2.4 Equivalence and relationships between models
5 min read•july 30, 2024
Computational models are like different languages for describing the same thing: computation. From simple to powerful , each model has its strengths. Understanding how they relate helps us grasp the nature of computation itself.
The equivalence between models is a big deal. It means results in one model often apply to others, making our lives easier. It also reinforces key ideas like vs and helps us explore the limits of what's computable.
Computational power comparison
Abstract machine models
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- Computational models represent abstract machines designed to study computation
- Finite automata recognize regular languages
- recognize context-free languages
- Turing machines recognize recursively enumerable languages
- provides a functional model of computation
- posits any algorithm can be computed by a Turing machine
- Establishes Turing machines as a universal model of computation
- Provides a formal definition of algorithm and computability
- Hierarchy of formal languages corresponds to increasingly powerful models
- Regular languages (finite automata)
- Context-free languages (pushdown automata)
- Context-sensitive languages (linear bounded automata)
- Recursively enumerable languages (Turing machines)
Extended computational models
- Random access machines (RAMs) more closely resemble modern computers
- Allow constant-time memory access
- Can perform arithmetic operations in constant time
- Random access stored program (RASP) machines extend RAMs
- Store program instructions in memory
- Allow self-modifying code
- Quantum computation models offer potential advantages for certain problems
- incorporate quantum superposition
- allow manipulation of quantum bits (qubits)
- Provide exponential speedup for some algorithms (factoring, database search)
- represent distributed parallel computation
- Consist of grid of cells with simple update rules
- Can simulate Turing machines (Rule 110)
- Used to model complex systems (Conway's Game of Life)
- Neural networks model parallel distributed processing
- Inspired by biological neural systems
- Can approximate arbitrary functions through learning
- Used for pattern recognition and machine learning tasks
- extend Turing machines with access to undecidable problems
- Allow querying an oracle for answers to undecidable problems
- Used to study relative computability and complexity
Turing machine equivalence
Tape and determinism equivalence
- Computational models considered equivalent if they can simulate each other
- Must solve the same set of problems
- May differ in efficiency or ease of use
- Single-tape and proven equivalent
- Multi-tape operations simulated on single tape by encoding
- Demonstrates adding tapes does not increase computational power
- Deterministic and proven equivalent
- Subset construction method used for proof
- Non-deterministic choices simulated by exploring all possibilities
- Shows does not add power for Turing machines
- Differs from finite automata where non- increases power
Equivalence with abstract models
- Turing machines and lambda calculus proven equivalent
- Universal Turing machine constructed to interpret lambda expressions
- Lambda calculus can encode Turing machine computations
- Demonstrates equivalence of imperative and functional computation
- Turing machines and shown equivalent
- Tag system rules simulate Turing machine transitions
- Turing machine constructed to simulate tag system evolution
- Proves Turing- of simple rewriting systems
- Turing machines and cellular automata demonstrated equivalent
- Cellular automaton constructed to simulate universal Turing machine
- Turing machine can simulate cellular automaton evolution
- Shows parallel local rules can achieve universal computation
Equivalence with decision problems
- equivalent to Turing machine halting
- Reduction from halting problem to PCP and vice versa
- Demonstrates undecidability of seemingly simple matching problem
- proven equivalent to Turing machine computation
- Tile sets constructed to simulate Turing machine transitions
- Turing machine can simulate tiling process
- Shows connection between computation and aperiodic tilings
Computational model relationships
Automata hierarchy relationships
- Time and trade-offs exist between models
- Non-deterministic models potentially offer exponential time savings
- Space-bounded models reveal relationships between determinism and non-determinism
- One-way and illustrate input processing effects
- Two-way automata more powerful for some languages
- All regular languages recognized by one-way automata
- Pushdown automata extend finite automata with stack memory
- Recognize context-free languages (balanced parentheses)
- Increased power comes at cost of algorithmic complexity
- added to pushdown automata yield Turing-equivalent model
- Two stacks sufficient for universal computation
- Demonstrates power of unrestricted memory access
Probabilistic and parallel models
- Probabilistic models offer potential efficiency advantages
- allow randomized algorithms
- Maintain equivalent computational power to deterministic models
- Enable efficient solutions for some problems (primality testing)
