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Turing Machines, Decidability and Undecidability Short Notes for GATE CS

Turing Machines, Decidability and Undecidability short notes for GATE CS: 12 study cards covering concepts, formulas, shortcuts and exam traps, plus solved pr

turing machines decidability and undecidability short notes

Last-Minute Revision: Perfect Shuffle

Essential Cheat Sheet Keep these rules locked in for the exam. 1. Core Mechanics Array length must be even (2 n). Splits into two halves of size n . Interleaves starting with the first half (Out-Shuffle). First element (index 0) and last element (index 2 n - 1) are fixed . 2. Position Formulas (0-based indexing) Single shuffle: f (i) = 2 i (mod 2 n - 1) (for 0 < i < 2 n - 1) Piecewise alternative: First half (i < n): moves to 2 i Second half (i \geq n): moves to 2 (i - n) + 1 3. Multiple Shuffles (k times) Position after k shuffles: i_{k} = (i_{0} \times 2^{k}) (mod 2 n - 1) 4. Restoration (Cycle Length) Minimum shuffles to restore original order: Smallest k where 2^{k} \equiv 1 (mod 2 n - 1) . 5. Critical Traps to Avoid Applying 0-based formulas to 1-based problems. Assuming the formula works for odd-length arrays. Confusing Out-Shuffle (starts with first half) with In-Shuffle (starts with second half).

Formula Matrix: Class Rules and Closure

#sc-s001 .grid { display: grid; grid-template-columns: 1fr 1fr; gap: 0.25rem 0.75rem; } #sc-s001 .cell { padding: 0.4rem 0; border-bottom: 1px solid var(--color-border); font-size: 0.85rem; line-height: 1.3; } #sc-s001 .cell.cond { color: var(--color-muted); font-size: 0.8rem; } #sc-s001 .cell.rule { color: var(--color-text); font-weight: 600; } #sc-s001 .amber { color: var(--color-primary); } L recursive ⟺ L and \overline{L} both RE Recursive closed \cup, \cap, \overline{x}, \cdot,^{*} RE closed \cup, \cap, \cdot,^{*} RE NOT closed \overline{x}, ∖ Subset of recursive \neq \Rightarrow recursive Superset of RE \neq \Rightarrow RE

Triggers: If You See This, Do This

#sc-s002 table { width: 100%; border-collapse: collapse; font-size: 0.85rem; } #sc-s002 th { text-align: left; padding: 0.4rem 0; border-bottom: 2px solid var(--color-primary); color: var(--color-primary); font-size: 0.75rem; text-transform: uppercase; letter-spacing: 0.05em; } #sc-s002 td { padding: 0.4rem 0; border-bottom: 1px solid var(--color-border); vertical-align: top; line-height: 1.3; } #sc-s002 td.trigger { color: var(--color-muted); } #sc-s002 td.action { color: var(--color-text); font-weight: 600; } Stem Signal First Move L and \overline{L} are RE L is recursive L is RE, \overline{L} not RE L is RE, not recursive "Every subset of L..." Check Σ^{*} counterexample TM halts on all inputs Language is recursive TM loops on non-members Language is RE, not recursive Complement of CFL Recursive (CFL \subset Recursive) Intersection of regular Regular ⟹ DCFL ⟹ Recursive

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Turing Machines, Decidability and Undecidability Short Notes for GATE CS

Turing Machines, Decidability and Undecidability short notes for GATE CS: 12 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

Last-Minute Revision: Perfect Shuffle

Essential Cheat Sheet

Keep these rules locked in for the exam.

1. Core Mechanics

Array length must be even ( $2 n$ ).
Splits into two halves of size $n$ .
Interleaves starting with the first half (Out-Shuffle).
First element (index 0) and last element (index $2 n - 1$ ) are fixed.

2. Position Formulas (0-based indexing)

Single shuffle: $f (i) = 2 i (mod 2 n - 1)$ (for $0 < i < 2 n - 1$ )
Piecewise alternative:
- First half ( $i < n$ ): moves to $2 i$
- Second half ( $i \geq n$ ): moves to $2 (i - n) + 1$

3. Multiple Shuffles ( $k$ times)

Position after $k$ shuffles: $i_{k} = (i_{0} \times 2^{k}) (mod 2 n - 1)$

4. Restoration (Cycle Length)

Minimum shuffles to restore original order: Smallest $k$ where $2^{k} \equiv 1 (mod 2 n - 1)$ .

5. Critical Traps to Avoid

Applying 0-based formulas to 1-based problems.
Assuming the formula works for odd-length arrays.
Confusing Out-Shuffle (starts with first half) with In-Shuffle (starts with second half).

Formula Matrix: Class Rules and Closure

L recursive

⟺

L and

\overline{L}

both RE

Recursive closed

\cup, \cap, \overline{x}, \cdot,^{*}

RE closed

\cup, \cap, \cdot,^{*}

RE NOT closed

\overline{x}, ∖

Subset of recursive

\neq \Rightarrow

recursive

Superset of RE

\neq \Rightarrow

Triggers: If You See This, Do This

Stem Signal	First Move
L and $\overline{L}$ are RE	L is recursive
L is RE, $\overline{L}$ not RE	L is RE, not recursive
"Every subset of L..."	Check $Σ^{*}$ counterexample
TM halts on all inputs	Language is recursive
TM loops on non-members	Language is RE, not recursive
Complement of CFL	Recursive (CFL $\subset$ Recursive)
Intersection of regular	Regular $⟹$ DCFL $⟹$ Recursive

Traps and Invariants

RE closed under complement (flipping accept/reject)

Check: Flipped recognizer loops on non-members.

Subsets inherit parent class (e.g., subset of recursive is recursive)

Check:

Σ^{*}

is recursive but contains non-RE subsets.

RE implies recursive (conflating recognizer with decider)

Check: Bidirectional gate fails if complement is not RE.

turing machines decidability and undecidability short notes

Last-Minute Revision: Perfect Shuffle

Essential Cheat Sheet

Formula Matrix: Class Rules and Closure

Triggers: If You See This, Do This

Login to view the complete short notes

Personalised first. High quality throughout.

Built around you, not around a syllabus PDF

Revision that hits your weak spots

Questions calibrated to the real exam

Notes written for recall, not for volume

One place for everything

Honest progress

Unlock the whole course

Turing Machines, Decidability and Undecidability Short Notes for GATE CS

Last-Minute Revision: Perfect Shuffle

Essential Cheat Sheet

Formula Matrix: Class Rules and Closure

Triggers: If You See This, Do This

Traps and Invariants

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