one for each stack• a function Nxt that links contexts of LI’s

Switching maskNxt function given by purple arrows

• Control state (h, M)h: current stack M: switching mask

• Move within a context: just update LI of stack h

• Es: M:

h=3

MPA transition from q5 to p5 on stack-3 symbol

(h is not changed)

Simulating MPA with SMs (1)

b1r1

g2

g1q1 q2 q2 q3

q3q4

q4q5

q1 q2 q2 q3

q3q4

q4 p5

q5

Simulating MPA with SMs (2)• Context-switch 1 (accumulated stack content

needed): add a new context to an existing LI

• Es: h=3, and MPA moves from q5 to p5 on a stack-2 symbol

M:

then h=2 and the SM is

b1r1

g2

g1q1 q2 q2 q3

q3q4

q4q5

q1 q2 q2 q3

q3q4

q4q5

q5 p5

Simulating MPA with SMs (3)• Context-switch 2 (accumulated stack content

no longer needed): start a new LI

• Es: h=3, and MPS moves from q5 to p5 on a stack-2 symbol

M:

then h=2 and the SM is

b1r1

g2

g1q1 q2 q2 q3

q3q4

q4q5

q1 q2

q3q4

q4q5q5 p5

PDA accumulating LIsGiven a PDA P over an alphabet , symbols ,#a k linear interface automaton (k-LIA) for P is a PDA s.t.

• input is over {,#} w11#w12#........#w1i1

w21#w22#........#w2i2 ....

• control states are h-LIs of P for hk

• on , simulates P on the last state of the LI

• on #, a new context is appended to the current LI (provided that it is a h-LI with hk-1)

• on , a new LI is started and stack is reset(a bottom-of-the-stack symbol is pushed onto the stack to avoid the use of previously pushed symbols)

(k)-LIs suffice for SMPA

Theorem. By restricting to k-scoped inputs,h-LIs with hk suffice to simulate the behavior of an MPA with switching masks

Thus, for each stack of an SMPA, we can restrict to k-LIAs

Determinization of SMPA (1) For an SMPA A

• construct the LIA Ah for each stack h

• construct Dh by determinizing each Ah as in [Alur-Madhusudan, STOC’04]

• construct the deterministic SMPA D (equiv. to A)– cross product of the Dh‘s

– parallel simulation of A with all the generated SMs (subset construction)

Determinization of SMPS (2)• a state of D is of the form (h, Q1,...,Qn, ) where

– h is the current stack– Q1,...,Qn is a state of the cross product

– is a set of switching masks

• within a context of stack h, D simulates Dh

(the Qh–component and all the switching masks in gets updated accordingly)

• on context-switching from stack h to stack i (a call/return of stack i is read), D simulates in parallel – Dh on either # or

– Di on the input symbol

the size of D is • exp in the size of A and • 2exp in the number of stacks

and the bound k

Comparisons

CSL

SMVPL

RMVPL[LPM10]

PMVPL[LMP07]

VPL[AM04]

OMVPL[BCCC96][MCP07][ABH08]

CFL

DCFL

TMVPL[LNP14]

Decision Problems

VPLCFLRMVPLSMVPLTMVPLPMVPLOMVPLCSL

Conclusions

• SMVPL form a robust theory of visibly languages(the largest among those closed under determinization)

• Sequentialization is nice for analysis purposes– Computations of MPA can be analyzed via

computations of PDA– used in software verification

• Scope-bounded words meaningfully extends to –words– Describe infinite on-going interaction among

different threads

Theory on infinite words?

• Little it is known on MPS over –words

• visibly pushdown Büchi automata [Alur-Madhusudan,J. ACM, 2009]

- the model is not determinizable

• emptiness for k-scoped Büchi MPA is PSPACE-complete [La Torre-Napoli,TCS’12]

• closure under union and intersection are simple