The Gold Standard of Universities | Tracing Reachable Modules When Alternative Prerequisites Contain Cycles

Suppose a student asks whether a named later module can ever become available through the supplied prerequisite rules. That is a reachability question. It differs from asking which term the module runs, whether a seat exists, whether it fits a timetable or whether it counts towards graduation.

For this first model, the only constraints are the Boolean prerequisite expressions. A module can be completed once its condition holds using earlier completed modules. Completion persists. Every newly permitted module can eventually be completed successfully, and there are no capacity limits, exclusions or calendar restrictions.

These assumptions are deliberately strong. They allow us to isolate the logical question before adding other planning conditions. A positive answer means a valid finite completion sequence exists within this fictional model. It is not an assurance that a student can register for the module at a particular institution or finish by a particular date.

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The condition A requires B AND C means both named passes must already be available. The condition A requires B OR C means either is sufficient. Inclusive OR also permits both; it does not mean that taking both is forbidden.

Write the connector in the ledger. A pair of arrows pointing into A can be ambiguous unless the diagram explains whether they represent two compulsory requirements or two alternatives. A clear expression is more useful than a beautiful diagram that loses the condition.

For a first paper check, put B in a completed set and leave C out. Ask whether each expression is true. The AND expression is false; the OR expression is true. This tiny comparison is worth stabilising before examining cycles. Otherwise, later graph terminology can conceal an elementary reading error that changes every subsequent conclusion.

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Compare B AND (C OR D) with (B AND C) OR D. If only D has been completed, the first expression is false and the second is true. Parentheses identify the grouping and therefore affect eligibility.

A learner should be able to explain each expression in ordinary language. The first requires B together with at least one of C and D. The second permits either the pair B and C or D alone. These are different rules despite containing the same module names.

Do not infer grouping from which arrangement would make a desired plan work. Preserve the official wording and obtain clarification when its meaning is unresolved. In our fictional examples, the grouping is supplied explicitly. That allows an independent check instead of turning an ambiguous catalogue sentence into an unsupported planning assumption.

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Let C₀ be the set of already established passes recognised for this exercise. A module belongs to that set only because its completion or applicable recognition has been supplied. Similar titles, general credit totals and hoped-for exemptions do not create entries.

We initially take C₀ to be empty. A foundation module can still become available if its stated prerequisite condition is true without earlier module passes. We write that condition as true, meaning no module prerequisite within this model.

The distinction matters. A foundation rule says that a module can be completed as a first step. An established pass says that completion is already part of the starting record. Both can initiate progress, but they occupy different places in the account. Neither says that a real course has no other admission, registration or administrative conditions.

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Here is the complete rule set for the main example. Every connector is inclusive and every module mentioned is part of this fictional catalogue.

Read one row at a time before tracing a route. A has an alternative entrance through C. D needs both A and B. G cannot use I alone because its rule also requires H.

The table separates the module’s identity from the expression that permits its next step. It contains no credit values or offering terms because those are not part of this first question. Leaving those quantities out explicitly is preferable to filling the table with invented scheduling information that the argument does not use.

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Given a completed set Cᵣ, evaluate every prerequisite expression against that set. Add the modules whose expressions are true, while retaining everything already present. Call the enlarged set Cᵣ₊₁.

In symbols, Cᵣ₊₁ = Cᵣ ∪ {m : Pm(Cᵣ) is true}. The symbol Pm means module m’s supplied prerequisite expression. The union keeps completion persistent.

All tests in a round use the previous set, not entries newly written during that same round. This convention makes the worked ledger reproducible and preserves the requirement for earlier completion. Another implementation can process additions one at a time, but it should record their valid order. Do not mix the two conventions and then interpret the number of rounds as an academic calendar. These rounds are stages of a logical calculation.

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From the empty set, C and I can be added because their rules are true. No other expression is satisfied. The first enlarged set is therefore C₁ = {C, I}.

A is not added in this first round because its test uses the earlier empty set. C becomes available as a result of the round, ready to support a later test. G also remains unavailable: I has not yet been added when the first tests are evaluated, and H is absent in any case.

