Lecture 09

Finite Languages $$\subset$$ Regular Languages $$\subset$$ Context-free Languages $$\subset$$ Decidable Languages $$\subset$$ Recognizable Languages

Power of Turning Machines

Theorem Every regular languages is decidable.

Proof Let $$M=\{Q,\Sigma,\delta,q_0,F\}$$ be a DFA that recognizes the regular language $$L$$.

Let $$T=\{Q',\Sigma',\Gamma',\delta',q_0',q_{acc},q_{rej}\}$$ be the Turning Machine by

• $$Q'= Q\cup\{q_{acc},q_{rej}\}$$
• $$q_0'=q_0$$
• $$\Sigma'=\Sigma$$
• $$\Gamma'=\Sigma\cup\{\_\}$$
• $$\delta'(q,a)=(\delta(q,a),a,R)$$ for every $$a\in\Sigma$$.
• $$\delta'(q,\_)=(q_{rej},\_,R)$$ if $$q\not\in\Sigma$$.
• $$\delta'(q,\_)=(q_{acc},\_,R)$$ if $$q\in\Sigma$$.

Theorem Every context-free languages is decidable.

Proof Wrong approach: simulate a PDA.

Right approach: simulate/implement the CFG that generates the CFL $$L$$.

Key Idea Implement a CFG in Chomsky Normal form for $$L$$. Let you enumerate all possible strings of length equal to the input generated by the grammar in finite time.

Variants of Turning Machines

VarTM

Definition Let VarTM be the extension of TM model where we have a finite number of variables that take values in $$\Gamma$$.

Theorem Every language decidable by VarTMs is also decidable by TMs.

Proof $$Q'=Q\times\Gamma^m$$ where $$m$$ is the number of variables in the VarTM.

SubTM

Definition Let SubTM be the extension of TM model where we can call other TMs and run them as a black box.

Theorem Every language decidable by SubTMs is also decidable by TMs.

Proof StackTM

Definition Let StackTM be the extension of TM model where the TMs have a stack.

Theorem Every language decidable by StackTMs is also decidable by TMs.

MultiTM

Definition Let MultiTM be the extension of TM model where TMs have a constant number $$k$$ of tapes.

Theorem Every language decidable by MultiTMs is also decidable by TMs.

Non-deterministic Turning Machines

Definition A non-deterministic TM is a 7-tuple $$(Q,\Sigma,\Gamma,\delta,q_0,q_{acc},q_{rej})$$ where $$\delta:Q\times\Gamma\rightarrow\mathcal{P}(Q\times\Gamma\times\{L,R\})$$

Definition The non-deterministic TM $$M$$ decides $$L$$ if

1. For every $$x\in L$$, at least one path leads to the accept state.
2. For every $$x\not\in L$$, no path leads to the accept states.
3. For every $$x\in\Sigma^*$$, all possible paths terminate (at $$q_{acc}$$ or $$q_{rej}$$ in a dead end).

Definition The non-deterministic TM $$M$$ recognizes $$L$$ if

1. For every $$x\in L$$, at least one path leads to the accept state.
2. For every $$x\not\in L$$, no path leads to the accept states.

Theorem If $$L$$ is decidable (recognizable) by a NTM, then it is also decidable(recognizable) by a TM.