Attack Calculus: A Typed, Evidence-Aware State-Transition Calculus for Cross-Domain Cybersecurity Reasoning

Cybersecurity frameworks classify adversary behavior, exchange threat intelligence, catalog defensive techniques, and generate attack graphs, yet incident reasoning still lacks a compact notation that states at once why an action occurs, how it occurs, which state enables it, which state it changes, and what evidence supports each claim. This paper proposes Attack Calculus Notation (ACN), a typed, evidence-aware state-transition calculus that connects analyst-readable incident reasoning to machine-checkable models. Each atomic action is an actor-labelled transition with explicit intent, mechanism, target, object, channel, valued guards, add and delete effects, and field-level evidence. Compact expressions denote finite trace languages and compile to a guarded predicate-transition model whose finite-ground reachability core reduces to typed STRIPS planning under stated invariant assumptions, which fixes the complexity of compromise checking. Evidence uncertainty defines a family of admissible models, over which ACN distinguishes possible from necessary compromise. Defensive controls are model transformations, and robust task-preserving synthesis selects the least-cost control bundle that eliminates every modeled prohibited outcome in every admissible model while keeping authorized-task utility above a floor in the worst admissible model. An AI-agent extension separates influence from authority and measures excess authority with a capability subsumption order. A complete worked example, reproduced by a published reference checker, shows a case where an analysis restricted to the most likely model selects a control that fails in an admissible alternative, and where attack-only minimization breaks a business task. The work is a formal specification and falsifiable research program; it claims no empirical superiority.