Parikh’s Theorem is a fundamental result in automata theory with numerous applications in computer science. These include software verification (e.g. infinite-state verification, string constraints, and theory of arrays), verification of cryptographic protocols (e.g. using Horn clauses modulo equational theories) and database querying (e.g. evaluating path-queries in graph databases), among others. Parikh’s Theorem states that the letter-counting abstraction of a language recognized by finite automata or context-free grammars is definable in Linear Integer Arithmetic (a.k.a. Presburger Arithmetic). In fact, there is a linear-time algorithm computing existential Presburger formulas capturing such abstractions, which enables an efficient analysis via SMT-solvers. Unfortunately, real-world applications typically require large alphabets (e.g. Unicode, containing a million of characters) — which are well-known to be not amenable to explicit treatment of the alphabets — or even worse infinite alphabets.

Symbolic automata have proven in the last decade to be an effective algorithmic framework for handling large finite or even infinite alphabets. A symbolic automaton employs an effective boolean algebra, which offers a symbolic representation of character sets (i.e. in terms of predicates) and often lends itself to an exponentially more succinct representation of a language. Instead of letter-counting, Parikh’s Theorem for symbolic automata amounts to counting the number of times different predicates are satisfied by an input sequence. Unfortunately, naively applying Parikh’s Theorem from classical automata theory to symbolic automata yields existential Presburger formulas of exponential size. In this paper, we provide a new construction for Parikh’s Theorem for symbolic automata and grammars, which avoids this exponential blowup: our algorithm computes and existential formulas in polynomial-time over (quantifier-free) Presburger and the base theory. In fact, our algorithm extends to the model of parametric symbolic grammars, which are one of the most expressive models of languages over infinite alphabets. We have implemented our algorithm and show it can be used to solve string constraints that are difficult to solve by existing solvers.