Building Artificial Languages To Probe The Boundaries Of Complexity Classes
Formal Languages as Probes
The mathematical formalisms known as formal languages provide a powerful set of tools for investigating the nature and boundaries of computational complexity classes. Complexity classes such as P, NP, and beyond categorize broad sets of computational problems by resource usage factors like time and memory. Formal language theory gives computer scientists a framework for constructing finely-tuned artificial languages that can inhabit and stress-test the properties of these classes.
By encoding problems into string sets and grammars, we can study how different generation and recognition rulesets interact with complexity measures. For instance, a regular language that recognizes all inputs conforming to a regular expression exist squarely within the deterministic P class. By tweaking the language’s components, we can probe at the borders of P and explore what lies beyond in probabilistic or nonlinear runtimes.
This ability for formal grammars to crisply characterize different complexity behaviors makes artificial languages a versatile empirical laboratory. As we shall see, building custom languages to match or mismatch different complexity attributes gives valuable insight into the expansiveness and limitations of models like the P vs NP question or the finer gradations of the Polynomial Hierarchy.
Crafting Languages that Target Complexity
The art of designing a formal language to illuminate complexity classes relies on astutely selecting compatible generation and recognition rulesets. The simplest but still nontrivial demonstrations come from languages that cleanly fall into either tractable P or intractable NP categories.
For example, consider a language over the binary alphabet {0, 1} that recognizes all even-length bitstrings with a mirrored palindrome sequence. We could generate this language with a context-free grammar by having a base production rule that concatenates copies of 0 and 1 subwords, wrapped in a palindromic envelope. The key insight is that while generating such inputs is easy, checking if a arbitrary string meets the syntactic constraints requires time quadratic in the length of the string. This places the language firmly in NP, showcasing the asymmetry between bounded polynomial generation and unbounded validation.
Designing more exotic languages that inhabit or skirt the fringes of complexity classes involves careful coordination of multiple generation and validation rule modules. Restricting ourselves to regular or context-free grammars keeps languages inside PSPACE, but moving to context-sensitive rules that count or reorder symbols expands possibilities dramatically. Linking these primal grammar engines with specialized external transducers yields intricate new languages stretching across polynomial hierarchy levels or other esoteric classes like PP.
Simulating Turing Machines with String Rewriting
To systematically scale the complexity of our languages, it helps to connect formal language concepts back to their parent automata models. Turing machines provide the canonical abstraction for encoding Decision problems with predictable resource costs. By simulating Turing machines with string rewrite systems, we can import a rich body of complexity results.
In this approach, the alphabet consists of symbols representing Instantaneous Descriptions (ID) of a Turing machine configuration, with special markers for the internal state, tape contents, and input/output areas. Simple rewrite rules then describe each step of the state transition function: consuming an input, writing a new square, and changing control states. Chaining these rules together thus encodes full machine computations within cascading strings.
With just a handful of ID symbols and rewrite templates we can bootstrap a small universal Turing machine, capable of carrying out any computation given an appropriate input program string. This grants enormous flexibility in embedding complexity attributes into artificial languages: we inherit by translation over 5000 research papers establishing scaling laws on space, time, nondeterminism, and much more!
New Frontiers in Language Design
Our ability to explicate complexity concepts through formal language theory highlights both past progress and future opportunities. Many advances in algorithms rely on subtle variations in models and measures that could benefit from linguistic framing. For example, active research into fine-grained complexity seeks sharper run time characterizations nested within traditional classes.
Connecting languages directly to semantic meanings also offers new vistas. Rather than modulating only surface syntax, we may one day be able to tweak conceptual relationships and truth conditions for targeted effects. Early work on neural networks that learn simple languages hints at co-evolving innate grammar and reasoning modules.
By crossing Chomsky Hierarchies with Complexity Zoo boundaries, we reinforce connections across logic, computer science, and linguistics. The future looks bright for elucidating more facets of computation through a menagerie of invented languages!
