PPES: A Probabilistic Execution Semantics for Foundation Models

PPES: A Probabilistic Execution Semantics for Foundation Models | Authorea try { document.documentElement.classList.add('js'); } catch (e) { } var _gaq = _gaq || []; _gaq.push(['_setAccount', 'G-8VDV14Y67G']); _gaq.push(['_trackPageview']); (function() { var ga = document.createElement('script'); ga.type = 'text/javascript'; ga.async = true; ga.src = ('https:' == document.location.protocol ? 'https://ssl' : 'http://www') + '.google-analytics.com/ga.js'; var s = document.getElementsByTagName('script')[0]; s.parentNode.insertBefore(ga, s); })(); Skip to main content Preprints Collections Wiley Open Research IET Open Research Ecological Society of Japan All Collections About About Authorea FAQs Contact Us Quick Search anywhere Search for preprint articles, keywords, etc. Search Search ADVANCED SEARCH SCROLL This is a preprint and has not been peer reviewed. Data may be preliminary. 2 April 2026 V2 Latest version Share on PPES: A Probabilistic Execution Semantics for Foundation Models Author : Huiwen Han 0009-0000-5852-5916 [email protected] Authors Info & Affiliations https://doi.org/10.22541/au.177212176.67413084/v2 194 views 87 downloads Contents Abstract Supplementary Material Information & Authors Metrics & Citations View Options References Figures Tables Media Share Abstract Foundation models are increasingly deployed as execution engines mediating the mapping from symbolic inputs to concrete actions. Traditional metaphors—treating models as programs, interpreters, or knowledge bases—fail to account for their stochastic, latent, and training-dependent execution behavior. We propose Probabilistic Program Execution Semantics (PPES), a formal framework in which foundation models are interpreted as probabilistic interpreters: runtime programs correspond to prompts, execution states include both observable outputs and hidden latent variables, and transitions are governed by a learned Markov kernel K𝜃 defined in the Giry measurable framework. Execution is resolved at infer-time, a distinct semantic phase absent in classical computing. PPES provides a rigorous account of key phenomena in foundation-model-based systems. We develop small-step probabilistic semantics formulated as a sub-probability transition measure (not a set-theoretic relation), define probabilistic reachability over execution traces, and prove four structural theorems: almostsure termination, trace monotonicity, Lipschitz continuity of semantics under parameter perturbation (in total variation, not KL divergence), and the Markov property of latent state. We also establish a notion of 𝜀-observational equivalence on prompts and prove a congruence theorem for sequential composition. Version note. This preprint (v2) revises v1 in four respects: (1) K𝜃 is now formally defined as a Giry-style Markov kernel with explicit measurability conditions; (2) the small-step rule is reformulated to avoid placing a sampling operation in an SOS premise; (3) the smoothness property uses total variation distance with an explicit Lipschitz condition; (4) all four structural properties are now stated as theorems with complete proofs. An extended theoretical development appears in [4]; engineering implications and the ArchHarness case study appear in [5]. Supplementary Material File (main-ppes-authorea-v2-20260402-02.pdf) Download 552.03 KB Information & Authors Information Version history V1 Version 1 26 February 2026 V2 Version 2 02 April 2026 Copyright This work is licensed under a Non Exclusive No Reuse License. Keywords computing and processing foundation models latent state non-determinism probabilistic program execution probabilistic semantics software engineering software engineering probabilistic semantics structural operational semantics Authors Affiliations Huiwen Han 0009-0000-5852-5916 [email protected] View all articles by this author Metrics & Citations Metrics Article Usage 194 views 87 downloads .FvxKWukQNSOunydq8rnd { width: 100px; } Citations Download citation Huiwen Han. PPES: A Probabilistic Execution Semantics for Foundation Models. Authorea . 02 April 2026. DOI: https://doi.org/10.22541/au.177212176.67413084/v2 If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download. For more information or tips please see 'Downloading to a citation manager' in the Help menu . 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