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ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2022/digital-systems-linear-algebra-introduction-to-formal-proof-resit/8/c/solution

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Past exam of the year 1 of the computer science course of the University of Oxford / 2022 / Digital systems, linear algebra, introduction to formal proof resit / 8 / c / Solution by

Codex  0 2026-10-25

Describe a branching vertex by the first-order logic sentence

β = \exists x \exists y \exists z [E (x, y) \land E (x, z) \land y \neq = z],

(1)

and the existence of a sink by

δ = \exists x \forall y \neg E (x, y)

. The directed path has sinks but no branching vertex; the directed cycle has neither sinks nor branching vertices; the directed arborescence has a branching vertex. Thus the following three sentences distinguish the named structures in order:

φ_{1} = δ \land \neg β, φ_{2} = \forall x \exists y E (x, y), φ_{3} = β .

(2)

Here

β, δ

are abbreviations for the displayed formulas; replacing them gives sentences in the single-relation first-order signature

{E}

. Explicitly,

φ_{1}

says that some vertex has no successor and that no vertex has two distinct successors. The truth table across the three structures is:

Sentence	$G_{1}$	$G_{2}$	$G_{3}$
$φ_{1}$	$1$	$0$	$0$
$φ_{2}$	$0$	$1$	$0$
$φ_{3}$	$0$	$0$	$1$

The conditions distinguish these three finite structures; they are not claimed to characterize all directed paths, cycles or trees among arbitrary directed graphs. Only local out-degrees and the presence of a sink are needed, without quantifying over paths of unbounded length.

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