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eduKateSG · Why Mathematics? · Binary Decision Diagrams for Reliability: Boolean Logic, Shannon Expansion, Variable Ordering and Exact Fault-Tree Probability
Start with the question, audit the worked example, or jump to the assumptions and practice. Every chapter stays open for uninterrupted reading.
Understand the question
Define the quantity and follow the mechanism.Route 2 · Chapters 6–10
Reproduce the example
Audit the numbers one step at a time.Route 3 · Chapters 11–15
Check assumptions
See where design and calibration matter.Route 4 · Chapters 16–20
Explore applications
Connect the method to a concrete investigation.Route 5 · Chapters 21–25
Study and verify
Practise, inspect failure cases and read further.
Full chapter index · Practice and checking · Mathematics Learning Hub
Why mathematics is important becomes clear when a fault tree or network has repeated events and too many combinations to list comfortably. A reduced ordered binary decision diagram, or BDD, represents a Boolean success or failure function as a directed acyclic graph. Each internal node asks whether one variable is false or true, and shared subgraphs avoid repeating equivalent calculations.
This guide develops Shannon expansion, reduction rules, variable ordering and exact probability recursion under independent basic events. It shows how BDDs prevent double counting in static fault-tree calculations, support sensitivity work and reveal computational trade-offs. It also explains the boundary: a compact graph does not validate event probabilities, independence, dynamic sequence rules or the original system logic.
Full chapter index
Chapters 1–5 · Understand the question
Chapters 6–10 · Reproduce the example
Chapters 11–15 · Check assumptions
Chapters 16–20 · Explore applications
Chapters 21–25 · Study and verify
n binary events create 2^n possible state combinations before structural simplification. Thirty events have more than one billion assignments. This is the central mathematical mechanism in why boolean enumeration grows. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. Not every problem requires explicit enumeration. Write the Boolean function first. The useful habit is to keep the physical question, the probability model and the numerical method in separate columns so that a correct computation is not mistaken for complete evidence.
Did You Know? A result can be numerically stable and still answer the wrong question if the failure event, denominator or transformation changes. Test one nearby case and predict the direction of change before recalculating. Save inputs, intermediate values and units so a second reader can audit the path.
For practice, turn the example into a sensitivity check. Change one defensible input while holding the others fixed, compare the new result with the original and explain why it moved. Then construct a counterexample in which the shortcut fails. End with two sentences: what the mathematics supports, and what remains uncertain.
Start with the operational question behind decision nodes. A BDD node tests one Boolean variable and has low and high outgoing edges. In the illustrative case, Low means x=0 and high means x=1 under the chosen convention. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. Reversing failure and success conventions changes interpretation. Label terminals and variables.
A student should be able to rebuild the result without hidden settings. Record the sampling rule or optimisation criterion, the exact failure definition, and the evidence used for every distribution or parameter. Compare a transparent baseline with the advanced method. If they disagree, investigate the mechanism before deciding which answer is more credible.
This chapter belongs to a larger decision, so translate the number carefully. State one action the evidence permits and one action it does not. A parent or educator can ask which school ideas are doing the work: algebra organises relationships, probability represents uncertainty, calculus describes local change, and computing repeats the documented rules at scale.
Treat terminal nodes as a model-checking problem rather than a vocabulary exercise. Terminal 0 and terminal 1 represent false and true values of the Boolean function. For a top-event BDD, terminal 1 can mean system failure. Recompute the example from the raw values and label every unit. A diagram without a declared top event is ambiguous. State the event in words.
The strongest check uses evidence the model did not choose for itself. Hold out later observations, an independent simulation set or a known-answer benchmark. Report errors where decisions are sensitive, especially near the failure boundary, instead of relying on one global accuracy score. A method that performs well on easy safe cases can still fail at the rare cases that motivated the analysis.
Now change one assumption and make a directional prediction before running the calculation. If the result moves the opposite way, inspect the sign convention, transformation, density ratio or constraint. This prediction-first routine develops transferable problem-solving skills because it connects symbolic work to a testable expectation.
