Is safe tuple relational calculus a turing complete language?
+3
A:
Let's forget about safety. By Codd's theorem, relational calculus is equivalent to first order logic. FOL is very limited, it can't express the fact that there's a route from a point A to point B in some graph (it can express the fact that there's a route from a point A to point B in limited length, for example ∃x ∃y ∃z ∃t route(a,x) and route(x,y) and route(y,z) and route(z,t) and route(t,b) means there's a route of length 4).
See descriptive complexity for a description of what is the strength of different logics.
sdcvvc
2010-01-10 17:32:50
Larry Watanabe
2010-01-10 17:49:14
You're saying that route is _smallest_ transitive relation satisfying edge(x,y) -> route(x,y). This definition requires least-fixed-point. To define a new relation in tuple calculus, you may use intersection, union, projection... but it is not possible to say "route it is a relation satisfying the following conditions...". You might also quantify over relations, but that's higher-order logic, and quantification is allowed only over individuals in FOL.
sdcvvc
2010-01-10 18:00:19
+1 thanks for the explanation sdcwc
Larry Watanabe
2010-01-10 20:31:46
Thanks for your reply..Its really a useful.
Sailaja
2010-01-15 13:41:58
+1
A:
According to Codd's Theorem, relational algebra and relational calculus are equivalent. It is well-known that relational algebra is not Turing Complete, so neither is relational calculus.
[Edit] You cannot, for instance, do aggregate operations (such as sum, max) or make recursive queries in relational algebra/calculus. See here (near the end).
BlueRaja - Danny Pflughoeft
2010-01-10 17:34:15
Either you are wrong or Larry Watanabe is. I have no idea of the topic, but this is getting interesting to watch! (goes to fetch popcorn)
Carl Smotricz
2010-01-10 17:36:35
Now relational algebra not being Turing Complete is more well-known :)
Larry Watanabe
2010-01-10 17:38:26
However, from reading another poster's link to Codd's theorem, relational algebra is not equivalent to relational calculus - relational algebra is essentially equivalent to propositional logic whereas relational algebra is equivalent to FOL.
Larry Watanabe
2010-01-10 17:44:00
@Larry: http://en.wikipedia.org/wiki/Relational_calculus - "The relational algebra and the relational calculus are essentially logically equivalent ... This result is known as Codd's theorem."
BlueRaja - Danny Pflughoeft
2010-01-10 18:22:18