- Parallel models explore relationship between parallelism and efficiency
- model shared-memory parallelism
- Circuit models capture parallel computation with bounded depth
- Parallel models can solve some problems faster, but not all
Space-bounded computation
- Space-bounded computation reveals model relationships
- relates deterministic and non-deterministic space
- Shows NSPACE() ⊆ DSPACE() for
- Implies = NPSPACE
- Linear bounded automata (LBAs) model context-sensitive languages
- Turing machines with tape limited to input length
- More powerful than pushdown automata, less than unrestricted Turing machines
- Logarithmic space computation studied for efficient algorithms
- define
- shows NL = coNL
Model equivalence implications
Complexity theory robustness
- Model equivalence allows choosing convenient model for proofs
- Results apply across equivalent models without loss of generality
- Simplifies analysis and proof techniques
- Robustness of complexity classes strengthened by model equivalence
- Classes like P, NP, PSPACE defined consistently across models
- Reinforces significance of these classes in characterizing difficulty
- Polynomial-time Church-Turing thesis extends original thesis
- Suggests any "reasonable" model simulatable by Turing machine with polynomial overhead
- Supports notion of polynomial-time as proxy for efficient computation
Simulation and transfer of results
- Universal simulators exist for equivalent models
- Allow efficient translation between model computations
- Complexity results transferred with only polynomial factors difference
- Model equivalence supports complexity-theoretic hierarchies
- Time hierarchy theorem holds across equivalent models
- Space hierarchy theorem applies to various space-bounded models
- Oracle machines and relativization demonstrate limits of proof techniques
- Some questions cannot be resolved using model-independent methods
- Motivates development of non-relativizing proof techniques
Implications for theory development
- Model equivalence crucial for developing lower bound techniques
- Allows focusing on inherent problem difficulty rather than model specifics
- Supports creation of general complexity measures (circuit complexity)
- Understanding equivalence helps identify barriers to proving separations
- Relativization barrier arises from model-independent simulation
- Natural proofs barrier relates to circuit lower bounds
- Equivalence informs design of new computational models
- Quantum computing developed with reference to classical models
- Motivates exploration of alternative computation paradigms (DNA computing)
Key Terms to Review (42)
Cellular Automata: Cellular automata are discrete, abstract computational systems that consist of a grid of cells, each of which can be in one of a finite number of states. The state of each cell in the grid evolves over time based on a set of rules that consider the states of neighboring cells. This simple yet powerful model can simulate complex behaviors and patterns, establishing important connections with other computational models and concepts.
Church-Turing Thesis: The Church-Turing Thesis posits that any computation that can be performed by an algorithm can also be executed by a Turing machine. This idea suggests a foundational equivalence between the intuitive notion of algorithmic computation and the formalized concept of computability established through Turing machines and lambda calculus. The thesis implies that deterministic and nondeterministic Turing machines, as well as other computational models, share this fundamental capability of computation.
Co-np: Co-NP is a complexity class that consists of the complements of decision problems in NP, meaning that a problem is in co-NP if its 'no' instances can be verified by a deterministic polynomial-time algorithm. This class is essential for understanding the broader landscape of computational complexity, especially in relation to NP and the hierarchy of complexity classes.
Completeness: Completeness refers to the property of a decision problem whereby if a problem is in a certain complexity class, it is as hard as the hardest problems in that class. This concept plays a vital role in understanding the relationships and boundaries of various complexity classes, indicating which problems can be efficiently solved and which cannot.
Cook's Theorem: Cook's Theorem states that the Boolean satisfiability problem (SAT) is NP-complete, meaning that it is as hard as the hardest problems in NP. This theorem establishes a foundational result in computational complexity theory, providing a benchmark for understanding the relationships among various complexity classes and the implications of problems that can be solved in polynomial time versus those that cannot.
Determinism: Determinism is the philosophical concept that all events, including moral choices, are determined completely by previously existing causes. In the context of computational complexity, it emphasizes how certain computational models behave in predictable ways based on their input and initial states. This idea connects to other essential concepts like computation power, time complexity, and problem-solving strategies, illustrating how different models can be compared based on their deterministic characteristics.
