- Automata Theory - Applications
- Automata Terminology
- Basics of String in Automata
- Set Theory for Automata
- Finite Sets and Infinite Sets
- Algebraic Operations on Sets
- Relations Sets in Automata Theory
- Graph and Tree in Automata Theory
- Transition Table in Automata
- What is Queue Automata?
- Compound Finite Automata
- Complementation Process in DFA
- Closure Properties in Automata
- Concatenation Process in DFA
- Language and Grammars
- Language and Grammar
- Grammars in Theory of Computation
- Language Generated by a Grammar
- Chomsky Classification of Grammars
- Context-Sensitive Languages
- Finite Automata
- What is Finite Automata?
- Finite Automata Types
- Applications of Finite Automata
- Limitations of Finite Automata
- Two-way Deterministic Finite Automata
- Deterministic Finite Automaton (DFA)
- Non-deterministic Finite Automaton (NFA)
- NDFA to DFA Conversion
- Equivalence of NFA and DFA
- Dead State in Finite Automata
- Minimization of DFA
- Automata Moore Machine
- Automata Mealy Machine
- Moore vs Mealy Machines
- Moore to Mealy Machine
- Mealy to Moore Machine
- Myhill–Nerode Theorem
- Mealy Machine for 1’s Complement
- Finite Automata Exercises
- Complement of DFA
- Regular Expressions
- Regular Expression in Automata
- Regular Expression Identities
- Applications of Regular Expression
- Regular Expressions vs Regular Grammar
- Kleene Closure in Automata
- Arden’s Theorem in Automata
- Convert Regular Expression to Finite Automata
- Conversion of Regular Expression to DFA
- Equivalence of Two Finite Automata
- Equivalence of Two Regular Expressions
- Convert Regular Expression to Regular Grammar
- Convert Regular Grammar to Finite Automata
- Pumping Lemma in Theory of Computation
- Pumping Lemma for Regular Grammar
- Pumping Lemma for Regular Expression
- Pumping Lemma for Regular Languages
- Applications of Pumping Lemma
- Closure Properties of Regular Set
- Closure Properties of Regular Language
- Decision Problems for Regular Languages
- Decision Problems for Automata and Grammars
- Conversion of Epsilon-NFA to DFA
- Regular Sets in Theory of Computation
- Context-Free Grammars
- Context-Free Grammars (CFG)
- Derivation Tree
- Parse Tree
- Ambiguity in Context-Free Grammar
- CFG vs Regular Grammar
- Applications of Context-Free Grammar
- Left Recursion and Left Factoring
- Closure Properties of Context Free Languages
- Simplifying Context Free Grammars
- Removal of Useless Symbols in CFG
- Removal Unit Production in CFG
- Removal of Null Productions in CFG
- Linear Grammar
- Chomsky Normal Form (CNF)
- Greibach Normal Form (GNF)
- Pumping Lemma for Context-Free Grammars
- Decision Problems of CFG
- Pushdown Automata
- Pushdown Automata (PDA)
- Pushdown Automata Acceptance
- Deterministic Pushdown Automata
- Non-deterministic Pushdown Automata
- Construction of PDA from CFG
- CFG Equivalent to PDA Conversion
- Pushdown Automata Graphical Notation
- Pushdown Automata and Parsing
- Two-stack Pushdown Automata
- Turing Machines
- Basics of Turing Machine (TM)
- Representation of Turing Machine
- Examples of Turing Machine
- Turing Machine Accepted Languages
- Variations of Turing Machine
- Multi-tape Turing Machine
- Multi-head Turing Machine
- Multitrack Turing Machine
- Non-Deterministic Turing Machine
- Semi-Infinite Tape Turing Machine
- K-dimensional Turing Machine
- Enumerator Turing Machine
- Universal Turing Machine
- Restricted Turing Machine
- Convert Regular Expression to Turing Machine
- Two-stack PDA and Turing Machine
- Turing Machine as Integer Function
- Post–Turing Machine
- Turing Machine for Addition
- Turing Machine for Copying Data
- Turing Machine as Comparator
- Turing Machine for Multiplication
- Turing Machine for Subtraction
- Modifications to Standard Turing Machine
- Linear-Bounded Automata (LBA)
- Church's Thesis for Turing Machine
- Recursively Enumerable Language
- Computability & Undecidability
- Turing Language Decidability
- Undecidable Languages
- Turing Machine and Grammar
- Kuroda Normal Form
- Converting Grammar to Kuroda Normal Form
- Decidability
- Undecidability
- Reducibility
- Halting Problem
- Turing Machine Halting Problem
- Rice's Theorem in Theory of Computation
- Post’s Correspondence Problem (PCP)
- Types of Functions
- Recursive Functions
- Injective Functions
- Surjective Function
- Bijective Function
- Partial Recursive Function
- Total Recursive Function
- Primitive Recursive Function
- μ Recursive Function
- Ackermann’s Function
- Russell’s Paradox
- Gödel Numbering
- Recursive Enumerations
- Kleene's Theorem
- Kleene's Recursion Theorem
- Advanced Concepts
- Matrix Grammars
- Probabilistic Finite Automata
- Cellular Automata
- Reduction of CFG
- Reduction Theorem
- Regular expression to ∈-NFA
- Quotient Operation
- Parikh’s Theorem
- Ladner’s Theorem
Undecidability in Automata Theory
As we know, "undecidability" is an important concept in Automata Theory. As per this concept, a problem cannot be solved by any algorithm, meaning there is no Turing Machine (TM) that can decide whether a given statement or problem is True or False. In this chapter, we will explain the basics of undecidability, including its key definitions and concepts, with examples for a better understanding.
