Past exam of the computer science course of the University of Oxford / 2015 / Knowledge representation and reasoning / 1 / c

With existential rules understood as function-free tuple-generating dependencies, $\boxed{\text{only (ii)}}$ belongs to the fragment. Split its antecedent to obtainThese two existential rules have exactly the same meaning as (ii): either category requires a supervisor, and when both hold the same supervisor may witness both requirements.

Sentence (i) requires a genuine choice between two positive alternatives and also an exclusion condition. Positive existential rules cannot enforce that exclusion: interpreting all their relations as full relations on a one-element domain always satisfies them, whereas that interpretation violates (i). Sentence (iii), in the supplied vocabulary, uses the nonconstant function $CountSupervise$ and is therefore outside the function-free fragment. Introducing a new relation for the counting comparison would change the representation and require additional axioms; it is not an equivalent sentence in the stated vocabulary.

For ordinary function-free Datalog, $\boxed{\text{none of the three sentences}}$ has the required form. Sentence (i) is not a definite first-order Horn clause: the category choice gives two positive alternatives, and its second clause is a negative constraint. Even allowing constraints would not remove the positive choice. Semantically, the category-choice condition fails closure under intersection of interpretations with fixed domain and functions: one model can choose only the undergraduate category, another only the masters category, and their intersection chooses neither.

Sentence (ii) needs an existentially supplied supervisor. Function-free Datalog cannot demand such a witness without naming or otherwise fixing it in the rule. Its Skolemized form is a first-order Horn clause with a function, rather than a Datalog rule. Sentence (iii) also contains a nonconstant function. Being Horn is therefore insufficient for membership in Datalog.

The standard relational guarded fragment contains $\boxed{\text{(i) and (ii)}}$. In (i), $Student(x)$ guards the only quantified variable:The constants in the course-taking relation do not introduce additional free variables that need guarding.

For (ii), use the two implications from part (c)(i). Each outer universal quantification is guarded by its unary category predicate. The inner existential quantification can be writtenwhose guard contains both $x$ and $y$. Sentence (iii) uses a nonconstant function and so is outside the standard relational guarded fragment. This conclusion uses the original vocabulary; it does not silently replace the counting term by a fresh relation.

In ALC, $\boxed{\text{only (ii)}}$ is representable over the given symbols. It is the general concept inclusionEquivalently, use one general concept inclusion for each category. Unary predicates become description logic concepts and the supervision predicate becomes a description logic role.

Sentence (i) depends on a ternary predicate evaluated at two named constants. Basic ALC has unary description logic concepts and binary description logic roles, with no constructor for that course-and-year slice of a ternary predicate. Sentence (iii) uses a counting function and a comparison relation that basic ALC cannot express. Replacing these conditions by freshly named description logic concepts would give an informal encoding, but would not define their meanings or establish equivalence in the supplied vocabulary.