Past exam of the computer science course of the University of Oxford / 2015 / Knowledge representation and reasoning / 2

An initial split into an acyclic definition and the remaining general concept inclusions isThe definition of $C$ is suitable for TBox unfolding: its right-hand side does not depend on $C$, and $C$ has only this defining left-hand side. The other displayed general concept inclusions do not initially have an atomic defined left-hand side. In particular, $A\sqcap B⊑\perp$ can be rewritten for TBox absorption, but is not already an atomic definition.

Absorb (4) using the equivalenceThus we can takeHere $A⊑\neg B$ is a primitive inclusion, handled by adding $\neg B$ whenever $A$ labels a node. For literal substitution-based TBox unfolding, introduce a fresh description logic concept $A_0$ and replace the primitive inclusion byThis preserves the original models after forgetting $A_0$: any model of the primitive inclusion extends by taking $A_0=A$, and any model of the new definition satisfies the primitive inclusion. The fresh name is important. Replacing $A$ by $\neg B$ alone would incorrectly assert that every non-$B$ element is an $A$ element.

Use the substitution version of TBox unfolding from part (a), and abbreviateThe notation $\overline{L}$ is the negation normal form of $\neg L$. The unfolded input description logic concept isFor the remaining TBox, turn each general concept inclusion $X⊑Y$ into $\neg X\bigsqcup Y$, and unfold all defined names. The global description logic concept to add at every generated node isIn the last line, negating $\exists R.(\exists R.A\bigsqcup \exists R.B)$ gives $\forall R.(\forall R.\neg A\sqcap \forall R.\neg B)$. Both levels of universal restriction are needed.

Start the tableau algorithm for ALC with a root $x$ labelled $D^{\prime }$ and $\Gamma$. Expand the logical conjunctions. For the disjunction from (2), choose $\forall R.\overline{L}$. For the disjunction from (3), choose $\forall R.L$. Apart from the compound formulas already processed, the root label isThere is no existential restriction at $x$, so no successor needs to be generated. In particular, $\forall R.\exists P.\top$ creates a $P$-successor only at an existing $R$-successor; it creates no $R$-successor itself. The branch is complete and has no tableau clash. There is $\boxed{\text{no directly blocked node}}$: the open completion graph has only its root. The universal restrictions can have incompatible fillers without creating a tableau clash when there are no successors.

For comparison, choosing the other alternative $\exists R.\exists R.L$ from (2) creates an $R$-successor $y$. The root restriction $\forall R.(L\sqcap B)$ puts $A_0,\neg B,B$ in $\mathcal{L}(y)$, giving a tableau clash immediately. That branch closes before direct blocking in description logic is needed. The figure records the processed formulas through the label $\Gamma$ and displays the decisive labels explicitly.

A concrete model confirms the open branch:All three remaining general concept inclusions hold because their relevant $R$-successor requirements are vacuous. The definition of $C$ and the disjointness condition hold as well. Thus $\boxed{D\text{ is satisfiable with respect to }\mathcal{T}}$.

Using the absorbed TBox with its primitive inclusion, translate description logic concepts as unary predicates and description logic roles as binary predicates. Concept satisfiability means that some element belongs to $A\sqcap C$, while every element satisfies the TBox axioms. The required first-order sentence isOnly ${\mathcal{T}}_u$ is included here, as requested. There is no requirement to satisfy (1), (2) or (3). For instance, one element with $A=C=\{d\}$, $B=\varnothing$ and $P=\{(d,d)\}$ satisfies this first-order sentence.

Alternatively, unfolding the absorbed TBox first gives the existential first-order sentence $\exists x\exists y(A_0(x)\wedge \neg B(x)\wedge P(x,y))$. The interpretation of the eliminated names is then recovered by $A=A_0\setminus B$ and $C=\{x:\exists yP(x,y)\}$.

For EL with bottom, the eligible axioms are $\boxed{(2),(4),(5)}$. Axioms (1) and (3) contain universal restrictions, and (3) also contains a disjunction, so they are outside its syntax. For plain EL, whose constructors are $\top$, atomic names, conjunction and existential restriction, the eligible axioms are $\boxed{(2),(5)}$; (4) is excluded because it uses $\perp$. These conventions should not be conflated.

