9.1 The diagonalization technique
4 min read•july 30, 2024
is a powerful proof technique in complexity theory, originating from Cantor's work on infinite sets. It's used to show the existence of problems unsolvable within certain resource bounds, exploiting the countability of Turing machines and uncountability of languages.
This technique is crucial for proving hierarchy theorems and establishing separations between complexity classes. It's applied to demonstrate the , , and even the existence of undecidable problems like the . However, it has limitations when dealing with closely related classes.
Diagonalization in complexity theory
Origins and fundamental concepts
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- Diagonalization proof technique developed by Georg Cantor demonstrates some infinite sets exceed others in size
- Complexity theory applies diagonalization to prove existence of problems unsolvable by specific algorithm types or within certain resource bounds
- Method constructs a problem differing from every problem in given enumeration by inverting i-th machine's behavior on i-th input
- Technique exploits Turing machine countability and language uncountability to establish separation results
- Particularly useful for proving hierarchy theorems showing increased resources (time or space) allow solving strictly more problems
- Applies to various computational models (Turing machines, circuits, oracle machines)
- Relies on self-referential constructions where machines simulate and contradict other machines' behavior
Applications in computational complexity
- Proves existence of undecidable problems (halting problem)
- Demonstrates time hierarchy theorem, showing more time allows solving more problems
- Establishes separations like ≠
- Applies to space complexity, leading to space hierarchy theorem
- Separates space complexity classes
- Useful for proving limitations of specific computational models
- Shows certain problems cannot be solved by restricted circuit classes
- Establishes lower bounds on resource requirements for solving certain problems
- Proves time lower bounds for specific algorithmic techniques
Diagonalization for proving complexity class limits
Proof construction and methodology
- Begins by assuming enumeration of all machines or algorithms in considered complexity class
- Constructs new language disagreeing with each machine in enumeration on at least one input
- Ensures language cannot be decided by any machine in the class
- Involves simulation step where new machine simulates i-th machine's behavior on i-th input and does opposite
- Carefully manages time bounds to keep diagonalizing machine within desired complexity class
- Must still allow simulation and contradiction of other machines
- Utilizes padding arguments to extend results to other complexity classes
- Employs delayed diagonalization for classes with tight time bounds
- Amortizes simulation cost over multiple inputs
Examples and applications
- Time hierarchy theorem proof using diagonalization
- Shows DTIME(n^2) ≠ DTIME(n)
- Space hierarchy theorem demonstration
- Establishes SPACE(n^2) ≠ SPACE(n)
- Proving existence of uncomputable functions
- Demonstrates uncountability of all functions vs. countability of computable functions
- Separating from
- Uses padded diagonalization to show EXPTIME ≠ PSPACE
- Establishing oracle separations
- Creates oracles relative to which P ≠ or NP ≠ coNP
Diagonalization limitations in complexity class separation
Challenges with closely related classes
- Diagonalization most effective for enumeratable classes, limiting applicability to certain complexity class separations
- Technique often fails when separating classes with similar resource requirements (P and NP)
- Struggles with classes defined by semantic conditions rather than syntactic resource usage restrictions
- Difficulties arise for probabilistic classes (ZPP, )
- Cannot directly handle classes with non-uniform computational models
- Limited effectiveness for circuit complexity questions
Barriers to diagonalization
- results demonstrate diagonalization alone cannot resolve P vs. NP question
- Baker, Gill, and Solovay's 1975 work shows existence of oracles with conflicting separations
- Diagonalization proofs typically relativize, remaining valid when both classes access same oracle
- Limits technique's power for separating classes that behave similarly under relativization
- , introduced by Razborov and Rudich, present another barrier to diagonalization-like techniques
- Shows limitations of certain proof strategies for separating complexity classes
- Algebraic relativization further extends relativization barrier to algebraic proof techniques
Alternative approaches and future directions
- developed to overcome some diagonalization limitations
- Used in and probabilistically checkable proofs
- Interactive proof methods provide new tools for complexity class separations
- Led to unexpected results like IP = PSPACE
- Non-relativizing techniques offer potential for progress on longstanding open problems
- Attempts to circumvent relativization barrier in circuit complexity
- Algebraic and topological methods explored for tackling complexity class separations
- Promise to provide new insights beyond traditional diagonalization approaches
- Quantum computing introduces new complexity classes and separation questions
- Requires adapting and extending classical proof techniques, including diagonalization
Key Terms to Review (27)
Alan Turing: Alan Turing was a pioneering mathematician and logician, best known for his foundational contributions to computer science and the concept of computation. He is famous for the Turing machine, a theoretical construct that helps understand the limits of what can be computed, as well as for his work on the Entscheidungsproblem and the concept of algorithmic processes. His contributions play a significant role in the diagonalization technique by illustrating undecidable problems and the boundaries of computability.