Deterministic Turing Machines: A Deterministic Turing Machine (DTM) is a theoretical model of computation that operates on an infinite tape using a finite set of states, with the property that for each state and tape symbol, there is exactly one action to take. This means that given the current state and the symbol it reads from the tape, the machine's next state, tape symbol to write, and direction to move the tape are uniquely determined. DTMs are significant in understanding computational complexity and serve as a foundational model for other computational structures.
Finite Automata: Finite automata are abstract computational models used to represent and manipulate a set of states to recognize patterns or accept certain strings of symbols. They play a critical role in the theory of computation, enabling the study of algorithmic processes and their complexities. Finite automata can be classified into deterministic (DFA) and non-deterministic (NFA) types, each having unique properties and applications in areas such as language recognition and parsing.
Hardness: Hardness in computational complexity refers to the difficulty of solving certain problems, particularly in the context of classifying problems based on how challenging they are to compute or verify. A problem is considered hard if it is at least as difficult as the hardest problems in a certain complexity class, indicating that if a solution for it can be efficiently found, then solutions for all problems in that class can also be efficiently solved. This idea connects to various models of computation and helps establish relationships between different classes, as well as grounding significant results like the Cook-Levin theorem.
Immerman–Szelepcsényi Theorem: The Immerman–Szelepcsényi theorem states that the complexity class NSPACE(log n) is equal to the complexity class P. This theorem establishes a crucial connection between space-bounded computation and polynomial-time computation, highlighting that problems solvable with logarithmic space can also be solved in polynomial time. It demonstrates that for certain computational models, particularly in the context of non-deterministic and deterministic machines, there exists a deep equivalence regarding their power in solving problems.
Lambda calculus: Lambda calculus is a formal system in mathematical logic and computer science for expressing computation based on function abstraction and application. It serves as a foundation for functional programming languages and is crucial in understanding how functions can be treated as first-class citizens in computation. By providing a minimalist framework, it allows for the exploration of concepts like variable binding and substitution, which relate to the equivalence of different computational models.
Linear Bounded Automata (LBA): Linear Bounded Automata (LBA) are a type of Turing machine that operates within a limited amount of tape, specifically proportional to the length of the input string. This restriction makes LBAs a powerful computational model, capable of recognizing a proper subset of context-sensitive languages. LBAs are crucial for understanding the boundaries of what can be computed with limited resources and play an essential role in the hierarchy of computational models.
Log-space reductions: Log-space reductions are a type of computational reduction that allows one problem to be transformed into another using logarithmic space in memory. This concept is vital when discussing the relationships between different computational models and complexity classes, as it helps determine how problems can be classified based on their computational requirements. By analyzing how a problem can be efficiently transformed, log-space reductions provide insight into the inherent difficulty and equivalence of problems within computational complexity.
Many-one reduction: Many-one reduction is a type of computational reduction where one problem can be transformed into another problem in such a way that a solution to the second problem directly provides a solution to the first. This concept is crucial for comparing the complexity of different problems and helps establish relationships between problems in terms of their difficulty, especially in classes like NP-completeness and PSPACE-completeness.
Multi-tape Turing machines: Multi-tape Turing machines are an extension of the standard Turing machine model, featuring multiple tapes and corresponding read/write heads, which allow for more efficient computation. With each tape capable of storing data independently, these machines can perform operations in parallel, effectively increasing their computational power compared to single-tape Turing machines. This enhanced functionality aids in exploring the equivalence and relationships between different computational models.
Multiple Stacks: Multiple stacks refer to the use of two or more stack data structures in computational models to manage information. This concept allows for greater complexity and flexibility in problem-solving by enabling operations on multiple stacks simultaneously, enhancing the ability to perform computations beyond what a single stack can achieve. Utilizing multiple stacks can lead to interesting relationships with other computational models, showcasing the diversity of approaches in computation.
Nl-completeness: NL-completeness refers to a classification of decision problems in the complexity class NP that can be solved by a non-deterministic Turing machine using logarithmic space. It highlights the most challenging problems in NL, indicating that if any NL-complete problem can be solved efficiently, then all problems in NL can also be solved efficiently. This concept is crucial for understanding the relationships and equivalences between various computational models.
Non-determinism: Non-determinism refers to a computational model where multiple possible outcomes or paths can occur from a given state without a predetermined sequence. This concept allows for the exploration of various computations simultaneously, leading to a broader understanding of problem-solving in computational theory. Non-determinism plays a crucial role in comparing different models of computation, classifying complex problems, and establishing relationships between decision problems and their solvability.
