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Decidability CS215, Lecture 7 c 1999 Mitsunori Ogihara Decidable Problems About Regular Languages The Acceptance Problem for DFA Define
^
is a DFA that accepts input string . Here we assume a fixed encoding scheme for
and . Theorem.
is decidable. Proof A Turing machine can, given an input
, try to decode
into an NFA
and a string
. If the decoding is successful then it can test whether
accepts by simulating
on . CS215, Lecture 7 c 1999 Mitsunori Ogihara The Acceptance Problem for NFA Define
^
is an NFA that accepts input string . Theorem.
^
is decidable. Proof Given an input
, try to decode
into an NFA
and a string . If “successful” then:
- Convert
to a DFA
.
- Run the machine for
^
on
. If the machine accepts, then accept ; otherwise reject . CS215, Lecture 7 c 1999 Mitsunori Ogihara 3 The Acceptance Problem for Regular Exp. Define
is a regular expression that produces . Theorem.
is decidable. Proof Given an input
, try to decode
into a regular expression
and a string
. If “successful” then: 1. Convert
to a DFA
.
- Run the machine for
on
. If the machine accepts, then accept ; otherwise reject . CS215, Lecture 7 c 1999 Mitsunori Ogihara 4
The Emptiness Problem for DFA Define
^
is a DFA that accepts no string . Theorem.
^
is decidable. Proof Given an input
, try to decode a DFA
out of
. If “successful” then: 1. Mark the start state of
.
Repeat until no new states are marked:
Mark any unmarked state that has a transition from a marked state
- Accept if no final state is marked ; reject otherwise. CS215, Lecture 7 c 1999 Mitsunori Ogihara The Equivalence Problem for DFA Define
^
and
are DFA that accept the same language . Theorem.
^
is decidable. Proof Given a string
, try to decode
into a pair of DFAs
and . If “successful” then construct a DFA
that accepts the symmetric difference of
^
and
,
^
^
^ ,
and test the emptiness of
.
CS215, Lecture 7 c 1999 Mitsunori Ogihara The Acceptance Problem for CFG Define
is a CFG that generates . Theorem.
is decidable. Proof Given an input
, try to decode
into a CFG
and a string . If “successful” then:
- Convert
to an equivalent Chomsky normal form grammar
%^ &
.
- List all derivations with
steps, where
.
- If any of the listed derivations generate , then accept ; otherwise, reject. CS215, Lecture 7 c 1999 Mitsunori Ogihara 7 The Emptiness Problem for CFG Define
is a CFG such that
. Theorem.
is decidable. Proof Given
, first try to decode a grammar
out of it. If “pass” then test the ability of generating terminal strings:
Mark all the terminals .
- Repeat the following until no new symbols are marked:
Mark any variables
with a production
such that all symbols in are marked .
Accept if the start symbol is marked; reject otherwise. CS215, Lecture 7 c 1999 Mitsunori Ogihara 8
