Equivalence of Computational Models

We’ve shown how to prove closure of a class of languages decidable by a certain model under an operation $\texttt{op}$:
- Given instances $M_1$ and $M_2$ of the model, we can construct a new instance $M$ of the same model such that: \[L(M) = L(M_1)\; \texttt{op}\; L(M_2)\]
- We proved DFA-decidable languages are closed under complement, union, and intersection
- We proved NFA-decidable languages are closed under union, concatenation, and Kleene star
We now turn to comparing the computational power of different models.
- If any regex can be simulated by an NFA, then NFAs are at least as powerful as regexes. \[ \text{"For all regexes } R, \text{ there exists an NFA } N \text{ such that } L(R) = L(N)\text{"} \]
- If any NFA can be simulated by a regex, then regexes are at least as powerful as NFAs.
- If both of these are true (they are!), then NFAs and regexes have equivalent power.

Simple Example: NFAs Are at Least as Powerful as DFAs

NFAs are generalizations of DFAs, so any DFA can be trivially simulated by an NFA.
A DFA is just an NFA with:
- no $\emptystring$ transitions and
- exactly one transition for each state and symbol pair.
To prove this formally:
- Let $D = (Q_D, \Sigma, \delta_D, s_D, F_D)$ be a DFA
- We can construct an equivalent NFA $N = (Q_N, \Sigma, \Delta_N, s_N, F_N)$ such that:
  - $Q_N = Q_D$
  - $\Delta_N(q, a) = \fragment{\{\delta_D(q, a)\}}$ for all $q \in Q_N$ and $a \in \Sigma$
  - $s_N = s_D$
  - $F_N = F_D$

Simulating NFAs

Trying to simulate an NFA with a DFA is less straight-forward.
Warm-up: which of the following are equivalent to plain NFAs?
- MultistartNFA: an NFA allowing more than one start state
- SingleAcceptNFA: an NFA allowing only a single accept state
- InjectiveNFA: an NFA for which the destination of each transition rule is either $\emptyset$ or a singleton $\{q_i\}$
- NoLoopNFA: an NFA for which no transition rule is of the form $q \stackrel{a}{\rightarrow} q$ for any $(q, a) \in Q \times \Sigma$
- NoEpsilonNFA: an NFA with no $\emptystring$ transitions.

Recall: Operation of an NFA

Example processing the string 010110:

010110

Begin in the state(s) $\{q_1\}$.
Read 0 and transition to state(s) $\{q_1\}$.

010110
Read 1 and transition to state(s) $\{q_1, q_2, q_3\}$.

010110
Read 0 and transition to state(s) $\{q_1, q_3\}$.

010110
Read 1 and transition to state(s) $\{q_1, q_2, q_3, q_4\}$.

010110
Read 1 and transition to state(s) $\{q_1, q_2, q_3, q_4\}$.

010110
Read 0 and transition to state(s) $\{q_1, q_3, q_4\}$.

010110
Since an accept state is highlighted, the NFA accepts the string.

What would a DFA need to keep track of in order to simulate this NFA?

Simulating NFAs

To simulate an NFA with $|Q|$ states operating on a string $w$, how many distinct sets of states must we keep track of in the worst case?

$|Q|^2$
$2^{|Q|}$
$|w|^2$
$2^{|w|}$

Example NFA to DFA Conversion

data/images/nfa/n4_simple.svg — Example NFA (no $\emptystring$ transitions)

Example NFA to DFA Conversion

data/images/nfa/n4.svg — Example NFA (with $\emptystring$ transitions)

This process usually does not generate “optimal” DFAs.
In this example, we can easily remove “dead” states.
For proving equivalence, we don’t care about optimality!

Example NFA to DFA Conversion

This process usually does not generate “optimal” DFAs.
In this example, we can easily remove “dead” states.
For proving equivalence, we don’t care about optimality!

There exist $k$-state NFAs for which the smallest equivalent DFA has $2^k$ states!
Example near this bound:
$k^\text{th}$-to-last symbol NFA

NFA to DFA Conversion: “Subset Construction”

Theorem: If a language is NFA-decidable, then it is also DFA-decidable.

Proof (no $\emptystring$ transitions)

Let $N = (Q_N, \Sigma, \Delta_N, s_N, F_N)$ be an NFA without $\emptystring$ transitions.
Construct $D = (Q_D, \Sigma, \delta_D, s_D, F_D)$ as follows:
- $Q_D = \powerset(Q_N)$
- $s_D = \{s_N\}$
- $F_D = \fragment{\setbuild{R \in Q_D}{\fragment{R \cap F_N \neq \emptyset}}}$
- For all $R \in Q_D$ and $b \in \Sigma$: \[\delta_D(R, b) = \bigcup_{q \in R} \Delta_N(q, b)\]
What about $\emptystring$ transitions? Convention:
- After reading each symbol, jump to all states reachable by any number of $\emptystring$ transitions.
- Also do this for the start state.

Following Epsilon Transitions

For any $R \subseteq Q_N$, define: \[ E(R) = \setbuild{q \in Q_N}{q \text{ is reachable from some $r \in R$ by a sequence of 0 or more $\emptystring$ transitions}} \]

data/images/nfa/multiple-eps-transitions.svg

$E(\{5\}) = \fragment{\{4, 5\}}$
$E(\{6\}) = \fragment{\{6\}}$
$E(\{1\}) = \fragment{\{1, 3, 4, 5, 6\}}$
$E(\{2\}) = \fragment{\{2\}}$
$E(\{3\}) = \fragment{\{3, 4, 5\}}$
$E(\{2, 5\}) = \fragment{\{2, 4, 5\}}$

NFA to DFA Conversion: “Subset Construction”

Theorem: If a language is NFA-decidable, then it is also DFA-decidable.

Proof

Let $N = (Q_N, \Sigma, \Delta_N, s_N, F_N)$ be an NFA.
Construct $D = (Q_D, \Sigma, \delta_D, s_D, F_D)$ as follows:
- $Q_D = \powerset(Q_N)$
- $\color{red} s_D = \fragment{E(\{s_N\})}$
- $F_D = \setbuild{R \in Q_D}{R \cap F_N \neq \emptyset}$
- For all $R \in Q_D$ and $b \in \Sigma$: \[\color{red} \delta_D(R, b) = \fragment{E\left(\bigcup_{q \in R} \Delta_N(q, b)\right)} \fragment{= \bigcup_{q \in R} E(\Delta_N(q, b))}\]

This convention for handling $\emptystring$ transitions is not the only possible one!

Corollary: a language is DFA-decidable if and only if it is NFA-decidable.

Title

Equivalence of Computational Models

Simple Example: NFAs Are at Least as Powerful as DFAs

Simulating NFAs

Recall: Operation of an NFA

Simulating NFAs

Example NFA to DFA Conversion

Example NFA to DFA Conversion

Example NFA to DFA Conversion

NFA to DFA Conversion: “Subset Construction”

Following Epsilon Transitions

NFA to DFA Conversion: “Subset Construction”