Non-deterministic Turing Machines: A non-deterministic Turing machine is a theoretical computational model that extends the capabilities of a standard Turing machine by allowing multiple possible actions for each state and input symbol. This means that, at any point during computation, the machine can 'choose' among different transitions, leading to potentially many different computational paths. This concept is crucial for understanding decision problems and the relationships between different computational models, especially in discussing complexity classes like P and NP.
NP: NP, or Nondeterministic Polynomial time, is a complexity class that represents the set of decision problems for which a solution can be verified in polynomial time by a deterministic Turing machine. This class is significant as it encompasses many important computational problems, including those that are computationally difficult, allowing us to explore the boundaries between what can be computed efficiently and what cannot.
Np-complete: NP-complete refers to a class of decision problems for which a solution can be verified quickly (in polynomial time) by a deterministic Turing machine, and every problem in NP can be reduced to it in polynomial time. This concept connects deeply with the nature of computational problems, growth rates of algorithms, and the relationships among various complexity classes.
Np-hardness: NP-hardness refers to a classification of problems that are at least as difficult as the hardest problems in NP, meaning that if any NP-hard problem can be solved in polynomial time, all NP problems can also be solved in polynomial time. This concept is crucial because it helps in understanding the boundaries of computational efficiency and the relationships between different computational problems. NP-hard problems do not need to be in NP themselves, and they are often used to demonstrate the complexity of other problems by showing reductions from known NP-hard problems.
One-way finite automata: One-way finite automata are theoretical computational models that process input strings in a linear fashion, moving in only one direction (from left to right) without the ability to return to previous input symbols. These automata are defined by a finite number of states and operate under a set of rules, transitioning between states based on the current input symbol and the state they are in. They play a crucial role in understanding the relationships between different computational models, especially in establishing the limits of computation power.
Oracle Machines: Oracle machines are theoretical computational models that extend the capabilities of standard Turing machines by incorporating an 'oracle,' which can instantly provide solutions to specific problems or decision questions. This concept allows researchers to explore the boundaries of computability and complexity, especially in analyzing the relationships between different complexity classes and the existence of NP-intermediate problems.
P: In computational complexity theory, 'p' refers to the class of decision problems that can be solved by a deterministic Turing machine in polynomial time. This class serves as a benchmark for comparing the efficiency of algorithms and lays the groundwork for understanding the relationships among various complexity classes.
P vs NP Problem: The P vs NP problem is a fundamental question in computer science that asks whether every problem whose solution can be quickly verified by a computer (NP) can also be quickly solved by a computer (P). This problem explores the relationship between two complexity classes, fundamentally impacting areas like algorithms, cryptography, and optimization.
Parallel Random Access Machines (PRAMs): Parallel Random Access Machines (PRAMs) are theoretical computing models that allow multiple processors to access a shared memory simultaneously. This model simplifies the design of parallel algorithms by abstracting away the complexities of real-world architectures, allowing researchers to focus on the efficiency of parallel computations and the relationships between different computational models.
Polynomial-time reduction: Polynomial-time reduction is a way to transform one problem into another in such a way that a solution to the second problem can be used to solve the first problem efficiently, specifically in polynomial time. This concept is fundamental in complexity theory as it helps establish relationships between problems, determining how hard they are relative to each other and identifying classes like P and NP.
Post's Correspondence Problem: Post's Correspondence Problem (PCP) is a decision problem that involves determining whether a sequence of strings can be matched to form the same string by concatenating pairs of tiles, where each tile consists of a pair of strings. This problem is significant in understanding computational models and their limitations, as it is an example of an undecidable problem, meaning there is no algorithm that can solve it for all possible instances. It also highlights important relationships between different computational models, revealing how certain problems transcend model boundaries.
Probabilistic Turing Machines: Probabilistic Turing Machines (PTMs) are theoretical models of computation that extend the classical Turing machine by incorporating randomness into their operation. These machines can make decisions based on random choices, allowing them to solve problems more efficiently than deterministic machines in some cases. The introduction of randomness leads to various complexity classes that describe the performance of these machines in terms of their ability to make correct decisions with high probability.
