Beyond Np-Completeness: Constructing Infinitely-Hard Languages

The Complexity Barrier

Classifying the complexity of computational problems is fundamental to computer science. The classes P and NP categorize problems by time complexity, whether they can be solved in polynomial time by a deterministic or non-deterministic Turing machine. P contains easy problems that can be solved quickly, while NP represents challenges that require exponential time in the worst case. NP-complete problems are the hardest problems in NP – if any NP-complete problem were to be solved in polynomial time, all problems in NP could be.

However, even NP-completeness has limitations for fully characterizing computational hardness. While the definition specifies worst-case exponential runtime, some problems exist that require infinite time to solve in all cases. These “infinitely-hard” problems form languages that cannot be decided by any Turing machine, so they lie outside the traditional complexity framework. Understanding infinite hardness allows pushing past barriers in complexity and better modeling infeasible challenges.

Building Infinitely-Hard Languages

Any language decidable given infinite computation time on a Turing machine constitutes an infinitely-hard problem. Such languages can interpret the symbols written on an infinite tape to recognize infinite patterns. While not physically realizable, this gedanken model allows investigating languages with non-computable properties.

Specific examples include variations of the halting problem that depend on infinitely long program executions. One such language contains all Turing machine encoding strings where the machine halted when simulated forever. This language encodes non-computable information yet remains definable. Its membership depends on properties observable given infinite computation resources.

Other infinitely-hard languages come from mathematics and logic, encoding definitions of non-computable functions. Despite only containing a finite number of symbols, the decision process implicitly references an infinite search. These constructions demonstrate that languages lie beyond NP-hardness and even Turing-decidability when granted infinite running time.

Hierarchy of Infinite Hardness

The capabilities of infinitary Turing machines can be classified into an arithmetical hierarchy based on the number of alternations of quantifiers needed to express a language’s membership condition. Each level contains strictly harder languages, requiring an extra infinity of time to decide membership.

Connections also emerge to ordinal notation systems which assign infinite integer labels to well-ordered sets. Defining a language in the nth arithmetical class can require searching a well-ordered set with order type omega to the nth power. This links the languages to TRANSFINITE runtime complexity.

In descriptive set theory, these infinitely-hard languages belong to the analytical hierarchy. This hierarchy categorizes infinite complexity beyond the arithmetical classes and leads to questions about the most powerful languages expressible when given access to infinite time Turing machines.

Infinite Hardness in the Real World

While physically unrealizable, infinitely-hard problems impact applied computer science, especially cryptography. Many cryptographic primitives rely on hardness assumptions believed to lie beyond NP-completeness.

For example, a language containing all composite numbers product of two primes can plausibly be placed at the third level of the arithmetical hierarchy. Factoring integers could then require quantification over three layers of infinite search. Cryptosystems such as RSA assume factoring possesses at least this level of transcomputational complexity.

Public key exchange, digital signatures, and homomorphic encryption all reduce to hard underlying problems like factoring. As a result, classifying these problems within a hierarchy of infinite complexity could bolster security definitions. Hardness rooted in infinity presents a more formidable barrier than finite exponential runtime.

Open Questions

Many open questions remain about infinitely-hard problems and languages:

• Can new infinite hierarchies be developed beyond the analytical hierarchy by considering different infinitely powerful Turing machines?
• Do efficiently computable functions exist for solving problems classified at higher transfinite levels like omega squared?
• What other cryptographic or mathematical problems harbor infinite complexity, and how does assuming different infinite hardness levels impact security?

This remains a rich area for future research at the boundaries of computability and complexity.

Summary

Infinitely-hard languages formalize problems with complexity beyond NP-completeness and even Turing-decidability, requiring transfinite time on more powerful machines. Hierarchies emerge classifying infinite hardness at higher ordinals and descriptive set theory levels. These languages have relevance to cryptography and assumptions on problem hardness. Many open questions remain about the implications and further classification of infinitely complex problems.