On Higher-Order Probabilistic Verification via the Weighted Relational Model of Linear Logic

2026-04-30

AI Summary

This work addresses the undecidable problem of almost-sure termination for higher-order probabilistic programs (PHORS) by proposing a novel approach grounded in the weighted relational semantics of linear logic. The authors reduce termination analysis to the study of algebraic properties of generating functions. By introducing a type system equipped with bounded exponents, they extend the class of affine PHORS and prove that the generating functions associated with this extended class are always algebraic. This result establishes the decidability of almost-sure termination for the new subclass. Consequently, the paper not only delineates a decidable fragment of probabilistic higher-order programs but also provides an effective verification mechanism based on solving algebraic equations.

Abstract

The problem of determining whether a probabilistic program terminates almost surely (i.e.~with probability one) is undecidable, and actually $Π^0_2$-complete. For this reason, a growing literature has explored classes of programs for which this and related problems can be shown (semi-)decidable. In this work we consider the termination problem for the language of Probabilistic Higher-Order Recursion Schemes (PHORS). Using the weighted relational semantics of linear logic, we translate this problem into the computation of suitable generating functions associated with the program interpreted. This way, we establish the decidability of almost sure termination for a class of programs that extends Li et al.'s affine PHORS via a type discipline with bounded exponentials. To achieve this, we show that the generating functions for such programs are always algebraic, that is, solutions of polynomial equations, yielding an effective method to answer the termination problem.

Problem

• probabilistic termination
• higher-order recursion schemes
• almost sure termination
• undecidability
• program verification

Innovation

• weighted relational semantics
• probabilistic higher-order recursion schemes
• almost sure termination
• algebraic generating functions
• bounded exponentials