PSPACE: PSPACE is the complexity class representing decision problems that can be solved by a Turing machine using a polynomial amount of space. It encompasses problems that, while potentially requiring exponential time to solve, can be managed within a reasonable space constraint, showcasing the intricate balance between time and space resources in computation.
Pushdown Automata: A pushdown automaton (PDA) is a type of computational model that extends finite automata with an additional memory structure known as a stack. This allows PDAs to recognize context-free languages, which are more complex than the regular languages recognized by finite automata. PDAs are essential in understanding the relationships between different computational models, particularly when comparing their expressive power and the types of languages they can recognize.
Quantum circuits: Quantum circuits are a model of quantum computation that represent algorithms as a sequence of quantum gates acting on qubits. Each gate performs a specific operation, transforming the qubits' states, and the arrangement of these gates in a circuit reflects the logic of the algorithm being executed. Quantum circuits allow for the representation of complex quantum algorithms and are essential for understanding the power and limitations of quantum computing in relation to other computational models.
Quantum Turing Machines: Quantum Turing Machines (QTMs) are theoretical models of computation that extend classical Turing machines by incorporating quantum mechanics principles. They operate on qubits instead of bits, allowing them to perform multiple calculations simultaneously through superposition and entanglement. This enhanced computational power has significant implications for understanding the equivalence and relationships between different computational models, particularly in assessing the boundaries of what can be computed efficiently.
Savitch's Theorem: Savitch's Theorem states that any problem that can be solved by a nondeterministic Turing machine using space $$S(n)$$ can also be solved by a deterministic Turing machine using space $$S(n)^2$$. This theorem shows the relationship between nondeterministic space complexity and deterministic space complexity, highlighting that nondeterminism provides a significant advantage in terms of space efficiency.
Single-Tape Turing Machines: A single-tape Turing machine is a theoretical computing model that consists of a tape divided into cells, a head that reads and writes symbols, and a finite set of states. This model is crucial in computational complexity as it helps establish the foundational concepts of decidability and computability. The single-tape configuration allows for the study of the efficiency of algorithms and complexity classes by analyzing how the machine processes input and manages its tape during computation.
Space Complexity: Space complexity refers to the amount of memory space required by an algorithm to run as a function of the length of the input. It is a crucial concept in computational complexity theory, as it helps evaluate how efficiently an algorithm uses memory resources, which is essential for understanding its performance alongside time complexity.
Tag Systems: Tag systems are a type of formal computational model that consists of a finite set of symbols and a set of rules for manipulating those symbols, which can lead to complex computations. They operate by reading a symbol from the end of a string, applying rules that determine how to modify the string, and potentially adding or removing symbols based on the current symbol read. This manipulation process can be understood in relation to other computational models, particularly when considering their equivalence and relationships with more traditional models like Turing machines.
Time Complexity: Time complexity is a computational concept that measures the amount of time an algorithm takes to complete as a function of the length of the input. It helps in evaluating and comparing the efficiency of different algorithms, especially as the size of input grows. Understanding time complexity is crucial for identifying which algorithms can handle larger inputs efficiently and plays a key role in determining the feasibility of solutions to computational problems.
Turing machines: A Turing machine is a theoretical computational model that consists of an infinite tape, a tape head for reading and writing symbols, and a set of rules for determining the machine's actions based on the current state and the symbol being read. This concept is central to understanding computation, as it provides a framework for defining what it means for a function to be computable. Turing machines are crucial in comparing the capabilities of different computational models and exploring the limits of algorithmic processes.
Two-way finite automata: Two-way finite automata (2DFA) are computational models that extend traditional finite automata by allowing the read head to move both left and right on the input tape, rather than just moving in one direction. This added flexibility enables 2DFAs to recognize a broader class of languages compared to their one-way counterparts, particularly when analyzing palindromes and other symmetric structures. The ability to move back and forth makes them a powerful tool in understanding the relationships between different computational models.
Wang Tiles: Wang tiles are square tiles that are used to create aperiodic tilings, meaning they can cover a plane without repeating patterns. They are defined by colored edges, where tiles can only be placed adjacent to each other if the colors of their touching edges match. This concept plays a crucial role in understanding computational systems and their relationships, particularly in the context of tiling problems and formal languages.