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Relational Calculus Chapter 4 Database Management Systems, R. Ramakrishnan 1 Relational Calculus Comes in two flavours: Tuple relational calculus (TRC) and Domain relational calculus (DRC). Calculus has variables, constants, comparison ops, logical connectives and quantifiers. – TRC: Variables range over (i.e., get bound to) tuples. – DRC: Variables range over domain elements (= field values). – Both TRC and DRC are simple subsets of first-order logic. Expressions in the calculus are called formulas. An answer tuple is essentially an assignment of constants to variables that make the formula evaluate to true. Database Management Systems, R. Ramakrishnan 2 First-Order Predicate Logic Predicate: is a feature of language which you can use to make a statement about something, e.g., to attribute a property to that thing. – Peter is tall. We predicated tallness of peter or attributed tallness to peter. – A predicate may be thought of as a kind of function which applies to individuals and yields a proposition. Proposition logic is concerned only with sentential connectives such as and, or, not. Predicate Logic, where a logic is concerned not only with the sentential connectives but also with the internal structure of atomic propositions. Database Management Systems, R. Ramakrishnan 3 First order predicate logic First-order predicate logic, first-order says we consider predicates on the one hand, and individuals on the other; that atomic sentences are constricted by applying the former to the latter; and that quantification is permitted only over the individuals First-order logic permits reasoning about propositional connectives and also about quantification. – All men are motal – Peter is a man – Peter is mortal Database Management Systems, R. Ramakrishnan 4 Tuple Relational Calculus Query: {T|P(T)} – T is tuple variable – P(T) is a formula that describes T Result, the set of all tuples t for which P(t) evaluates True. – Find all sailors with a rating above 7. – {S | S Sailors S .rating 7} Atomic formula – R Re l – R.a op S.b , op is one of , , , , , – R.a op constant Database Management Systems, R. Ramakrishnan 5 TRC Formula – Any atomic formula – p, p q, p q, p q – R( p( R)) – R( p( R)) Example – Find the names and ages of sailors with a rating above 7 {P | S Sailors( S .rating 7 P.name S .name P.age S .age)} Database Management Systems, R. Ramakrishnan 6 TRC Find the names of sailors who have reserved all boats {P | S SailorsB Boats (R Re serves( S.sid R.sid R.bid B.bid P.sname S .sname)) Find sailors S such that for all boats B there is a Reserves tuple showing that sailor S has reserved boat B Database Management Systems, R. Ramakrishnan 7 Domain Relational Calculus Query has the form: x1, x2,..., xn | p x1, x2,..., xn Answer includes all tuples x1, x2,..., xn that make the formula p x1, x2,..., xn be true. Formula is recursively defined, starting with simple atomic formulas (getting tuples from relations or making comparisons of values), and building bigger and better formulas using the logical connectives. Database Management Systems, R. Ramakrishnan 8 DRC Formulas Atomic formula: – x1, x2,..., xn Rname , or X op Y, or X op constant – op is one of , , , , , Formula: – an atomic formula, or – p, p q, p q , where p and q are formulas, or – X ( p( X )) , where variable X is free in p(X), or – X ( p( X )) , where variable X is free in p(X) Database Management Systems, R. Ramakrishnan 9 Free and Bound Variables The use of quantifiers X and X in a formula is said to bind X. – A variable that is not bound is free. Let us revisit the definition of a query: x1, x2,..., xn | p x1, x2,..., xn There is an important restriction: the variables x1, ..., xn that appear to the left of `|’ must be the only free variables in the formula p(...). Database Management Systems, R. Ramakrishnan 10 Find all sailors with a rating above 7 I, N, T , A | I, N, T , A Sailors T 7 The condition I, N, T , A Sailors ensures that the domain variables I, N, T and A are bound to fields of the same Sailors tuple. The term I, N, T , A to the left of `|’ (which should be read as such that) says that every tuple I, N, T , A that satisfies T>7 is in the answer. Modify this query to answer: – Find sailors who are older than 18 or have a rating under 9, and are called ‘Joe’. Database Management Systems, R. Ramakrishnan 11 Find sailors rated > 7 who’ve reserved boat #103 I, N, T , A | I, N, T , A Sailors T 7 Ir, Br, D Ir, Br, D Re serves Ir I Br 103 We have used Ir , Br , D . . . as a shorthand for Ir Br D . . . Note the use of to find a tuple in Reserves that `joins with’ the Sailors tuple under consideration. Database Management Systems, R. Ramakrishnan 12 Find sailors rated > 7 who’ve reserved a red boat I, N, T , A | I, N, T , A Sailors T 7 Ir, Br, D Ir, Br, D Re serves Ir I B, BN, C B, BN, C Boats B Br C ' red ' Observe how the parentheses control the scope of each quantifier’s binding. This may look cumbersome, but with a good user interface, it is very intuitive. (Wait for QBE!) Database Management Systems, R. Ramakrishnan 13 Find sailors who’ve reserved all boats I, N, T , A | I, N, T , A Sailors B, BN,C B, BN,C Boats Ir, Br, D Ir, Br, D Re serves I Ir Br B Find all sailors I such that for each 3-tuple B, BN,C either it is not a tuple in Boats or there is a tuple in Reserves showing that sailor I has reserved it. Database Management Systems, R. Ramakrishnan 14 Find sailors who’ve reserved all boats (again!) I, N, T , A | I, N, T , A Sailors B, BN, C Boats Ir, Br, D Re serves I Ir Br B Simpler notation, same query. (Much clearer!) To find sailors who’ve reserved all red boats: ..... C 'red ' Ir, Br, D Re serves I IrBr B Database Management Systems, R. Ramakrishnan 15 Unsafe Queries, Expressive Power It is possible to write syntactically correct calculus queries that have an infinite number of answers! Such queries are called unsafe. – e.g., S | S Sailors It is known that every query that can be expressed in relational algebra can be expressed as a safe query in DRC / TRC; the converse is also true. Relational Completeness: Query language (e.g., SQL) can express every query that is expressible in relational algebra/calculus. Database Management Systems, R. Ramakrishnan 16 Summary Relational calculus is non-operational, and users define queries in terms of what they want, not in terms of how to compute it. (Declarativeness.) Algebra and safe calculus have same expressive power, leading to the notion of relational completeness. Database Management Systems, R. Ramakrishnan 17

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Relational calculus, relational algebra, Tuple relational calculus, atomic formula, tuple calculus, the Sailors, predicate calculus, domain calculus, Domain relational calculus, relational model

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posted: | 5/8/2010 |

language: | English |

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