Ask the learner to justify both an inclusion and an exclusion. Explaining why C enters is only half the check. Explaining why G stays out reveals whether AND has been preserved. This practice makes a closure table a record of reasons rather than a list of letters copied from an answer.

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At the next round, C is already present. A’s condition B OR C is true even though B is still absent. Therefore C₂ = {A, C, I}.

In the following round, B can be added because A is present. Now C₃ = {A, B, C, I}. The apparent A–B cycle has not prevented progress. A entered through a separate permitted alternative, then supported B.

This is exactly where a flattened all-arrows diagram can mislead. If it treats every incoming edge as compulsory, it wrongly requires B before A and declares the pair blocked. If it rejects a whole catalogue merely because any cycle is visible, it reaches the same mistaken conclusion. The correct account preserves the rule and supplies an explicit entrance sequence: C, then A, then B.

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With both A and B established, D enters at round four. D then satisfies E’s alternative at round five, and E permits F at round six. The resulting set is {A, B, C, D, E, F, I}.

One more evaluation adds nothing. G still needs H, while H still needs G. Having I is insufficient to start that pair. The calculation has stabilised.

Read the outcome precisely. Seven of the nine modules are reachable from the supplied starting record and foundation rules. G and H are not reachable under those rules. The catalogue has both a cycle with an entrance and a cycle without one. A single label such as cyclic therefore loses the difference that matters to the student’s question. The full trace shows where progress becomes possible and where it genuinely stops.

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The rows below record cumulative sets, not just newly added modules. Keeping that convention visible helps another reader reproduce the calculation.

Every addition has a witness in the previous row. C and I use foundation rules; A uses C; B uses A; D uses A and B; E uses D; F uses E. No row assumes G or H in order to justify the other.

This witness check is a practical defence against accidental same-round reasoning. It also makes the table useful in discussion: a reviewer can point to the exact row and prerequisite that supports a proposed progression step.

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There are only nine distinct modules. Each strict enlargement adds at least one previously absent module, and no module is removed. Therefore there can be at most nine strict enlargements before the process stabilises.

This finite bound does not claim that a student needs nine academic terms. It concerns iterations of a small logical procedure. In the worked example, two foundations enter together and some modules never enter, so the actual number of strict enlargements is smaller.

The argument generalises to any finite catalogue with the stated persistent-completion rules. A calculation that repeatedly changes without adding a genuinely new module has either failed to use set semantics or introduced another kind of state. Checking what can change is often a clearer termination argument than trusting that a loop will eventually stop because the example looks small.

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For expressions built only from positive module conditions, AND, OR, true and false, adding completed modules cannot turn a true expression false. This is monotonicity with respect to set inclusion.

An AND expression stays true because both required parts remain true. An OR expression stays true because at least one previously true part remains true. Repeating that reasoning through nested expressions establishes the same property for the whole condition.

This is why completing every currently eligible module is harmless within our simplified model. It cannot remove another module’s logical eligibility. Real preclusions, alternative-credit restrictions or limited opportunities can change that conclusion, and we examine that boundary later. State the positive-rule assumption before relying on the procedure. A proof based on monotonicity does not automatically apply to a catalogue containing conditions whose truth can be lost.

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A fixed point is a set that does not change when the addition rule is applied. The final seven-module set is one. The set containing all nine modules is also a fixed point: if G and H are already assumed present, their conditions introduce nothing new.

But that larger fixed point does not establish how G and H could first be completed. It contains a self-supporting assumption. Starting from the empty record never produces either member.

This distinction answers a subtle error in a plausible worksheet. A learner may check that every module’s prerequisite appears somewhere in the final list and call the list feasible. The list may still lack a valid order from the actual starting record. A final membership check must be accompanied by a progression account. Circular support within a proposed list is not evidence of an available first step.

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Take any fixed point that contains the supplied initial passes. It must contain every foundation or module that the rules permit from those passes. Therefore it contains the first calculated enlargement.

If it contains Cᵣ, monotonicity means it must also contain every module newly permitted from Cᵣ. Since it is already closed under the rules, it contains Cᵣ₊₁. Repeating this argument shows that it contains every calculated round and therefore the stabilised set.

