相关论文: Infinitude of Primes Using Formal Language Theory

We give a simple new proof that regular languages defined by first-order sentences with no quantifier alteration can be defined by such sentences in which only regular atomic formulas appear. Earlier proofs of this fact relied on arguments…

We provide a semi-grammatical description of the set of normal proofs of positive formulae in minimal predicate logic, i.e. a grammar that generates a set of schemes, from each of which we can produce a finite number of normal proofs. This…

We survey facts mostly emerging from the seminal results of Alan Cobham obtained in the late sixties and early seventies. We do not attempt to be exhaustive but try instead to give some personal interpretations and some research directions.…

Some aspects of the physical nature of language are discussed. In particular, physical models of language must exist that are efficiently implementable. The existence requirement is essential because without physical models no communication…

A regular language $L$ is said to be prime, if it is not the product of two non-trivial languages. Martens et al. settled the exact complexity of deciding primality for deterministic finite automata in 2010. For finite languages, Mateescu…

We provide two new proofs of the infinitude of prime numbers, using the additive Ramsey-theoretic result known as Folkman's theorem (alternatively, one can think of these proofs as using Hindman's theorem). This adds to the existing…

In this expository article, the real numbers are defined as infinite decimals. After defining an ordering relation and the arithmetic operations, it is shown that the set of real numbers is a complete ordered field. It is further shown that…

Logic languages based on the theory of rational, possibly infinite, trees have much appeal in that rational trees allow for faster unification (due to the safe omission of the occurs-check) and increased expressivity (cyclic terms can…

We define a notion of randomness for individual and collections of formal languages based on automatic martingales acting on sequences of words from some underlying domain. An automatic martingale bets if the incoming word belongs to the…

We found a regularity of the behavior of primes that allows to represent both prime and natural numbers as infinite matrices with a common formation rule of their rows. This regularity determines a new class of infinite cyclic groups that…

As transformers have gained prominence in natural language processing, some researchers have investigated theoretically what problems they can and cannot solve, by treating problems as formal languages. Exploring such questions can help…

I propose a class of non-positional numeral systems where numbers are represented by Dyck words, with the systems arising from a recursive extension of prime factorization. After describing two proper subsets of the Dyck language capable of…

We characterize the infinite words determined by indexed languages. An infinite language $L$ determines an infinite word $\alpha$ if every string in $L$ is a prefix of $\alpha$. If $L$ is regular or context-free, it is known that $\alpha$…

The atoms of a regular language are non-empty intersections of complemented and uncomplemented quotients of the language. Tight upper bounds on the number of atoms of a language and on the quotient complexities of atoms are known. We…

We consider languages defined by signed grammars which are similar to context-free grammars except productions with signs associated to them are allowed. As a consequence, the words generated also have signs. We use the structure of the…

Regular expressions in an Automata Theory and Formal Languages course are mostly treated as a theoretical topic. That is, to some degree their mathematical properties and their role to describe languages is discussed. This approach fails to…

Formal languages theory is useful for the study of natural language. In particular, it is of interest to study the adequacy of the grammatical formalisms to express syntactic phenomena present in natural language. First, it helps to draw…

The theory of finite term algebras provides a natural framework to describe the semantics of functional languages. The ability to efficiently reason about term algebras is essential to automate program analysis and verification for…

We prove that all standard subregular language classes are linearly separable when represented by their deciding predicates. This establishes finite observability and guarantees learnability with simple linear models. Synthetic experiments…

Traditional linguistic theories have largely regard language as a formal system composed of rigid rules. However, their failures in processing real language, the recent successes in statistical natural language processing, and the findings…