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arXiv · 2611.07781

Reliability-Aware Z-Grammars and Z-Automata over a Credible-Grade Semiring

Sherzod Turaev·Nor Haniza Sarmin·Wan Heng Fong
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Abstract

Fuzzy grammars and weighted automata usually attach to each word a single grade, which records how strongly the word is generated but not how far that record can be trusted. We study weighted grammars and finite automata whose weights are scalar grade-reliability pairs inspired by Z-numbers. Sequential composition multiplies both components, while aggregation selects a derivation by its credible grade, the product of grade and reliability, with ties resolved by grade. On the canonical carrier, in which the weights of zero grade or zero reliability are identified with zero, the resulting structure is a commutative selective semiring, isomorphic to a subsemiring of a lexicographically ordered product, and its credible-grade projection is a homomorphism onto the Viterbi semiring. We compare several alternative combination rules and identify the carriers on which the relevant algebraic laws hold. Standard results on weighted automata then yield a rational characterization and closure properties. We prove that positive credible-grade cuts and selected-run reliability filters of finite automata are regular and effectively constructible for rational weights and thresholds, and we exhibit distinctions that the scalar projection cannot see: a reliability filter that no scalar reweighting of the same one-state automaton implements, nonmonotonicity under the addition of runs, and an unambiguous automaton with no deterministic equivalent although its scalar projection is deterministic. A count-based example illustrates an attestation penalty for grammar rules. The construction concerns scalar pairs, and its extension to arbitrary fuzzy-valued Z-numbers remains open.

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Sherzod Turaev, Nor Haniza Sarmin, Wan Heng Fong. 2026-11-05. Reliability-Aware Z-Grammars and Z-Automata over a Credible-Grade Semiring. https://arxiv.org/abs/2611.07781

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