Attention Meets Reachability: Structural Equivalence and Efficiency in Grammar-Constrained LLM Decoding
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arXiv
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| Natura: | Preprint |
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2026
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| author | Alpay, Faruk Senturk, Bilge |
| author_facet | Alpay, Faruk Senturk, Bilge |
| contents | We study grammar-constrained decoding (GCD) as a coupling between an autoregressive next-token distribution and a reachability oracle over a pushdown system compiled from a context-free grammar (CFG). We prove an oracle invariance theorem: language-equivalent grammars induce identical admissible next-token sets for every prefix, hence identical logit masks, yet can yield provably different compiled state spaces and online ambiguity costs. We give exact control-state blowup counts for the canonical $a^n b^n$ language under redundant nonterminal delegation, and introduce a left-to-right structural ambiguity cost (SAC) measuring incremental packed-parse-forest growth per token. For two equivalent grammars over all finite strings, SAC is $O(1)$ per token under right-recursion but $Θ(t^2)$ per token and $Θ(n^3)$ cumulatively under concatenation. We establish engine-independent lower bounds: any sound, retrieval-efficient, parse-preserving online masking engine must incur $Ω(t^2)$ work per token on a specific constant-size CFG family, unconditionally within this model. We define decoding-cost equivalence classes of grammars and prove existence of minimal-SAC representatives within bounded rewrite families. Finally, we characterize the true conditional sampler via a Doob $h$-transform and derive sharp one-step KL and total-variation distortion bounds for hard-masked decoding in terms of survival-probability spread among admissible next tokens. We integrate these results with Transformer and Mixture-of-Experts architectures, derive latency envelopes in terms of vocabulary size, active state sets, and beam width, and connect SAC to instrumentation-based predictive performance models and automated grammar optimization. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_05540 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Attention Meets Reachability: Structural Equivalence and Efficiency in Grammar-Constrained LLM Decoding Alpay, Faruk Senturk, Bilge Computation and Language Formal Languages and Automata Theory Machine Learning 68Q45, 68T50, 68Q25, 68T07, 60J10 F.4.3; F.1.1; F.2.2; I.2.7; I.2.6 We study grammar-constrained decoding (GCD) as a coupling between an autoregressive next-token distribution and a reachability oracle over a pushdown system compiled from a context-free grammar (CFG). We prove an oracle invariance theorem: language-equivalent grammars induce identical admissible next-token sets for every prefix, hence identical logit masks, yet can yield provably different compiled state spaces and online ambiguity costs. We give exact control-state blowup counts for the canonical $a^n b^n$ language under redundant nonterminal delegation, and introduce a left-to-right structural ambiguity cost (SAC) measuring incremental packed-parse-forest growth per token. For two equivalent grammars over all finite strings, SAC is $O(1)$ per token under right-recursion but $Θ(t^2)$ per token and $Θ(n^3)$ cumulatively under concatenation. We establish engine-independent lower bounds: any sound, retrieval-efficient, parse-preserving online masking engine must incur $Ω(t^2)$ work per token on a specific constant-size CFG family, unconditionally within this model. We define decoding-cost equivalence classes of grammars and prove existence of minimal-SAC representatives within bounded rewrite families. Finally, we characterize the true conditional sampler via a Doob $h$-transform and derive sharp one-step KL and total-variation distortion bounds for hard-masked decoding in terms of survival-probability spread among admissible next tokens. We integrate these results with Transformer and Mixture-of-Experts architectures, derive latency envelopes in terms of vocabulary size, active state sets, and beam width, and connect SAC to instrumentation-based predictive performance models and automated grammar optimization. |
| title | Attention Meets Reachability: Structural Equivalence and Efficiency in Grammar-Constrained LLM Decoding |
| topic | Computation and Language Formal Languages and Automata Theory Machine Learning 68Q45, 68T50, 68Q25, 68T07, 60J10 F.4.3; F.1.1; F.2.2; I.2.7; I.2.6 |
| url | https://arxiv.org/abs/2603.05540 |