Our calculated fixed point is consequently the smallest one containing the starting record. This is the least-fixed-point interpretation of the rules. The proof is useful because it prevents an unsupported larger solution from replacing the reachable one. It also gives the learner a reason for the procedure, rather than asking them to memorise repeated addition as an arbitrary trick.

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Each newly added module has its condition satisfied by an earlier round. We can therefore list modules in round order and, within a round, any order. Every prerequisite used as a witness already appears earlier. The worked table supplies a valid finite sequence in our model.

Conversely, consider any valid finite completion sequence from the initial record. Its first added module satisfies a condition using initial passes. It must belong to the closure. If every earlier completion belongs to the closure, monotonicity and closure ensure the next valid module belongs too.

Thus the calculation includes every module that some valid finite sequence could reach, and every module it includes has a valid sequence. These two directions matter. A method that finds a few valid modules but misses others is incomplete; a method that admits unsupported circular modules is unsound.

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Change only A’s rule from B OR C to B AND C. From the empty record, C and I still enter, but A cannot enter without B, and B cannot enter without A. D, E and F consequently remain blocked as well.

The reachable set now contains only C and I. A small connector change has altered the downstream result substantially. It did not merely redraw a line in a diagram.

Now make the opposite error at G: replace H AND I with H OR I. Once I enters, G can follow, then H. That enlarged result belongs to the changed rule, not the original catalogue. These paired comparisons expose both directions of mistranslation. Turning OR into AND can falsely block a route; turning AND into OR can falsely admit one. Accurate reasoning begins before graph analysis, with faithful reading.

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Return to the original ledger and suppose H is explicitly supplied as already recognised in the initial record. This is a new fictional scenario, not a recommendation to award an exemption.

At the first round C and I enter, alongside the retained H. At the second, A and G enter: A uses C, while G uses H and I. B, D, E and F follow through their earlier chain. All nine modules become reachable.

The recognition record changes the starting information, not the prerequisite text. Keep those two operations separate. Rewriting G’s AND condition to obtain the same convenient outcome would misdescribe the scenario. In an actual programme, the responsible institution decides whether a prior qualification, pass or approved arrangement satisfies a named requirement. A planning calculation can use a supplied decision; it cannot manufacture that decision.

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Suppose the only additional recognised pass is I. In the original catalogue, I already has a foundation route. Starting with I can change when it appears in the logical rounds, but it still does not supply G or H.

G continues to need H, and H continues to need G. The final reachable set remains the original seven modules. An apparently helpful recognition record therefore need not change the eventual reachability of the target.

This example makes advice questions more precise. Ask which named condition the proposed recognition would satisfy and what still blocks the next step. A general statement that more recognised credits must solve the problem is too weak. Additional credit can be valuable for another requirement while making no difference to this particular logical barrier. The target and the recognition’s documented effect should remain connected.

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Consider a separate three-module catalogue: J requires K AND L; K requires J AND L; L requires J AND K. No foundation rule is supplied.

With no initial passes, nothing starts. With only J established, K still needs L and L still needs K. The same reasoning holds if the single starting pass is K or L. One recognised entrance is insufficient.

Any two established passes permit the third. Therefore two are necessary and sufficient to reach all three in this separate fictional model. All three modules lie in one mutually connected graph region, yet one starting pass does not unlock it. A graph’s strongly connected grouping can help organise investigation, but it does not replace the AND/OR conditions. Preserve the expressions when assessing what kind of initial support would actually change the result.

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The closure says which modules are eventually reachable under the supplied rules. It need not provide the fewest completed modules needed for one target, the cheapest option or the earliest calendar finish.

For example, if a target requires A OR Z and both are reachable, closure can include both. A particular target plan may need only one. Choosing between them requires an objective and the relevant costs or timing assumptions.

The round table is also not necessarily a minimal learning plan for an actual student. It grants eventual successful completion of every eligible module and ignores workloads. The existing guide to prerequisites and limited offering terms develops the separate scheduling account. Establish logical possibility first, then add the conditions needed for the student’s actual planning question.

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A module may help satisfy a prerequisite expression and still have a different role in a graduation checklist. A recognised pass might clear one condition without supplying a required number of credits in a particular category.

Conversely, a module that counts towards a credit total does not automatically clear a named prerequisite. The two records should be connected through the documented programme rules, not treated as interchangeable sets of numbers.

