9.1 The diagonalization technique

Diagonalization 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 time hierarchy theorem, space hierarchy theorem, and even the existence of undecidable problems like the halting problem. 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 P ≠ EXP
• 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 EXPTIME from PSPACE
  • Uses padded diagonalization to show EXPTIME ≠ PSPACE
• Establishing oracle separations
  • Creates oracles relative to which P ≠ NP 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, BPP)
• Cannot directly handle classes with non-uniform computational models
  • Limited effectiveness for circuit complexity questions

Barriers to diagonalization

• Relativization 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
• Natural proofs, 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

• Arithmetization techniques developed to overcome some diagonalization limitations
  • Used in interactive proof systems 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