Expert-level formal methods covering formal specification, model checking, theorem proving, type theory, program verification, and formal languages.
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TLA+: temporal logic of actions, specifies concurrent and distributed systems. Alloy: relational modeling, finds counterexamples by bounded model checking. Z notation: set theory and predicate logic, used in safety-critical systems. VDM: Vienna Development Method, pre/post conditions and invariants.
State space exploration: exhaustively check all reachable states. LTL: linear temporal logic — always, eventually, until, next. CTL: computation tree logic — for all paths, there exists a path. BDD: binary decision diagrams — symbolic state space representation. Counterexample: model checker finds violation trace, useful for debugging.
Coq: dependent types, constructive logic, extract verified code. Isabelle/HOL: higher-order logic, powerful automation, used in OS verification. Lean: modern proof assistant, mathlib library, growing adoption. Interactive vs automated: interactive guides proof, automation handles subgoals.
Simply typed lambda calculus: types prevent certain errors, Curry-Howard correspondence. Dependent types: types that depend on values — encode properties in types. Linear types: track resource usage, ensure resources used exactly once. Gradual typing: mix static and dynamic typing safely.
Hoare logic: pre-condition, command, post-condition triples. Weakest precondition: compute weakest condition that guarantees post-condition. Loop invariants: property maintained through each iteration. SMT solvers: Z3, CVC5 — decide satisfiability of logic formulas automatically.
| Pitfall | Fix |
|---|---|
| State explosion in model checking | Use abstractions and symmetry reduction |
| Spec that verifies but is wrong | Validate spec against expected behaviors |
| Automation fails on complex goals | Break into lemmas, guide automation |
| Forgetting fairness conditions | Liveness properties require fairness assumptions |
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Expert-level abstract algebra knowledge. Use when working with groups, rings, fields, modules, Galois theory, representation theory, or algebraic structures. Also use when the user mentions 'group', 'ring', 'field', 'homomorphism', 'isomorphism', 'normal subgroup', 'quotient group', 'Sylow theorem', 'Galois theory', 'polynomial ring', 'ideal', 'module', 'vector space', or 'representation theory'.
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Expert-level acoustics knowledge. Use when working with sound waves, acoustic propagation, resonance, standing waves, Doppler effect, room acoustics, ultrasound, noise control, musical acoustics, or architectural acoustics. Also use when the user mentions 'sound wave', 'decibel', 'frequency', 'resonance', 'standing wave', 'Doppler effect', 'acoustic impedance', 'reverberation', 'ultrasound', 'noise control', 'harmonics', or 'sound intensity'.
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Expert-level aerodynamics covering subsonic and supersonic flow, lift and drag, airfoil theory, boundary layers, compressible flow, and CFD methods.
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Expert-level air quality engineering covering pollutant sources, atmospheric dispersion, air quality standards, emission controls, and air quality modeling.
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Expert-level aircraft structures covering structural design philosophy, wing and fuselage analysis, fatigue and damage tolerance, composite structures, and certification.
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Expert-level algorithms and complexity theory covering sorting, searching, graph algorithms, dynamic programming, greedy algorithms, NP-completeness, and algorithm design paradigms.
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