The mathematics of shannon expansion becomes useful only when its mechanism is explicit. Any Boolean function satisfies f=x f|x=1 OR not-x f|x=0. Consider the worked illustration: The diagram recursively conditions on each variable. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. The probability recursion needs mutually exclusive branches and consistent logic. Verify expansion on a truth table.
There are three uncertainties to keep apart. Sampling or numerical uncertainty comes from finite computation. Parameter uncertainty comes from estimating inputs. Model-form uncertainty comes from simplified physics, distributions or dependence. Reducing the first does not automatically reduce the other two. A credible report names all three and shows which one drives the decision.
For a practical investigation, create a small reproducible notebook or spreadsheet with a frozen data sheet, formula sheet, diagnostics sheet and decision note. Another student should be able to change one input and obtain the same update without guessing what the model meant.
Primary source: https://doi.org/10.1109/TC.1986.1676819. Read it for the method and its assumptions; do not transfer its performance claims automatically to another model.
Use ordered bdds to practise sensitivity analysis. An OBDD uses one global variable order along every path. Variables might be tested A then B then C. Before accepting the result, identify the quantity that would change the conclusion most. A poor order can make the graph exponentially larger. Record the chosen order.
A simple baseline is essential. It reveals whether the advanced method is improving accuracy or only adding complexity. Compare model calls, wall time, uncertainty and failure-mode coverage under one declared budget. Repeat stochastic calculations with several seeds; a favourable single run is not evidence of dependable efficiency.
Students can extend the example by adding correlation, a second failure mode or a modest measurement error. Explain which part of the formula changes and which remains fixed. This exercise shows how mathematics transfers from a clean textbook case to engineering and computing without pretending that the classroom model is a certified design tool.
Build a small counterexample for reduction rule one. Merge nodes that test the same variable and have identical low and high children. The initial example is: Shared subgraphs store repeated residual functions once. Then create a nearby case where the usual shortcut becomes unreliable. Merging non-equivalent functions corrupts logic. Use canonical hashing or a verified package.
Counterexamples are positive learning tools. They reveal the conditions that make a method work and help students recognise when a software output deserves a pause. Keep the counterexample small enough to plot or enumerate. Check sign, support, bounds and conservation of probability before adding complexity.
When writing the conclusion, separate a documented fact from an interpretation and from an illustrative scenario. Cite the primary method paper for the factual mechanism, use the local calculation for the illustration, and label any recommendation as conditional on the stated model. This keeps career and engineering advice age-appropriate and honest.
Connect reduction rule two to a real decision while protecting the boundary between calculation and judgement. Eliminate a node whose low and high children are identical. If the result no longer depends on x, the test is unnecessary. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. Removing a relevant node by mistake changes the function. Check equivalence after reduction.
The audit should include the original question, input provenance, failure convention, algorithm settings, convergence or stopping rule, and a sensitivity result. For an adaptive method, also preserve the sequence of choices. For an optimisation, preserve every constraint. For Bayesian updating, preserve the prior and observation model. These records turn a result into something another person can challenge and improve.
A final transfer question is useful: where else would the same mathematical idea appear? Density ratios appear in survey correction and machine learning, probabilistic constraints in planning, and conditional probability in medical testing. The context changes, but transparent assumptions and reproducible checks remain valuable.
For a fixed variable order, a reduced OBDD is a canonical representation of a Boolean function. Equivalent functions reduce to the same graph under the same order. This is the central mathematical mechanism in canonicity. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. Different orders can produce different graph sizes. Compare only after fixing order. The useful habit is to keep the physical question, the probability model and the numerical method in separate columns so that a correct computation is not mistaken for complete evidence.
Did You Know? A result can be numerically stable and still answer the wrong question if the failure event, denominator or transformation changes. Test one nearby case and predict the direction of change before recalculating. Save inputs, intermediate values and units so a second reader can audit the path.
For practice, turn the example into a sensitivity check. Change one defensible input while holding the others fixed, compare the new result with the original and explain why it moved. Then construct a counterexample in which the shortcut fails. End with two sentences: what the mathematics supports, and what remains uncertain.
Start with the operational question behind independent probability recursion. For independent event x with probability p, node probability is p times high-child probability plus (1-p) times low-child probability. In the illustrative case, Terminals have probabilities 1 and 0 for the represented event. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. Dependence invalidates simple local weighting. State the joint model.
