- 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
Limitations of Finite Automata
Simple finite machines can be formed through finite automata. In this chapter, we will see some of the limitations of finite automata and highlight some cases where the simple finite automata fails and where we need to use other more complex automata.
Finite Automata and its Features
Though we will discuss the limitations of finite automata, but we must recap what this exactly is. As we know the finite automaton is a computational device that reads input symbols one at a time and changes its internal state based on the input.
It consists of a finite set of states, input symbols, transition function, initial state, and final states. It has a wide range of applications include logic analysis, communication systems, text processing, and hardware modelling, etc.
Along with their versatile applications, the finite automata have inherent limitations that restrict their applicability to certain classes of problems. Let us discuss them one by one in the following few sections.
Computational Power
With respect to computation, there are certain limitations of finite automata. Finite automata are connected to regular languages, which is the simplest class of formal languages in the Chomsky hierarchy. These languages can be recognized by finite automata or described by regular expressions.
They include useful patterns like strings with even 0s, strings ending with "ing", and valid email addresses, etc. But for a little complex problem like common patterns matching as we see in balanced parentheses checking, arbitrary palindromes, and strings with equal symbols. The finite automata fail due to its finite memory.
Comparison with More Powerful Computational Models
To understand the limitations of finite automata in a better way, it's needed to compare them with more powerful computational models −
- Pushdown Automata (PDA) − These are extending version of finite automata with a stack inside them. It allows recognizing context-free languages. PDAs can handle nested structures, making them suitable for tasks like parsing programming languages.
- Linear Bounded Automata (LBA) − More powerful automata after PDA are LBA. LBAs can recognize context-sensitive languages, offering more power than finite automata while still having some restrictions.
- Turing Machines − In Chomskys hierarchy TM are the most powerful model. Turing machines have an infinite tape for memory and can compute any algorithmically solvable problem.
Limitations of Finite Automata Considering Memory Constraints
The main problem for finite automata is that finite automata have limited memory beyond the current state. This means they can only remember a limited amount of information about the input they've seen.
Once transitioning to a new state, they lose all information except for the current one. This makes it impossible for finite automata to handle tasks that require tracking an unbounded amount of information. Due to this issue they are not capable of counting unbounded quantities.
Inability to Count Unbounded Quantities
Finite automaton can be designed to count up to any specific number (finite), it cannot count to an arbitrary limit determined by the input.
For example −
- A finite automaton can be constructed to accept strings with exactly 100, 0s.
- However, no finite automaton can be accepting strings of the form anbn for all values of n.
Let us see some other examples where finite automata fails.
Examples of Languages beyond Finite Automata Capabilities
Let's consider the following examples of languages that cannot be recognized by finite automata −
- {anbn | n ≥ 0} − Strings with an equal number of 'a's followed by 'b's. This requires counting and comparing the number of 'a's and 'b's, which is impossible with finite memory.
- {ww | w is any string} − Strings that consist of two identical halves. Recognizing this language would require remembering the entire first half of the string to compare it with the second half.
- {anbmcn | n, m ≥ 0} − Strings where the number of 'a's matches the number of 'c's, with any number of 'b's in between. This language requires counting and remembering two separate quantities across an arbitrary distance.
Conclusion
Finite automata is a powerful concept used to model and analyze computational processes, however it has some limitations.
Finite automata suffers from finite memory problem and it cannot keep anything in memory except the current state. We demonstrated this with the help of a set of examples where the finite automata fails. In all of these examples one thing is common, there are some common pattern that should be remembered for later processing. The finite automata fails at that moment.