Arithmetization Techniques: Arithmetization techniques involve converting logical or computational statements into arithmetic ones, allowing problems that are originally defined in terms of languages or sets to be analyzed within the framework of number theory. This method is particularly useful for demonstrating the limits of computability and understanding the relationships between different complexity classes through the use of numerical representations. The ability to translate problems into arithmetic form creates powerful connections between logic and computation, enabling diagonalization arguments to effectively show the existence of uncomputable functions or undecidable problems.
Bpp: BPP, or Bounded-error Probabilistic Polynomial time, is a complexity class that represents decision problems solvable by a probabilistic Turing machine in polynomial time, where the algorithm has a bounded probability of error. This means that for problems in BPP, there exists an algorithm that can make random choices to find a solution, and the algorithm will produce the correct answer with high probability. The significance of BPP arises in various contexts, including comparisons with deterministic classes and the exploration of randomness in computation.
Cantor's Diagonal Argument: Cantor's Diagonal Argument is a mathematical proof that demonstrates the uncountability of the real numbers by showing that there is no way to list all real numbers. This argument uses a diagonalization technique to construct a new real number that differs from each number in an infinite list at least at one decimal place, proving that the set of real numbers is larger than the set of natural numbers.
Closure Properties: Closure properties refer to the behavior of a set of languages or computational problems under certain operations, such as union, intersection, and complementation. These properties help in understanding how different classes of problems relate to each other and provide insight into the limits and capabilities of computational models. Closure properties are crucial for analyzing complexity classes like P, exploring the polynomial hierarchy, and applying diagonalization techniques to separate different complexity classes.
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.
Complete Problems: Complete problems are a special category of decision problems that are both in NP and as hard as any problem in NP, meaning that if a polynomial-time solution exists for any complete problem, it can be used to solve all problems in NP efficiently. They serve as benchmarks for the computational complexity of various algorithms and play a critical role in understanding the limits of what can be computed quickly. Identifying complete problems allows researchers to classify and compare the difficulty of other problems.
Decidable Problems: Decidable problems are decision problems for which an algorithm can be constructed that will always provide a correct yes or no answer in a finite amount of time. This concept plays a crucial role in understanding the limits of computation and helps categorize problems based on their solvability. In computational complexity, knowing whether a problem is decidable informs us about the resources required to solve it, including space and time, which connects to more complex ideas such as the space hierarchy theorem and diagonalization techniques.
Diagonalization: Diagonalization is a mathematical technique used to show the existence of languages or problems that cannot be solved by certain computational models. This technique often demonstrates that there are more languages than machines that can recognize them, establishing a hierarchy among languages and machines. It plays a crucial role in proving limits on computation, particularly in the context of time and space complexity.
Diagonalization Theorem: The diagonalization theorem is a fundamental concept in computational complexity theory that shows how to construct a language that is not recognized by any enumerator, thus establishing the existence of languages that are not recursively enumerable. This theorem helps in proving the limits of what can be computed by Turing machines and is crucial for understanding the hierarchy of languages and problems in complexity theory.
Exp: The term 'exp' refers to exponential time complexity, denoting algorithms that run in time proportional to $2^{cn}$ for some constant $c > 0$, where $n$ is the size of the input. This classification is important as it distinguishes problems that may be computationally feasible from those that become impractical to solve as input sizes grow, especially in relation to how resources are measured and utilized in computation.
Exptime: Exptime, short for exponential time, refers to the complexity class of decision problems that can be solved by a deterministic Turing machine in time that is exponential in the size of the input. This class is significant as it represents problems that are solvable, but not efficiently, contrasting with lower complexity classes such as polynomial time and even nondeterministic classes like NP. Understanding Exptime helps in exploring the relationships between various complexity classes, particularly in terms of hierarchy and the nature of solvable versus unsolvable problems.
