- 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
Decision Problems for Automata and Grammars
In theory of computation, we have seen several automata and grammars. A Turing Machine (TM) uses recursive enumerable grammars. In recursion theory and computational complexity theory, a decision problem is a yes-or-no question on specified sets of inputs.
Decision problems are important in understanding the capabilities and limitations of computational models such as automata and grammars. Automata and grammars are mathematical models used to define formal languages. Here in this article we will see several decision problems and their details for a clear understanding.
Automata and Decision Problems
Finite automata are simple computational models which is used to recognize regular languages. They have finite number of states, transitions between states, and a set of input symbols.
A finite automaton processes an input string by moving through states according to the input symbols, and it accepts or rejects the string based on its final state.
Decision Problems for Finite Automata
Following are the decision problems for finite automata −
- Emptiness Problem − The emptiness problem checks whether the language recognized by a given finite automaton is empty or not, so whether it accepts any strings at all or not. This is a decision problem where the answer is "yes" if the language is empty and "no" otherwise.
- Universality Problem − The universality problem checks whether a given finite automaton accepts all possible strings over its input alphabet or not. This problem can also be framed as a decision problem where the answer is "yes" if the automaton accepts every possible string and "no" otherwise.
- Equivalence Problem − The equivalence problem for finite automata checks whether two given automata recognize the same language. This is another example of a decision problem where the answer is "yes" if the languages are the same and "no" if they are
Grammars and Decision Problems in Context-Free Grammars
Context-free grammars (CFGs) are used to generate context-free languages. A CFG consists of a set of production rules that describe how strings in the language can be generated from a start symbol.
Decision Problems for Context-Free Grammars
Following are the decision problems for context-free grammars −
- Membership Problem − The membership problem checks whether a given string belongs to the language generated by a specific context-free grammar or not. This is a decision problem with the answer "yes" if the string can be generated by the grammar and "no" otherwise.
- Emptiness Problem − This is similar to finite automata, the emptiness problem for CFGs checks whether the language generated by the grammar is empty or not. The answer to this decision problem is "yes" if the language is empty and "no" otherwise.
- Equivalence Problem − The equivalence problem for CFGs checks whether two grammars generate the same language or not. This problem is more complex than the equivalence problem for finite automata. The answer is "yes" if the languages are identical and "no" if they differ.
Examples of Decision Problems in Automata and Grammars
Go through the following examples to get a better understanding of how decision problems matter in automata and grammars.
Example 1: Membership Problem in Finite Automata
Consider a finite automaton A that recognizes the language of strings over the alphabet {0,1} that end in "01." The decision problem here is to determine whether a given string, say "1101", belongs to the language recognized by A.
Example 2: Emptiness Problem in Context-Free Grammars
Let G be a context-free grammar with production rules that generate strings containing an equal number of 'a's and 'b's. The emptiness problem asks whether G generates any strings at all.
If the production rules are such that no string can be generated (for example, if there is a missing rule to produce 'b'), then the answer to this decision problem is "yes," indicating that the language is empty.
Example 3: Universality Problem in Finite Automata
Consider a finite automaton B that recognizes all strings over the alphabet {0,1}. The universality problem asks whether B accepts every possible string over {0,1}. Since B is designed to accept all strings, the answer to this decision problem is "yes."
Example 4: Equivalence Problem in Context-Free Grammars
Suppose we have two context-free grammars, G1 and G2, where G1 generates strings of balanced parentheses, and G2 generates strings that match the same pattern but with a different set of production rules.
The equivalence problem asks whether the languages generated by G1 and G2 are the same. If both grammars generate identical strings, the answer to this decision problem is "yes". However, if there is any difference, the answer is "no".
Computational Complexity of Decision Problems
Here it categorises decidable decision problems by how difficult they are to solve. "Difficult," in this sense, is described in terms of the computational resources needed by the most efficient algorithm for a certain problem.
For automata and grammars, decision problems are often classified into different complexity classes, such as P, NP, and PSPACE.
- Class P − The complexity class P is the set of decision problems that can be solved by a deterministic Turing machine in polynomial time. For example, the membership problem for finite automata is in class P, as it can be solved efficiently.
- Class NP − The class NP includes decision problems for which a solution can be verified in polynomial time. Some decision problems for context-free grammars, such as the membership problem, are in NP.
- Class PSPACE − The class PSPACE includes decision problems that can be solved using a polynomial amount of memory. Certain decision problems for grammars, like the equivalence problem, may fall into this category due to their complexity.
Undecidable Decision Problems
Some decision problems in the context of automata and grammars are undecidable, so no algorithm can solve them for all possible inputs.
For example, the equivalence problem for context-free grammars is undecidable in general, as there is no algorithm that can determine whether two arbitrary CFGs generate the same language.
Conclusion
In this chapter, we covered the concept of decision problems in automata theory and grammars. They involve yes-or-no questions that determine properties such as language membership, emptiness, and equivalence. We need to understand the decision problems which also help in classifying languages and automata into different complexity classes.