Diagnosis of Failure Modes

• Diagnosis of failure modes is the systematic identification, isolation, and explanation of deviations in system performance using formal and data-driven models.
• It utilizes algorithmic methods like backward fault propagation, conflict detection, and logical reasoning to derive minimal, non-contradictory diagnoses from observed failures.
• Practical applications span cyber-physical, software, and socio-technical systems, where compositional analysis and rigorous modeling drive enhanced safety and targeted remediation.

Diagnosis of failure modes is the systematic identification, isolation, and explanation of the specific ways a system, component, or process deviates from its intended functional behavior. This encompasses formal, algorithmic reasoning steps, data-driven methodologies, and simulation frameworks designed to map system outputs back to minimal sets of internal causes, typically to enable safety, reliability, or targeted remediation. Modern diagnosis of failure modes spans cyber-physical, software, machine learning, and socio-technical systems, and is characterized by both rigorous mathematical modeling and scalable tool support (Jahanian, 2022).

1. Theoretical Foundations of Failure Mode Diagnosis

Diagnosis of failure modes entails determining which internal faults, under a system’s structure and dynamics, suffice to explain observed deviations at system outputs. Theoretical frameworks formalize this as a combination of the following concepts:

• System function model: The baseline for reasoning, typically specified as $y = f(x, p)$, where $x$ represents inputs, $p$ parameters, and $y$ outputs over a state space $V$ (Boolean or real variables) (Jahanian, 2022).
• Faults and failure modes: Faults correspond to deviations of variables from their intended values. Each variable $x$ is associated with an intended value $\bar x$; $x$ is considered fault-free if $x = \bar x$, and faulty otherwise. Failure at an output $y$ is defined by $x$0.
• Abstract failure modes: Rather than tracking all numerical errors, diagnosis leverages abstractions into finite sets of symbolic modes, e.g., $x$1 (for high, low, true, false, and nominal, respectively) to encode deviations for both real and Boolean variables, facilitating generalization across system scale (Jahanian, 2022).
• Failure scenarios: Diagnosis operates on logical relationships of the form $x$2 and seeks minimal logical explanations that entail observed output failures.
• Component composition: System-level failure scenarios are constructed by composing local models of function blocks, allowing diagnosis to scale from low-level program analysis to system-level architectures.

This logical, formal approach contrasts with purely expert-driven or data-centric frameworks and explicitly facilitates compositionality, minimality of diagnoses, and unbiased rule-based inference.

2. Algorithmic Methods for Failure Mode Diagnosis

Diagnosis is realized as a multi-stage, algorithmic process that combines rule-based inference, propagation, and logical reasoning:

1. Model Declaration: Enumerate all component functions and derive their local failure models as implication rules of the form $x$3, recording both conjunctive (certain-cause) and disjunctive (minimum-condition) implications as appropriate.
2. Hypothesis Generation: Observe system outputs with failure, mark known good (certain) variables, and initialize the agenda with these observations.
3. Backward Fault Propagation: Apply local component failure rules in reverse, recursively inferring the minimal combinations of input faults that could cause observed output failures. Conjunctive rules yield direct diagnoses; disjunctive rules expand the search space.
4. Conflict Detection and Pruning: Use exclusivity axioms—such as “a variable cannot occupy two distinct fault modes simultaneously”—to prune contradictory assignments (e.g., the same parameter marked both ‘high’ and ‘low’) and discard diagnoses inconsistent with known good variables.
5. Diagnosis Assembly: Combine all minimal, non-contradictory local causes into global diagnoses using logical operations (conjunction for serial dependencies, disjunction for parallel/OR structures).
6. Output: The algorithm yields a set of minimal diagnoses—disjoint logical explanations for observed failure modes—that can serve as root causes or actionable entry points for remediation (Jahanian, 2022).