A student should be able to rebuild the result without hidden settings. Record the sampling rule or optimisation criterion, the exact failure definition, and the evidence used for every distribution or parameter. Compare a transparent baseline with the advanced method. If they disagree, investigate the mechanism before deciding which answer is more credible.
This chapter belongs to a larger decision, so translate the number carefully. State one action the evidence permits and one action it does not. A parent or educator can ask which school ideas are doing the work: algebra organises relationships, probability represents uncertainty, calculus describes local change, and computing repeats the documented rules at scale.
Treat a two-event or example as a model-checking problem rather than a vocabulary exercise. For top event A OR B, probability is pA+(1-pA)pB=pA+pB-pA pB. The conditioned branches avoid double counting A and B together. Recompute the example from the raw values and label every unit. The formula assumes independence in its product term. Verify with four states.
The strongest check uses evidence the model did not choose for itself. Hold out later observations, an independent simulation set or a known-answer benchmark. Report errors where decisions are sensitive, especially near the failure boundary, instead of relying on one global accuracy score. A method that performs well on easy safe cases can still fail at the rare cases that motivated the analysis.
Now change one assumption and make a directional prediction before running the calculation. If the result moves the opposite way, inspect the sign convention, transformation, density ratio or constraint. This prediction-first routine develops transferable problem-solving skills because it connects symbolic work to a testable expectation.
The mathematics of repeated basic events becomes useful only when its mechanism is explicit. Fault trees can reuse one basic event in several branches, creating dependence through identity. Consider the worked illustration: A BDD tests the shared variable once along each path. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Treating repeated appearances as separate independent events is wrong. Use one canonical variable per event.
There are three uncertainties to keep apart. Sampling or numerical uncertainty comes from finite computation. Parameter uncertainty comes from estimating inputs. Model-form uncertainty comes from simplified physics, distributions or dependence. Reducing the first does not automatically reduce the other two. A credible report names all three and shows which one drives the decision.
For a practical investigation, create a small reproducible notebook or spreadsheet with a frozen data sheet, formula sheet, diagnostics sheet and decision note. Another student should be able to change one input and obtain the same update without guessing what the model meant.
Use fault-tree conversion to practise sensitivity analysis. AND, OR and voting logic can be compiled into a BDD for exact static top-event probability. Minimal cut sets can guide or validate the construction. Before accepting the result, identify the quantity that would change the conclusion most. Sequence-dependent gates are not ordinary static Boolean logic. Classify dynamic behaviour first.
A simple baseline is essential. It reveals whether the advanced method is improving accuracy or only adding complexity. Compare model calls, wall time, uncertainty and failure-mode coverage under one declared budget. Repeat stochastic calculations with several seeds; a favourable single run is not evidence of dependable efficiency.
Students can extend the example by adding correlation, a second failure mode or a modest measurement error. Explain which part of the formula changes and which remains fixed. This exercise shows how mathematics transfers from a clean textbook case to engineering and computing without pretending that the classroom model is a certified design tool.
A second primary source is https://doi.org/10.1016/0951-8320(93)90060-C. Compare the problem definition, estimator and validation rather than quoting one result in isolation.
Build a small counterexample for network success functions. Edges or components can be Boolean variables in a connectivity success function. The initial example is: A BDD can represent whether a source remains connected to a sink. Then create a nearby case where the usual shortcut becomes unreliable. State explosion may still occur for difficult graph orderings. Exploit topology and decomposition.
Counterexamples are positive learning tools. They reveal the conditions that make a method work and help students recognise when a software output deserves a pause. Keep the counterexample small enough to plot or enumerate. Check sign, support, bounds and conservation of probability before adding complexity.
When writing the conclusion, separate a documented fact from an interpretation and from an illustrative scenario. Cite the primary method paper for the factual mechanism, use the local calculation for the illustration, and label any recommendation as conditional on the stated model. This keeps career and engineering advice age-appropriate and honest.
Connect variable ordering heuristics to a real decision while protecting the boundary between calculation and judgement. Place related or influential variables to reduce repeated residual functions. A depth-first order following fault-tree structure may help some cases. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. No single heuristic is optimal for every model. Report node counts across candidates.