Here is a normalized TBox for the bottom-inclusive version. Introduce one fresh description logic concept $X$ for $\exists R.A$, and useThese are the usual normalized forms: an existential restriction on the left with an atomic right-hand side, an atomic left-hand side with an existential restriction on the right, or a conjunction of atomic names implying $\perp$. Here $\top$ is permitted as a filler.

The first two general concept inclusions force $X\equiv \exists R.A$. The third then says $\exists R.A⊑\exists R.\exists R.A$, exactly (2). Conversely, any model of (2) extends by $X=(\exists R.A{)}^{\mathcal{I}}$ to satisfy all three normalized general concept inclusions. The last two are the two directions of (5). For plain EL, delete just $A\sqcap B⊑\perp$ from this normalized TBox.

The standard semantics of ALC and EL do not impose the unique name assumption. Therefore the two individual names may denote the same element. A minimum-size model of either normalized TBox from part (d), together with the ABox, isThe $R$ self-loop supplies the ABox fact and makes both $\exists R.A$ and $\exists R.X$ true at $d$. The $P$ self-loop supplies exactly the existential restriction defining $C$. Disjointness holds because $B$ is empty. Hence every normalized general concept inclusion is satisfied. A domain must be nonempty, so $\boxed{|{\Delta }^{\mathcal{I}}|=1}$ is minimal.

If a separate unique name assumption were imposed, the minimum would instead be two. One can take $a\ne b$, $A=\{a\}$, $C=\{b\}$, $R=\{(a,b)\}$, $P=\{(b,b)\}$ and $B=X=\varnothing$. There is then no $R$-edge ending at an $A$ element, so the normalized requirements involving $X$ hold vacuously. That additional convention is not part of standard description logic semantics.

For plain EL, $\boxed{\text{every concept is satisfiable with respect to every EL TBox}}$. On a one-element domain, interpret every atomic description logic concept as the whole domain and every description logic role as the full binary relation. A structural induction makes every EL description logic concept denote that element. Every general concept inclusion is consequently true, including cyclic ones. The decision procedure can therefore always answer yes on well-formed inputs.

For EL with bottom, concept satisfiability with respect to general TBoxes is $\boxed{\text{P-complete}}$ under the usual logarithmic-space reductions. Polynomial-time completion propagates atomic inclusions, conjunction consequences and existential requirements; deriving $\perp$ for the queried description logic concept detects unsatisfiability. There are polynomially many such labels and normalized relationships to propagate.

For the lower bound, encode a monotone Boolean circuit with a starting description logic concept $S$. Give each input gate labelled one an inclusion $S⊑G_i$. Encode an AND gate by $G_i\sqcap G_j⊑G_k$, and an OR gate by $G_i⊑G_k$ and $G_j⊑G_k$. Finally add $G_{\mathrm{out}}⊑\perp$. Then $S$ is unsatisfiable exactly when the circuit output is one. The complement version is also P-complete, since P is closed under complement. Thus allowing $\perp$ changes a trivial satisfiability problem into a nontrivial polynomial-time one. The bottom-inclusive convention is also reflected in the ^[ref].

The complete numbered axioms in the rule fragment RL description logic are $\boxed{(1),(3),(4)}$. Their translations to Datalog with constraints areThe first rule is (1). For (3), its existential antecedent needs a two-edge $R$ path ending at an $A$ or $B$ element. Split that disjunction into two Datalog rules, and use a separate variable $w$ for an arbitrary immediate $R$-successor of $x$. The conclusion applies to every such successor, not merely the path witness $y$.

The last rule is a constraint: a ground instance with both premises true makes the knowledge base inconsistent. If an implementation permits only positive Datalog heads, replace $\perp$ by a fresh nullary predicate $Bad$ and separately check that it is never derived. Merely deriving $Bad$ without that check would not express disjointness.

Axiom (2) requires an existential witness in its head and is outside RL description logic. Axiom (5), considered as the whole equivalence, is outside for the same reason in the direction $C⊑\exists P.\top$. If definitions are split before selecting eligible general concept inclusions, its reverse direction is eligible and adds $\boxed{P(x,y)\to C(x)}$. Keep the difference between selecting the complete equivalence and retaining its rule-expressible half explicit. The permitted positions of existential and universal restrictions are specified in the ^[ref].