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Compute an arc-consistency closure by revising every directed constrained pair in complete sweeps until one sweep deletes nothing. The first sweep is unconditional. With $v$ variables, maximum constraint domain size $d$, and $e$ constrained pairs, at most $vd$ deletions permit at most $vd+1$ sweeps. At $O(d^2)$ per revision plus $O(v)$ per emptiness scan, a direct time complexity bound is $O((vd+1)(e{d}^2+v))$. AC-3 uses a work list to avoid scanning unrelated arcs after each local change.

Forward checking makes minimum remaining values respond to current restrictions rather than the original constraint domain sizes. After $X=v$, its neighbors may have fewer choices, become forced or lose all choices. Minimum remaining values then selects the newly constrained variable early, while an empty constraint domain triggers immediate failure. The two mechanisms expose failure sooner and reduce wasted backtracking.

Forward checking only tests an unassigned neighbor against the newly assigned value. It need not check whether two still-unassigned variables have mutually supporting values. Arc consistency requires every value at either end of each binary constraint to have support in the other constraint domain, and AC-3 propagates deletions until this condition stabilizes or a constraint domain becomes empty.^[ref]

For example, let $X,Y,Z$ initially have constraint domains $\{1,2\}$ with constraints $X\ne Y$ and $Y\ne Z$. Assigning $X=1$ makes forward checking reduce $D_Y$ to $\{2\}$, but leaves $D_Z=\{1,2\}$. Arc consistency also deletes $2$ from $D_Z$, leaving $\{1\}$. Neither procedure alone decides every general constraint satisfaction problem: a triangle of two-valued inequalities satisfies arc consistency but is unsatisfiable.

The translation gives a sound and complete route through an optimized SAT solver; it trades access to general-purpose SAT machinery against encoding cost and the possible loss of constraint-specific structure. Modern SAT solvers use efficient unit propagation, adaptive branching and clause learning. Learned Boolean clauses can summarize conflicts involving many original variables, allowing later branches to avoid repeating the same failed combination. The reduction also offers a standard, reusable interface: improvements to a SAT solver can benefit many finite-domain constraint satisfaction problems.

The Boolean formula may be much larger than the original constraint satisfaction problem, especially for a succinct high-arity relation. One original variable becomes many indicators, and general branching may lose the interpretation that the minimum remaining values and degree heuristic exploit directly. Native constraint propagation can exploit a relation without enumerating all its forbidden tuples. Even in the binary case, stronger propagation does not follow just from switching to SAT.

For example, take $D_X=\{1,2\}$, $D_Y=\{a,b,c\}$, and let the only allowed pairs be $(1,c),(2,a),(2,b)$. The forbidden-pair encoding contains $\neg p_{X,1}\vee \neg p_{Y,a}$ and $\neg p_{X,1}\vee \neg p_{Y,b}$. If $p_{Y,c}$ becomes false, arc consistency deletes $X=1$ because it has no support. Yet the direct encoding has no unit Boolean clause forcing $\neg p_{X,1}$: both $a,b$ remain possible individually, so plain unit propagation does not make that deletion. A SAT solver can eventually derive it through additional reasoning or search. Conversely, clause learning can combine information beyond the local consistency enforced by AC-3. Neither approach uniformly dominates: performance depends on representation, propagation strength and the instance's structure; both still face exponential worst-case search on general inputs.

The first assertion is true. Each deletion removes a value with no support in an adjacent constraint domain. If a complete solution still used that value, the solution’s neighboring value would supply exactly such a support, a contradiction. By mathematical induction, AC-3 never deletes a value used by an original solution. An empty constraint domain therefore proves that no solution exists.

The second assertion is false. Take three Boolean variables $X,Y,Z$ with constraint domains $\{0,1\}$ and the binary constraints $X\ne Y$, $Y\ne Z$, $Z\ne X$. Every value has the opposite value as support across each edge, so AC-3 deletes nothing and returns true. Yet $X\ne Y$ and $Y\ne Z$ imply $X=Z$, contradicting $Z\ne X$. Arc consistency checks local support, not compatibility of all supports in one complete assignment.

The third assertion is true under the usual explicit-input, constant-cost constraint-checking model. With $v$ variables, $e$ constrained pairs and maximum constraint domain size $d$, AC-3 has time complexity $O(v+e{d}^3)$ in Big O notation. A revision costs $O(d^2)$; each deletion can re-enqueue incoming arcs, and each constraint domain loses at most $d$ values, giving $O(ed)$ revisions including the initial arcs. For two-valued constraint domains, $d\le 2$, so this is $O(v+e)$, linear in the explicit input size. This uses fixed-size relations, or already combined constant-time tests per pair. “Binary” normally describes constraint arity, not two-valued constraint domains; the latter restriction is the extra hypothesis needed here. As usual, initially empty constraint domains should be rejected before entering the algorithm, including isolated variables.

Interpret MyAC as doing at least one initial sweep, then repeating while the preceding sweep deleted a value. Its loop guard otherwise has no initialized truth condition. Both procedures apply the same support-based deletion rule, so both preserve every solution and terminate because finite constraint domains only shrink. Both return false on a detected empty constraint domain; neither successful return establishes satisfiability.

If a sweep changes no constraint domain, every directed arc has been revised against unchanged constraint domains, so all arcs satisfy arc consistency. Conversely, AC-3 places every potentially affected incoming arc back on its work list after a deletion. Its empty work list therefore also establishes arc consistency. When both run to successful completion, they compute the same arc-consistency closure, independently of revision order. To see maximality, any subdomains of the initial constraint domains that satisfy arc consistency survive every revision: each of their values retains a support inside those subdomains. Thus neither procedure can delete a value belonging to any such fixed point. Early failure may leave different intermediate constraint domains, but both failures correctly prove impossibility.

Their scheduling costs differ. AC-3 directs work toward arcs that may have lost supports, with standard time complexity $O(v+e{d}^3)$. The full-sweep arc consistency method revisits all $2e$ arcs after even one remote change. There are at most $vd$ value deletions, so at most $vd+1$ sweeps; a direct implementation including a scan for empty constraint domains costsIf empty-domain detection is maintained during revisions rather than rescanned, the familiar comparison is $O(ve{d}^3)$ versus $O(e{d}^3)$, apart from initialization. These are upper bounds, not exact costs on every input.

In practice MyAC is simple and may have low overhead or benefit from a fortunate sweep order. It can also waste many scans after changes propagate slowly opposite to that order. AC-3 incurs work-list bookkeeping but normally avoids those unrelated revisions. Both inherit the inability of arc consistency to decide arbitrary constraint satisfaction problems.