The audit should include the original question, input provenance, failure convention, algorithm settings, convergence or stopping rule, and a sensitivity result. For an adaptive method, also preserve the sequence of choices. For an optimisation, preserve every constraint. For Bayesian updating, preserve the prior and observation model. These records turn a result into something another person can challenge and improve.
A final transfer question is useful: where else would the same mathematical idea appear? Density ratios appear in survey correction and machine learning, probabilistic constraints in planning, and conditional probability in medical testing. The context changes, but transparent assumptions and reproducible checks remain valuable.
BDD evaluation is linear in the number of graph nodes once built, but construction can be expensive. A compact BDD enables fast probability updates. This is the central mathematical mechanism in computational complexity. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. Worst-case size remains exponential. Monitor memory and intermediate graphs. The useful habit is to keep the physical question, the probability model and the numerical method in separate columns so that a correct computation is not mistaken for complete evidence.
Did You Know? A result can be numerically stable and still answer the wrong question if the failure event, denominator or transformation changes. Test one nearby case and predict the direction of change before recalculating. Save inputs, intermediate values and units so a second reader can audit the path.
For practice, turn the example into a sensitivity check. Change one defensible input while holding the others fixed, compare the new result with the original and explain why it moved. Then construct a counterexample in which the shortcut fails. End with two sentences: what the mathematics supports, and what remains uncertain.
Start with the operational question behind sensitivity derivatives. Differentiate the probability recursion with respect to a basic-event probability to obtain importance information. In the illustrative case, A large derivative shows where a small probability change strongly affects the top event under the model. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. Importance is not the same as causal intervention value. Pair with feasibility and uncertainty.
A student should be able to rebuild the result without hidden settings. Record the sampling rule or optimisation criterion, the exact failure definition, and the evidence used for every distribution or parameter. Compare a transparent baseline with the advanced method. If they disagree, investigate the mechanism before deciding which answer is more credible.
This chapter belongs to a larger decision, so translate the number carefully. State one action the evidence permits and one action it does not. A parent or educator can ask which school ideas are doing the work: algebra organises relationships, probability represents uncertainty, calculus describes local change, and computing repeats the documented rules at scale.
Treat uncertainty propagation as a model-checking problem rather than a vocabulary exercise. Sample or bound basic-event probabilities and reuse the BDD for rapid top-event recalculation. The graph stays fixed while probabilities vary. Recompute the example from the raw values and label every unit. Structural uncertainty cannot be repaired by parameter sampling. Test alternative logic models.
The strongest check uses evidence the model did not choose for itself. Hold out later observations, an independent simulation set or a known-answer benchmark. Report errors where decisions are sensitive, especially near the failure boundary, instead of relying on one global accuracy score. A method that performs well on easy safe cases can still fail at the rare cases that motivated the analysis.
Now change one assumption and make a directional prediction before running the calculation. If the result moves the opposite way, inspect the sign convention, transformation, density ratio or constraint. This prediction-first routine develops transferable problem-solving skills because it connects symbolic work to a testable expectation.
The mathematics of dependent events becomes useful only when its mechanism is explicit. Common causes or statistical dependence require joint conditional probabilities, extra variables or other methods. Consider the worked illustration: Two failures driven by one power supply are not independent leaves. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Plugging marginal probabilities into the independent recursion is biased. Model shared causes explicitly.
There are three uncertainties to keep apart. Sampling or numerical uncertainty comes from finite computation. Parameter uncertainty comes from estimating inputs. Model-form uncertainty comes from simplified physics, distributions or dependence. Reducing the first does not automatically reduce the other two. A credible report names all three and shows which one drives the decision.
For a practical investigation, create a small reproducible notebook or spreadsheet with a frozen data sheet, formula sheet, diagnostics sheet and decision note. Another student should be able to change one input and obtain the same update without guessing what the model meant.
Use dynamic systems to practise sensitivity analysis. Priority gates, spares, repairs and timing cannot always be represented by a static Boolean function. Failure A before B differs from B before A. Before accepting the result, identify the quantity that would change the conclusion most. An ordinary BDD loses order information. Use dynamic fault trees, Markov or temporal methods.