Decidable Problems
We already learnt them, but here we will see a basic recap. A problem is considered decidable if there exists a Turing Machine that can provide a definitive "yes" or "no" answer for every possible input within a finite amount of time.
In other words, the Turing Machine must halt on every input, either accepting or rejecting it.
Example − The problem of determining whether a given string is accepted by a Deterministic Finite Automaton (DFA) is decidable. A Turing Machine can simulate the DFA and decide whether to accept or reject the string.
Recognizable Problems
A problem is recognizable or Turing Recognizable, if there exists a Turing Machine that will accept every string in the language but may either reject or run indefinitely on strings not in the language.
Unlike decidable problems, recognizable problems do not guarantee that the machine will halt for all inputs or not.
Undecidable Problems
A problem is considered undecidable if no Turing Machine can be constructed that will always halt with a correct "yes" or "no" answer for every input. In other words, there is no algorithm that can decide the problem in all cases.
Example − The Halting Problem, which asks whether a given Turing Machine will halt on a given input, is a classic example of an undecidable problem.
The Halting Problem
The Halting Problem is one of the most well-known undecidable problems. It asks whether a Turing Machine, for a given input, will halt (stop executing) or continue to run forever.
Proof of Undecidability
Suppose there is a Turing Machine H that can decide the Halting Problem. Now, construct a new Turing Machine D that does the following −
- For an input x, if H(x, x) (where x is both the machine and input) halts, D(x) will run indefinitely.
- If H(x, x) does not halt, then D(x) will halt.
This leads to a contradiction because if D runs on its own description, it both halts and does not halt. Therefore, H cannot exist, proving that the Halting Problem is undecidable.
The Post Correspondence Problem (PCP)
Another interesting problem is the Post Correspondence Problem; it is about finding a match between two lists of strings. We have two lists of strings, the question is whether there is a sequence of indices that, when applied to both lists, produces the same string.
Proof of Undecidability
The undecidability of PCP is proven by reducing it to another undecidable problem. Suppose there exists a solution to PCP, then it can be used to solve the Halting Problem, which we know is undecidable. Since solving the Halting Problem is impossible, PCP must also be undecidable.
The Blank Tape Halting Problem
Another example of undecidability is the Blank Tape Halting Problem, which asks whether a Turing Machine will halt when started with a blank tape.
Proof of Undecidability
The problem is reduced from the Halting Problem. If we could decide whether a Turing Machine halts on a blank tape, we could decide whether it halts on any arbitrary input. However, since the Halting Problem is undecidable, the Blank Tape Halting Problem is also undecidable.
The Virus Detection Problem
Another interesting problem is Virus Detection Problem which is also undecidable. It asks whether there exists an algorithm that can decide if a program is a virus or not.
Proof of Undecidability
If a virus detection program could perfectly identify whether a program halts (runs to completion) or runs indefinitely (spreads like a virus), it could be used to solve the Halting Problem. However, since the Halting Problem is undecidable, virus detection is also undecidable.
Implications of Undecidability
Following are some of the implications of undecidability −
- Limits of Computation − Undecidability shows the limits of what can be computed and what now. Not every problem can be solved by an algorithm, which means that some problems must be approached differently, often through approximation, heuristics, or probabilistic methods.
- Impact on Formal Verification − Formal verification tries to prove the correctness of algorithms and systems. Undecidability says that not all properties of a system can be verified, placing a boundary on what formal methods can achieve.
- Practical Considerations − Many problems are theoretically undecidable, but in practice, they may be solvable for specific instances or under certain constraints. This understanding leads to the development of practical algorithms that work in many, though not all, cases.
Conclusion
In this chapter, we explained the concept of undecidability. We saw the basics of decidable and undecidable problems and presented several examples to underline the idea of undecidability clearly.
Undecidable Problems include the Halting Problem, the Post Correspondence Problem, the Blank Tape Halting Problem, and the Virus Detection Problem. But all of them can be transferred to Halting Problems. These examples show the limitations of computation and the boundaries of algorithmic problem-solving.