Descriptive Complexity Approach To Separating P And Np

Defining the Complexity Classes P and NP

The complexity classes P and NP formalize the intuitive notion of “easy” and “hard” problems in computer science. The class P consists of decision problems that can be solved in polynomial time by a deterministic Turing machine. Specifically, a problem X is in P if there exists an algorithm that can decide whether any given input instance of X is a “yes” or “no” instance in a number of computation steps bounded by a polynomial in the size of the input.

In contrast, the class NP consists of decision problems where a “yes” instance can be verified in polynomial time by a non-deterministic Turing machine. That is, a problem Y is in NP if there is an algorithm such that given any input and a certificate vouching for its “yes”-ness, the algorithm can check that the input and certificate pair indeed represent a “yes” instance in polynomial time. The certificate represents a proof that the problem instance is solvable.

Polynomial-Time Verifiability of NP Problems

The key distinction between P and NP is the notion of polynomial-time verifiability. For NP problems, solutions can be efficiently verified even though finding the solutions may not be computationally easy. Many important problems in computer science, including constraint satisfaction, graph problems, and propositional satisfiability, exhibit this asymmetry. For example, verifying that a Sudoku puzzle is solved involves checking whether each row, column and subgrid contains the numbers 1 through 9 just once. This can be done quickly even for large puzzles by scanning through them. However, solving arbitrary Sudoku puzzles is believed to be hard.

Likewise, checking whether a given truth assignment satisfies a Boolean formula in conjunctive normal form (CNF) can be done efficiently by plugging the assignment into the formula to verify it makes the formula evaluate to TRUE. But finding a satisfying assignment or showing none exists appears difficult in the worst case.

Implications of P = NP or P ≠ NP

The fundamental question is whether all problems with polynomial-time verifiable solutions also have polynomial-time algorithms for finding those solutions, i.e. does P = NP? Or are there problems in NP that are intrinsically harder, i.e. P ≠ NP?

If P = NP, then there are no inherently difficult computational problems, since all NP problems would be solvable in polynomial time. This would have wide-ranging practical implications like being able to efficiently solve scheduling, planning, search, and optimization problems on large inputs where exhaustive search is infeasible. But it would also imply unexpected mathematical consequences, potentially allowing polynomial-time algorithms for solving problems in number theory and cryptography that are believed to be intractable.

On the other hand, if P ≠ NP, it would confirm widely held conjectures about logically simpler problems being computationally harder. Computer scientists could then focus efforts on approximating solutions to NP-hard problems efficiently. Philosophically, P ≠ NP means some formal mathematical statements are easier to check than to find or to prove from scratch.

Using Descriptive Complexity to Separate Complexity Classes

Logic-Based Descriptive Complexity

Descriptive complexity offers an alternative lens for studying computational complexity using mathematical logic instead of Turing machine models. The key insight is linking complexity classes to the expressive power of different logical formalisms. Less succinct logics correspond to more complex problems.

For instance, problems expressible as first-order logic (FO) formulas with a fixed number of variables capture the class P. The number of variables bounds the number of entities that needs to be considered, allowing construction of polynomial-time algorithms. In contrast, problems expressible in second-order logic, which includes quantification over relations, correspond to more complex problems in NP and beyond.

Fagin’s Theorem: NP Captured by Existential Second-Order Logic

A seminal result in descriptive complexity is Fagin’s theorem, which states that a problem X is in NP if and only if X is expressible as an existential second-order logic (∃SO) formula. These formulas allow existential quantification over relations but not universal quantification. For example, the NP-complete problem of checking graph 3-colorability can be expressed as:

∃R1,∃R2,∃R3 such that [R1, R2 and R3 partition the vertices] AND [no edge has both endpoints in any Ri]

Fagin’s theorem implies NP corresponds exactly to the set of problems that have short witnesses confirming yes instances, mirroring the polynomial-time verifiability characterization. Furthermore, it demonstrates second-order existential quantification captures the complexity of NP problems.

Separating NP from P using Descriptive Complexity

With the expressive power linked to complexity classes, separation results showing a logic cannot define problems in a simpler class demonstrate complexity separations. For instance, using Ehrenfeucht-Fraïssé games, Kleene’s theorem states FO sentences with just three variables cannot in general define properties checkable with only two FO variables.

More significantly, proving Fagin’s theorem required showing that full NP cannot be captured by first-order logic. This implies P ≠ NP since P corresponds exactly to FO. Intuitively, FO cannot express computational paths involving unbounded chains of relations the way NP verifications require. This logic-based separation bolsters evidence for P ≠ NP.

Open Questions and Future Research Directions

While descriptive complexity has related logics to complexity classes, resolving the P vs NP question remains open. Further work lies in establishing tighter connections to aid this effort. Other directions include studying the power of different logics for capturing complexity, developing model checking algorithms, and applying descriptive complexity to find optimal logics to express problems simply and declaratively.

Bridging the conditional hardness and universality results from diagonalization with the unconditional separations from logic could open new angles of attack. Developing simpler logical formalisms that capture intriguing complexity classes like NP intersect coNP could also elucidate their structure. Ultimately, the descriptive lens offers another rich toolkit to uncover the precise frontiers of complexity.