A simple baseline is essential. It reveals whether the advanced method is improving accuracy or only adding complexity. Compare model calls, wall time, uncertainty and failure-mode coverage under one declared budget. Repeat stochastic calculations with several seeds; a favourable single run is not evidence of dependable efficiency.
Students can extend the example by adding correlation, a second failure mode or a modest measurement error. Explain which part of the formula changes and which remains fixed. This exercise shows how mathematics transfers from a clean textbook case to engineering and computing without pretending that the classroom model is a certified design tool.
Build a small counterexample for practice activity. Build reduced BDDs for A OR B and A AND (B OR C), then calculate probabilities by recursion. The initial example is: Try two variable orders and count nodes. Then create a nearby case where the usual shortcut becomes unreliable. The small example assumes independent events. Check with complete truth tables.
Counterexamples are positive learning tools. They reveal the conditions that make a method work and help students recognise when a software output deserves a pause. Keep the counterexample small enough to plot or enumerate. Check sign, support, bounds and conservation of probability before adding complexity.
When writing the conclusion, separate a documented fact from an interpretation and from an illustrative scenario. Cite the primary method paper for the factual mechanism, use the local calculation for the illustration, and label any recommendation as conditional on the stated model. This keeps career and engineering advice age-appropriate and honest.
Connect parent and educator guidance to a real decision while protecting the boundary between calculation and judgement. BDDs connect logic, conditional probability, recursion, graphs and algorithms. Ask why sharing a subgraph avoids counting the same condition twice. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. A small truth table should precede software. Use three variables first.
The audit should include the original question, input provenance, failure convention, algorithm settings, convergence or stopping rule, and a sensitivity result. For an adaptive method, also preserve the sequence of choices. For an optimisation, preserve every constraint. For Bayesian updating, preserve the prior and observation model. These records turn a result into something another person can challenge and improve.
A final transfer question is useful: where else would the same mathematical idea appear? Density ratios appear in survey correction and machine learning, probabilistic constraints in planning, and conditional probability in medical testing. The context changes, but transparent assumptions and reproducible checks remain valuable.
BDD does not mean Bayesian decision diagram, compactness does not prove the model, and repeated fault-tree leaves are not new independent events. Canonicity depends on variable order and reduction. This is the central mathematical mechanism in common misconceptions. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. Exact arithmetic can answer an incorrect top event. Audit logic and data separately. The useful habit is to keep the physical question, the probability model and the numerical method in separate columns so that a correct computation is not mistaken for complete evidence.
Did You Know? A result can be numerically stable and still answer the wrong question if the failure event, denominator or transformation changes. Test one nearby case and predict the direction of change before recalculating. Save inputs, intermediate values and units so a second reader can audit the path.
For practice, turn the example into a sensitivity check. Change one defensible input while holding the others fixed, compare the new result with the original and explain why it moved. Then construct a counterexample in which the shortcut fails. End with two sentences: what the mathematics supports, and what remains uncertain.
Start with the operational question behind when bdds are not the best tool. Very large graphs, strong dependence, continuous variables and sequence-aware dynamics may need decomposition, SAT, simulation or state-space methods. In the illustrative case, A repairable standby network has temporal transitions. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. Forcing a Boolean snapshot can omit mission history. Choose the representation from the mechanism.
A student should be able to rebuild the result without hidden settings. Record the sampling rule or optimisation criterion, the exact failure definition, and the evidence used for every distribution or parameter. Compare a transparent baseline with the advanced method. If they disagree, investigate the mechanism before deciding which answer is more credible.
This chapter belongs to a larger decision, so translate the number carefully. State one action the evidence permits and one action it does not. A parent or educator can ask which school ideas are doing the work: algebra organises relationships, probability represents uncertainty, calculus describes local change, and computing repeats the documented rules at scale.
Treat research and next reading as a model-checking problem rather than a vocabulary exercise. Bryant established graph-based algorithms for Boolean function manipulation, while Rauzy developed BDD-based fault-tree analysis algorithms. Read reduction and ordering details before trusting a package. Recompute the example from the raw values and label every unit. Algorithmic exactness does not validate event probabilities. Reproduce OR and voting benchmarks.