Halting Problem: The halting problem is a decision problem that asks whether a given program will finish running or continue indefinitely when provided with a specific input. This problem is crucial because it reveals fundamental limits of computation, illustrating that there are certain questions about program behavior that cannot be answered by any algorithm. Understanding this concept helps in comprehending the boundaries of what can be computed and plays a significant role in different areas of computational theory.
Interactive Proof Systems: Interactive proof systems are a framework in computational complexity theory where a computationally limited verifier interacts with a more powerful prover through a series of messages to verify the validity of a statement. This setup emphasizes the role of communication and interaction in proving the correctness of computational problems, highlighting the contrasts between traditional proofs and those requiring interaction.
Intractable Problems: Intractable problems are computational problems for which no efficient solution algorithm is known. These problems are typically characterized by their high complexity, making them infeasible to solve in a reasonable amount of time as the size of the input grows. Understanding intractable problems is crucial, as they often illustrate the limitations of computation and highlight the boundaries of what can be achieved with algorithms, especially in relation to defined classes of problems and techniques like diagonalization.
Natural proofs: Natural proofs are a class of techniques in computational complexity theory used to demonstrate that certain functions or languages cannot be efficiently computed by specific classes of algorithms. They aim to provide a simple and intuitive framework for proving lower bounds on complexity classes, particularly for showing that certain properties hold for a wide range of problems. However, these proofs face limitations when it comes to establishing results related to the famous P vs NP question.
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-completeness: NP-completeness refers to a class of decision problems for which no efficient solution algorithm is known, yet if a solution is provided, it can be verified quickly (in polynomial time). This concept plays a crucial role in understanding the limits of computational problems and helps in categorizing problems based on their difficulty, particularly in relation to other complexity classes like P and NP.
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: P vs NP is a fundamental question in computational complexity theory that asks whether every problem whose solution can be quickly verified (NP) can also be solved quickly (P). This question is central to understanding the limits of what can be computed efficiently, with implications for cryptography, algorithm design, and the nature of computational problems.
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.
Reduction: Reduction is a fundamental concept in computational complexity theory that involves transforming one problem into another problem in a way that preserves the properties of interest. It is used to show relationships between problems, particularly to demonstrate the hardness of problems by relating them to known difficult problems. This concept is key for understanding complexity classes, comparing algorithms, and establishing problem classifications, especially when determining if a problem is NP-complete or #P-complete.
Relativization: Relativization is a technique in computational complexity theory that involves studying the behavior of computational classes by providing them with access to external information or oracles. This approach allows researchers to investigate questions like whether P equals NP by examining how complexity classes behave when given these oracles. By using relativization, theorists can identify limitations of certain proof techniques, particularly in relation to key open problems in the field.
Space Hierarchy Theorem: The Space Hierarchy Theorem states that for any reasonable function $$f(n)$$ that grows faster than a logarithmic function, there exist languages that can be decided by algorithms using space $$O(f(n))$$ but not by algorithms using space $$o(f(n))$$. This theorem highlights the existence of a hierarchy of complexity classes based on the amount of memory used, connecting to various concepts like diagonalization and relativization, as well as broader implications for understanding computational limits.
Stephen Cook: Stephen Cook is a prominent computer scientist known for his foundational work in computational complexity theory, particularly for formulating the concept of NP-completeness. His groundbreaking contributions established a framework for understanding the relationship between problems that can be solved efficiently and those for which solutions can be verified efficiently, influencing numerous areas in computer science and mathematics.
Time Hierarchy Theorem: The time hierarchy theorem states that given a reasonable model of computation, such as a Turing machine, for any time-constructible function $$f(n)$$, there exists a language that can be decided in time $$f(n)$$ but not in any smaller time complexity $$g(n)$$, where $$g(n)$$ is asymptotically smaller than $$f(n)$$. This theorem showcases the richness of time complexity classes and highlights the differences in computational power based on varying time constraints.
Uniformity: Uniformity refers to a property of computational models where the processes that generate a family of functions or circuits are consistent and predictable across all instances. This concept is significant because it ensures that the rules or algorithms applied do not vary arbitrarily, which allows for a structured way to analyze complexity classes and the power of computational resources. In various contexts, uniformity helps to establish relationships between different computational paradigms, making it easier to understand how different classes relate to each other.