Algorithmic Software Verification

IV. Regularity of configs of a PDA; modeling programs using PDA

Pushdown automata(, Q, q_in, E, F)

Q – finite set of states

q_in -- initial state

-- stack alphabet

E: { q –a, push d q’ a , e

q –a, pop d q’

q –a q’ }

ConfigurationsA = (, Q, q_in, E, F)

Configuration of A: (q, w) -- q Q, w *

Configuration graph of A:

(q, w) (q’, w) if there is a trans q –a q’

(q, w) (q’, w d) if there is a trans q –a,push dq’

(q, w d) (q’, w) if there is a trans q –a,pop dq’

Note that this is an infinite graph.

A config (q,w) is reachable if there is a path from (q_in, ε)in the config graph.

ConfigurationsA = (, Q, q_in, E, F)

Let CA = { wq | (q,w) is a reachable config }

We want to show that CA is regular.

Also, the size of the automaton accepting CA

is cubic in |A|.

Automaton accepting CA

A = (, Q, q_in, E, F)

We will construct automaton B accepting CA

Let states of CA be Q.

Alphabet of CA is ; initial state is q_in; final states: Q

Edges: The smallest set of edges such that: If q –a q’ then q – ε q’ must belong to

B If q –a,push dq’ then q –d q’ must belong

to B If q –a, pop d q’ and q’’ --a* q

then q’’ – ε q’ must belong to B

Constructing B in O(n^3) time A = (, Q, q_in, E, F) stack:= emptyset; hash:= emptyset; for every q in Q, put (q,q) on stack for every q,q’, set (q, q’).c-direct := emptyset; (q, q’).c-trans := emptyset; for every transitions u –push a q, q –pop a v put (u,v) on (q,q’).c-direct for every q, q’ in Q, q diff from q’, for every t in Q, put [ (q’, t) (q, t) ] and [ (t, q) (t, q’) ] in (q,q’).c-trans while stack is not empty e:= pop(stack) put e in hash (if e is not already in hash) transfer e.c-direct into stack for every [ f g ] in e.c-trans if f is in hash, then put g on stack else put g on f.c-direct

Constructing B in O(n^3) time Stack – unprocessed epsilon edges

Hash -- processed epsilon edges

(q,q’).c-direct has (u,v):

if (q, q’) is an eps edge, then (u,v) is an eps edge too.

(q,q’).c-trans has (u1, v1) (u2, v2):

Given eps transition (q,q’), if (u1, v1) is

an epsilon edge, then (u2, v2) is also

an epsilon edge.

Emptiness L(A) is nonempty

if and only if there is some reachable configuration

of the form (q_f, w), q_f in F

if and only if q_f is reachable from q_in

Hence L(A) emptiness can be decided in O(n^3) time

Homework II 1. Take the translation of PDA emptiness

to finite state games. This works only

if emptiness of PDA is with respect

to empty stack: i.e. PDA must read a word and at

the end have an empty stack in order to accept the

word.

Modify the translation such that it works for PDA that accept by finite state.

[ You are not allowed to transform the PDA into one

accepting by empty stack ]

Homework II 2. Take any reasonable non-empty PDA.

Translate the PDA emptiness problem into a game,

solve the game and find the winning strategy.

Then, give the word(s) that are accepted by the PDA

which correspond to the winning strategy you have

chosen.

(PDA should be reasonable – should not be regular!)