The strongest check uses evidence the model did not choose for itself. Hold out later observations, an independent simulation set or a known-answer benchmark. Report errors where decisions are sensitive, especially near the failure boundary, instead of relying on one global accuracy score. A method that performs well on easy safe cases can still fail at the rare cases that motivated the analysis.
Now change one assumption and make a directional prediction before running the calculation. If the result moves the opposite way, inspect the sign convention, transformation, density ratio or constraint. This prediction-first routine develops transferable problem-solving skills because it connects symbolic work to a testable expectation.
Continue with the Mathematics Learning Hub and the previous reliability article. These links own broader probability ideas and the immediately preceding rare-event geometry.
The mathematics of frequently asked questions becomes useful only when its mechanism is explicit. FAQs should cover nodes, terminals, Shannon expansion, ordering, reduction, probability recursion, repeats and dynamic limits. Consider the worked illustration: Each answer should state the top-event convention. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Avoid claiming dependence is handled by default. Publish the variable order and node count.
There are three uncertainties to keep apart. Sampling or numerical uncertainty comes from finite computation. Parameter uncertainty comes from estimating inputs. Model-form uncertainty comes from simplified physics, distributions or dependence. Reducing the first does not automatically reduce the other two. A credible report names all three and shows which one drives the decision.
For a practical investigation, create a small reproducible notebook or spreadsheet with a frozen data sheet, formula sheet, diagnostics sheet and decision note. Another student should be able to change one input and obtain the same update without guessing what the model meant.
Quick answers
What should I calculate first? Start with the estimand or failure event, then a simple baseline.
Does the advanced method guarantee accuracy? No. It needs input, numerical and model checks.
Which school mathematics matters? Algebra, graphs, probability, functions and careful units all contribute.
How can a student practise safely? Use fictional or open benchmark data and preserve every intermediate value.
What makes a career claim responsible? Describe how the mathematics is used without promising admission, salary or employment.
Extended practice laboratory
Practice case 1: Why Boolean enumeration grows
Setup: Use this chapter’s illustrative case: Thirty events have more than one billion assignments.
Task: Write the Boolean function first.
Checking target: Not every problem requires explicit enumeration.
Practice case 2: Decision nodes
Setup: Use this chapter’s illustrative case: Low means x=0 and high means x=1 under the chosen convention.
Task: Label terminals and variables.
Checking target: Reversing failure and success conventions changes interpretation.
Practice case 3: Terminal nodes
Setup: Use this chapter’s illustrative case: For a top-event BDD, terminal 1 can mean system failure.
Task: State the event in words.
Checking target: A diagram without a declared top event is ambiguous.
Practice case 4: Shannon expansion
Setup: Use this chapter’s illustrative case: The diagram recursively conditions on each variable.
Task: Verify expansion on a truth table.
Checking target: The probability recursion needs mutually exclusive branches and consistent logic.
Practice case 5: Ordered BDDs
Setup: Use this chapter’s illustrative case: Variables might be tested A then B then C.
Task: Record the chosen order.
Checking target: A poor order can make the graph exponentially larger.
Practice case 6: Reduction rule one
Setup: Use this chapter’s illustrative case: Shared subgraphs store repeated residual functions once.
Task: Use canonical hashing or a verified package.
Checking target: Merging non-equivalent functions corrupts logic.
Practice case 7: Reduction rule two
Setup: Use this chapter’s illustrative case: If the result no longer depends on x, the test is unnecessary.
Task: Check equivalence after reduction.
Checking target: Removing a relevant node by mistake changes the function.
Practice case 8: Canonicity
Setup: Use this chapter’s illustrative case: Equivalent functions reduce to the same graph under the same order.
Task: Compare only after fixing order.
Checking target: Different orders can produce different graph sizes.
Practice case 9: Independent probability recursion
Setup: Use this chapter’s illustrative case: Terminals have probabilities 1 and 0 for the represented event.
Task: State the joint model.
Checking target: Dependence invalidates simple local weighting.
Practice case 10: A two-event OR example
Setup: Use this chapter’s illustrative case: The conditioned branches avoid double counting A and B together.
Task: Verify with four states.
Checking target: The formula assumes independence in its product term.