The existing article on allocating modules to overlapping requirement categories addresses another feasibility question: whether distinct completed modules can fill required slots without forbidden double use. Our closure calculation establishes neither that allocation nor graduation. A dependable plan keeps each conclusion attached to the requirement it actually checks, so progress in one account does not silently approve another.

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The main model requires prerequisites to have been completed earlier. A genuine permission for concurrent study changes that temporal relationship. We cannot represent it simply by assuming each module in a cycle has already been passed.

For a separate fictional permission, two named modules might be allowed as an approved joint registration after a foundation pass. That joint action needs its own explicit rule and its other conditions. A strict earlier-pass closure would correctly reject the cycle under the original rules but would be incomplete for the changed joint-action system.

Read the applicable meaning before choosing the representation. Corequisite terminology can have institution-specific details. The useful learning outcome is recognising which relationship must be clarified, not treating an attractive simultaneous plan as proof that the institution permits it. A different permission requires a different supplied model.

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Suppose a separate rule permits Q only while X has not been completed. Completing X can then turn Q’s condition from true to false. The positive-rule monotonicity argument no longer applies.

An indiscriminate procedure that completes every currently eligible option may lose a route that another sequence preserves. A state-based planning model would need to account for choices and their consequences. Simply adding a NOT symbol to the old table does not preserve the old proof.

This is an explicit limit, not a claim that every real preclusion has that exact meaning. Institutions define their own conditions, which must be read accurately. When a new rule enters a planning account, ask which earlier proof depended on its absence. This habit makes mathematical transfer more reliable than treating a successful algorithm as universally applicable to any catalogue language.

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If a catalogue line is unclear, record the uncertainty and the question that would resolve it. Do not treat the unknown as true because the target is desirable, or as false and then report a definitive impossibility.

For a fictional sensitivity check, calculate one scenario where the disputed alternative is permitted and another where it is not. If the target is reachable in both, that particular uncertainty may not affect the reachability conclusion. If the results differ, the conclusion remains conditional.

These are scenario calculations, not authority to interpret the real rule. Keep the source wording, applicable programme version and responsible advice route with the plan. The aim is to arrive at a precise question: does this documented condition permit the entrance used by this sequence? That is more useful than asking whether the whole course is flexible.

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For each added module, record its round, the rule version and the earlier passes that satisfy the chosen alternative. For an AND expression, preserve witnesses for every required part. For an OR expression, identify at least one actual satisfied route.

Keep established recognition records separately from newly constructed completions. This makes it possible to see whether a route depends on an official supplied clearance, a foundation rule or an ordinary earlier pass.

An audit should also list the blocked targets and their unresolved conditions. Do not include only the successful route. A useful record explains why G and H remain unavailable in the main example, even though I is present. With that explanation attached, another reader can assess exactly what a changed rule or starting record would need to alter.

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A fictional learner writes every module into a final set and checks that every prerequisite appears somewhere in it. They conclude that all nine can be completed from the empty record.

Ask for the first completion among G and H. G requires H earlier, while H requires G earlier. Neither can be that first completion under the original rules. The final membership check has mistaken a closed assumption for a reachable history.

Repair the answer by constructing rounds from the supplied initial set and writing a witness for each addition. The revised conclusion keeps seven reachable modules and excludes the unsupported pair. This is a focused correction: it preserves the useful idea of checking conditions while adding the missing earlier-completion requirement. The learner now has a test that transfers to another circular-looking catalogue.

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Use five new modules. P is a foundation. Q requires R OR P. R requires Q. S requires Q AND T. T requires S. Start with no recognised passes.

Calculate the closure before reading the next chapter. Identify the order in which the reachable modules enter. Then answer whether the presence of P makes every cycle escapable, and explain what still blocks S or T.

For a second version, change only T’s rule to S OR R. Recompute from the same empty starting record. Do not carry a completion from the first calculation into the second unless it is re-established by the new trace. The two versions deliberately share most of their wording. Their difference tests whether the learner follows the changed connector and its consequences rather than memorising the shape of a previous diagram.

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In the first version, round one adds P, round two adds Q through P, and round three adds R through Q. Nothing else enters. S needs T as well as Q, and T needs S. The final reachable set is {P, Q, R}.