A sketch of the pseudocode illustrating these steps is detailed in (Jahanian, 2022): $V$3

3. Mathematical Frameworks and Diagnostic Formalisms

Diagnosis frameworks provide rigorous definitions and operators facilitating both formal reasoning and implementation:

• Failure mode formalization: For real $x$4, failure mode $x$5 ($x$6), $x$7 ($x$8), $x$9 ($p$0); for Boolean $p$1, $p$2 ($p$3True, $p$4False), $p$5 ($p$6False, $p$7True), $p$8 ($p$9).
• Failure scenario relation: For each component, relation $y$0 encodes the allowed mode propagations, permitting algebraic manipulation and automated reasoning.
• Dual and logical dual: Operators for inverting failure mode (e.g., $y$1) and logical duals (swapping $y$2 and inverting modes), enabling systematic construction and analysis of diagnosis scenarios.
• Impact index: Defines the effect of mode-changes on output faults. For a change from mode $y$3 to $y$4, the function $y$5 quantifies (in a finite set $y$6) how likely the change repairs or worsens the failure, supporting ranking of candidate repairs or interventions.
• Axioms: Ensures mutually exclusive diagnosis assignments, such as Axiom 1: no variable can be in two distinct non-nominal modes simultaneously, enabling logical consistency and minimality (Jahanian, 2022).
• Diagnosis as logical explanation: Formally, a diagnosis is a set (or logical combination) of input-mode propositions that, when propagated upward through system structure, entail the observed failure and are logically consistent with all known facts.

4. Practical Application, Scalability, and Example

Diagnosis of failure modes by formal reasoning has direct applicability in safety-critical and complex systems, supported by scalable tool implementations:

• Input: System modeled as a network of component functions, each with failure-modes defined as logical rules.
• Process: Given a specific output failure, backward-propagate implications, applying conflict detection and pruning rules, and combine minimal explanations.
• Example (Range Checker): In a system $y$7, an observed failure $y$8 in the “commission” mode ($y$9) reduces to just two minimal diagnoses: $V$0 has failed high, or $V$1 has failed low.
• Scalability: The formalism can be incorporated into program analysis tools ingesting logic (e.g., Ladder Logic), auto-deriving failure scenarios, and outputting diagnoses. Combinatorial explosion is mitigated by pruning paths with certain-cause or certainty-based reasoning and logical simplification.
• Assumptions: Models require accurate functional logic, all causes localized to input/parameter domains, and a pre-failure “healthy” state for all variables.
• Limitations: Diagnosis can become intractable if the number of combinatorial explanations grows rapidly; meaningful application depends on pruning and simplification heuristics (Jahanian, 2022).

5. Guidelines for Systematic Diagnosis of Failure Modes

Rigorous diagnosis requires a disciplined methodology:

1. Model the system as compositional functions $V$2 (code, logic, arithmetic).
2. Assign variable roles, distinguishing “suspicious” from “certain” (healthy) domains.
3. Derive explicit local failure models using formal reasoning—prefer certain-cause rules where possible.
4. Observe and encode deviations at outputs as diagnosis targets.
5. Apply the backward-propagation algorithm for mode-based inference, adhering to certainty constraints.
6. Prune contradictory/inconsistent hypotheses, respecting exclusivity and certainty.
7. Assemble minimal, non-redundant diagnoses through logical simplification.
8. Validate diagnoses by further testing or controlled parameter changes; rank candidates by impact index to guide intervention.

Implementing this process systematically replaces ad hoc or bias-prone reasoning with a repeatable, logic-driven mechanism that accelerates uncovering root causes from observed failures, bridges the gap from specification to diagnostic action, and enables targeted engineering remedies (Jahanian, 2022).

6. Extensions and Connections to Broader Contexts

Formal failure mode diagnosis as described in Failure Mode Reasoning (Jahanian, 2022) underpins diverse applications ranging from safety instrumented systems in industrial settings (e.g., power plant control) to simulation-driven analysis and integration into model-based safety frameworks. Key connections include:

• Automated tool support that parses real-world program logic and generates system-specific failure mode diagnoses at scale.
• Compositional integration with higher-level safety analysis methods such as HiP-HOPS or component fault trees, augmenting input-side FMR with logic-CPU and hardware failure models (Jahanian et al., 2020).
• Bridging specification and design: Backward reasoning from outputs ensures diagnoses are both precise and justified by actual system structure, supporting transparent traceability and auditability, which is essential in regulated and safety-critical domains.

This approach serves as a reference for modern diagnosis of failure modes, establishing a rigorous mathematical, algorithmic, and practical foundation for scalable, unbiased fault analysis in complex engineered systems.