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This overview explains how advice affects exact decision-diagram
and satisfiability lifecycles with checked results and fallback.

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A smaller diagram or a changed search order does
not by itself establish a faster cold or resident request.

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A decision diagram for and can test A first.

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A zero immediately determines output zero.

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When A is one, the answer equals B.

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The structure encodes both possibilities rather than one fixed assignment.

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Input zero one reaches zero without needing B to determine the answer.

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Input one one follows the A-one path and then the B-one path to one.

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The shorter path for one assignment does not imply that B is irrelevant

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to the whole function.

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For a multiplexer, S selects whether the result equals A or B.

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Testing S first exposes that choice immediately.

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Other variable orders organize the remaining cases differently and can change the size of

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a reduced diagram.

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The useful order is a structural question that needs exact preservation and cost checks.

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E1 and E2 put advice in front of exact
BDD-order and SAT or equivalence lifecycles.

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E1 and E2 put advice in front of exact
BDD-order and SAT or equivalence lifecycles.

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An ordering may change diagram size without changing the function.

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SAT advice may change search without
changing what counts as a valid witness.

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For a conceptual equivalence test, compare
A AND B with NOT of NOT A OR NOT B.

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Build a miter that exclusive-ors their outputs and ask whether it can be true.

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Exhaustive enumeration finds no satisfying assignment,
establishing equivalence on this finite domain.

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Change the second rule to A OR B and the miter has a witness at 01.

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A fresh BDD build,

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a resident restriction session, a fresh SAT solve
and an assumptions session have different costs.

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The retained learned advice did not establish
a general timing win and remains unpromoted.

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Exact witness checking and advice-off fallback stay authoritative.

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Work a second complete case with A 1, B 0.

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Keep this assignment fixed throughout the calculation.

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Each symbol retains the meaning already introduced; a new value does not create a new logical rule.

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And returns zero because B is false.

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Or returns one because A is true.

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Their exclusive-or is one, so this assignment satisfies the miter and demonstrates disagreement.

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The witness is useful because it can be substituted back into both original expressions and checked directly.

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The two original formulas are not both true on this assignment: and is zero and or is one.

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Their miter is true because those truth values differ.

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Satisfiable means that some satisfying assignment exists.

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Both and and or are individually satisfiable, for example at one one; that does not make their functions equivalent.

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The calculated answer belongs to this complete assignment under the stated rule.

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To compare F and its De Morgan expression, construct their exclusive-or, called a miter.

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H is one exactly where the two original expressions disagree.

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Asking whether H can be one is therefore an exact equivalence question.

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The two output columns agree in all four rows, so the miter is always zero.

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A sound satisfiability procedure reporting unsatisfiable for the miter establishes that no

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disagreement assignment exists in the declared universe.

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Replace G with or and the input zero one makes the miter one.

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Substitute that witness into the original expressions: and returns zero

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while or returns one.

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A satisfying miter assignment explains the actual disagreement.

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Reusing resident diagrams or solver state has separate preparation and per-query semantics

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that must be declared.

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Measure the caller's actual lifecycle and objective.

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Measure the caller's actual lifecycle and objective.

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Keep exact witness checks and advice-off fallback authoritative.