In the second version, P, Q and R enter in the same first three rounds. T can then enter through R at round four, and S follows at round five. All five are reachable. The changed alternative supplies a genuine entrance into the previously blocked pair.

The explanation should name that entrance. Saying only that the second graph works misses the mechanism. For an extension, retain the first version but supply T as an initial recognised pass. Q still needs its route from P, and S can enter only once Q is established. A seed does not erase other AND requirements.

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Another fictional module permits any two of P, Q and R. Its condition can be written (P AND Q) OR (P AND R) OR (Q AND R). With only P complete, the condition is false. With P and R complete, it is true. All three also satisfy the inclusive condition.

This is still a positive expression, so the same closure reasoning applies. The translation is useful because it exposes what a numerical phrase requires instead of treating any two as an unexplained label.

Keep the condition distinct from any two credits or any two activities. The named items and the threshold’s scope matter. A formal expression preserves a supplied rule; it cannot decide which real courses the institution recognises as members of the permitted set.

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Consider the fictional condition P AND (P OR Q). Once P is established, the expression is true whether Q is present or absent. It is logically equivalent to P alone.

The repeated appearance of P does not require two separate completions of the same module. Our model uses a set of named passes, not one token consumed for every occurrence in an expression. This matters when a diagram displays the same foundation along several routes.

An independent check can evaluate all four possibilities for the presence of P and Q. The expression and P agree in each case. A real programme can have separate rules about distinct credits or double counting; those need their own representation. Do not import token consumption into this Boolean prerequisite account merely because a module name appears twice.

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Completion persists during one calculation, but the rule set is fixed throughout that calculation. If a prerequisite expression changes, begin a fresh analysis using the applicable new rules and the actual established starting record.

Do not retain a hypothetical completion solely because it appeared in an earlier version’s planning trace. It was a consequence of that version, not an independently supplied pass. Conversely, an actual recognised pass should not be removed just because a teaching example has changed its letters.

Keep rule versions and evidence roles visible. This separates a student’s real record from a sequence that was merely shown to be possible. It also makes a changed conclusion explainable: identify the altered condition, recompute its consequences and preserve the earlier account as a different scenario.

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The repair route begins with small completed sets and individual expressions. Stabilise AND, inclusive OR and grouping before introducing a large catalogue. Ask the learner to justify one inclusion and one exclusion.

The stabilisation route builds round tables and witnesses. Change a single connector or starting pass, then explain why the final closure changes or remains the same. The student should be able to reconstruct the trace without copying the earlier letters.

The extension route proves leastness, compares valid sequences and examines limits such as preclusions or concurrent actions. It also asks what evidence is needed for a real academic conversation. The Education parent guide connects this approach with independent transfer: understanding a method includes knowing when its assumptions no longer fit.

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Does any cycle make a programme impossible?

No such conclusion follows from a flattened graph with alternatives. The main example has cycles that become reachable through explicit OR entrances. Some other cycles remain blocked. Preserve the rules and the starting record.

Does a final set satisfying every rule prove a valid beginning?

Not necessarily. A larger fixed point can contain self-supporting assumptions. A trace from established passes distinguishes the reachable closure from those unsupported additions.

Are rounds academic terms?

They are logical stages here. Actual offering calendars, workloads, places and durations require a separate scheduling model and current institutional information.

Can this calculation grant a prerequisite waiver?

No. A recognised clearance can be supplied as an input only when the relevant decision exists. The model does not authorise it.

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For institutional and programme questions, return to Universities. Continue with offering-term scheduling and requirement-category allocation for their distinct feasibility questions.

To read actual documents, use a university module description, a programme handbook and a course registration guide. These reading routes support accurate interpretation; they do not replace the responsible institution’s answer.

UBC Okanagan's registration guidance distinguishes prerequisite requirements and its waiver route. Its calendar introduction directs readers to current course sections and schedules. These are source-specific examples, not the invented rules above. For a formal computing connection, a Berkeley course project on relational algebra discusses monotone fixed-point evaluation of positive rules. Sources checked on 24 October 2026. Every catalogue, trace, counterexample and proof here is an original educational construction. Explore How X Works for wider learning routes.

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