Temporal Relational Calculus

MAIN TEXT A natural temporal extension of the relational calculus allows explicit variables and quantification over a given time domain, in addition to the variables and quantifiers over a data domain of uninterpreted constants. The language is simply the two-sorted version (variables and constants are temporal or non-temporal) of first-order logic over a data domain D and a time domain T . The syntax of the two-sorted first-order language over a database schema ρ = {R1, . . . , Rk} is defined by the grammar rule: Q ::= R(ti, xi1 , . . . , xik) | ti < tj | xi = xj | Q ∧Q | ¬Q | ∃xi.Q | ∃ti.Q In the grammar, ti’s are used to denote temporal variables and xi’s to denote data (non-temporal) variables. The atomic formulae ti < tj provide means to refer to the underlying ordering of the time domain. Note that the schema ρ contains schemas of timestamped temporal relations (see the entry Point-stamped Temporal Models). Given a point-timestamped database DB and a two-sorted valuation θ, the semantics of a TRC query Q is defined in the standard way (similarly to the semantics of relational calculus) using the satisfaction relation DB, θ |= Q: DB, θ |= Rj(ti, xi1 , . . . , xik) if Rj ∈ ρ and (θ(ti), θ(xi1), . . . , θ(xik)) ∈ R j DB, θ |= ti < tj if θ(ti) < θ(tj) DB, θ |= xi = xj if θ(xi) = θ(xj) DB, θ |= Q1 ∧Q2 if DB, θ |= Q1 and DB, θ |= Q2 DB, θ |= ¬Q1 if not DB, θ |= Q1 DB, θ |= ∃ti.Q1 if there is s ∈ T such that DB, θ[ti 7→ s] |= Q1 DB, θ |= ∃xi.Q1 if there is a ∈ D such that DB, θ[xi 7→ a] |= Q1 where R j is the interpretation of the predicate symbol Rj in the database DB. The answer to a query Q over DB is the set Q(DB) of valuations that make Q true in DB. Namely, Q(DB) := {θ|FV (Q) : DB, θ |= Q} where θ|FV (Q) is the restriction of the valuation θ to the free variables of Q.