# Higher-Order Operator Precedence Languages

## TL;DR

This paper introduces Higher-Order Operator Precedence (HOP) languages, extending OP languages by using k-tuples of terminal symbols, resulting in a more expressive deterministic language family with closure properties and hierarchical structure.

## Contribution

It extends Floyd's OP languages to higher orders using locally testable regular languages, creating a broader class called HOP with proven closure and hierarchy properties.

## Key findings

- HOP languages properly include OP languages.
- HOP languages form an infinite hierarchy ordered by parameter k.
- Max-languages are defined by a simple cancellation rule.

## Abstract

Floyd's Operator Precedence (OP) languages are a deterministic context-free family having many desirable properties. They are locally and parallely parsable, and languages having a compatible structure are closed under Boolean operations, concatenation and star; they properly include the family of Visibly Pushdown (or Input Driven) languages. OP languages are based on three relations between any two consecutive terminal symbols, which assign syntax structure to words. We extend such relations to k-tuples of consecutive terminal symbols, by using the model of strictly locally testable regular languages of order k at least 3. The new corresponding class of Higher-order Operator Precedence languages (HOP) properly includes the OP languages, and it is still included in the deterministic (also in reverse) context free family. We prove Boolean closure for each subfamily of structurally compatible HOP languages. In each subfamily, the top language is called max-language. We show that such languages are defined by a simple cancellation rule and we prove several properties, in particular that max-languages make an infinite hierarchy ordered by parameter k. HOP languages are a candidate for replacing OP languages in the various applications where they have have been successful though sometimes too restrictive.