I need to express this constrain in relational algebra:
I have some table with one column with all possible values: ALL_VAL
and table with some values from ALL_VAL that not mach some rule: NOT_FIT_VAL
and I can calculate FIT_VAL = ALL_VAL - NOT_FIT_VAL
what i need is a constraint: in FIT_VAL there minimum one item.
I am using not equal sign with empty group:
ALL_VAL,
NOT_FIT_VAL
FIT_VAL = ALL_VAL - NOT_FIT_VAL
FIT_VAL <> {empty}
but I am not sure that <>(not equal) is allowed at all in relational algebra
there is not a single book or article that shows example or saying that I can use it.
I would like some clarification about it, and the correct expression.
thank you
Strictly speaking, the expression "FIT_VAL <> {empty}" is not a relational expression (it does not produce a relation, but a truth value instead), therefore it is a bit problematic to consider such expressions as being "valid relational algebra expressions".
But that is strictly speaking, and I don't know how much slack your textbook cuts its readers/users in that area. Under the "strictly speaking" approach, it is even outright impossible to use relational algebra to define constraints, because the definition of a constraint must produce a boolean result (does the database satisfy it or not) almost by definition. That is probably the reason why it is so highly exceptional to see relational algebra being used to express/define database constraints !
Another approach for defining database constraints using relational algebra, is to define a relational expression that plays the role of "faults expression" and then implicitly, tacitly assume that the rule is that the result of evaluating this expression must at all times be empty. But that's (AFAIK) an entirely private approach of mine, and I'd be surprised if you had also found it in a textbook.
Related
I have around 200 dummies, and wish to run a constrained OLS regression where I impose that the sum of all coefficients on the dummies is equal to 1.
One option is to type:
constraint define 1 dummy_1+dummy_2 +...+dummy_200=1
cnsreg y x_1 x_2 dummy_1-dummy_200, c(1)
...but typing the constraint out would obviously be very painful.
Is there a way to quickly define such a large constraint? The matrix form would be very quick and straightforward, but after much reading online and in Stata guide, it is not clear to me how to do constraints in matrix form, and if they are even possible.
There are at least two sides to this, how to do it and whether it will work in any statistical sense.
How to do it seems easier than you fear as the difficult bit is just inserting "+" signs between the variable names, and that's string manipulation. Something like
unab myvars : dummy_*
local myvars : subinstr local myvars " " "+", all
mac li
constraint 1 `myvars' = 1
should get you started. The macro list is so you can see what you did, especially if it is not what you want.
Whether it will work for you statistically is outside the scope of this forum, but if that's the only constraint note that it's consistent with all kinds of negative and positive coefficients. Perhaps there are special features of your problem that make it a natural constraint, but my intuition is that such a model will be hard to estimate.
I would take a completely different approach. Such constraints typically occur when trying out a different coding scheme for a set of indicator variables. If that is the case then I would use Stata's factor variables, combined with margins with the contrast operators.
I am having problems understanding the difference between validity and satisfiability.
Given the following:
(i) For all F, F is satisfiable or ~F is satisfiable.
(ii) For all F, F is valid or ~F is valid.
How do I prove which is true and which is false?
Statement (i) is true, as for all F, F will either be satisfiable, or ~F will be satisfiable (truth table). However, how do I go about solving for statement (ii)?
Any help is highly appreciated!
Aprilrocks92,
I don't blame you for being confused, because actually logicians, mathematicians, heck even those philosopher types use the words validity differently sometimes.
Trying not to overcomplicate, I'll give you a thin definition: a conclusion if valid when it is true whenever the premises are true. We also say, given a suitably defined logic, that the conclusion follows as a "logical consequence" of the premises.
On the other hand, satisfisability means that there exists a valuation of the non logical symbols in the formula F that makes the formula true in the logic.
So I should probably mention the difference between semantics and syntax to explain. The syntax of your logic is all those logical and non logical symbols, and the deductive rules that enable you to make "steps" towards proof in the logic. My definition of satisfisability above mentioned the word "valuation"- now how does that fit? Well the answer is that you need to supply a semantics: in short this is the structure that the formula of the logic are expressions of, usually given in set theory, and a valuation of a given F is a function that maps all the non logical symbols in F to sets and members of sets, which a given semantics for the logic composes into a truth value.
Hmm. I'm not sure that's the best explanation, but I don't have much time.
Either way, that should help you understand the difference. To answer your question about the difference between (i) and (ii) without giving too much away, think: what's the relationship between the two? Well given that as above an F' is true given a valuation that sends the sentence to true. So you could "rewrite" my definition of validity as: a conclusion is valid iff whenever the premises are satisfisable the conclusion is satisfisable.
Now, with regards your requirement to prove these things, I strongly suspect you've got a lot more context about your logic you're not telling us and your teacher or text book has intimated a context in which to answer these, as actually taken in the general sense your question doesn't make complete sense.
XPath 2.0 has some new functions and syntax, relative to 1.0, that work with sequences. Some of theset don't really add to what the language could already do in 1.0 (with node sets), but they make it easier to express the desired logic in ways that are more readable. This increases the chances of the programmer getting the code correct -- and keeping it that way. For example,
empty(s) is equivalent to not(s), but its intent is much clearer when you want to test whether a sequence is empty.
Correction: the effective boolean value of a sequence is in general more complicated than that. E.g. empty((0)) != not((0)). This applies to exists(s) vs. s in a boolean context as well. However, there are domains of s where empty(s) is equivalent to not(s), so the two could be used interchangeably within those domains. But this goes to show that the use of empty() can make a non-trivial difference in making code easier to understand.
Similarly, exists(s) is equivalent to boolean(s) that already existed in XPath 1.0 (or just s in a boolean context), but again is much clearer about the intent.
Quantified expressions; e.g. "some $x in expression satisfies test($x)" would be equivalent to boolean(expression[test(.)]) (although the new syntax is more flexible, in that you don't need to worry about losing the context item because you have the variable to refer to it by).
Similarly, "every $x in expression satisfies test($x)" would be equivalent to not(expression[not(test(.))]) but is more readable.
These functions and syntax were evidently added at no small cost, solely to serve the goal of writing XPath that is easier to map to how humans think. This implies, as experienced developers know, that understandable code is significantly superior to code that is difficult to understand.
Given all that ... what would be a clear and readable way to write an XPath test expression that asks
Does value X occur in sequence S?
Some ways to do it: (Note: I used X and S notation here to indicate the value and the sequence, but I don't mean to imply that these subexpressions are element name tests, nor that they are simple expressions. They could be complicated.)
X = S: This would be one of the most unreadable, since it requires the reader to
think about which of X and S are sequences vs. single values
understand general comparisons, which are not obvious from the syntax
However, one advantage of this form is that it allows us to put the topic (X) before the comment ("is a member of S"), which, I think, helps in readability.
See also CMS's good point about readability, when the syntax or names make the "cardinality" of X and S obvious.
index-of(S, X): This one is clear about what's intended as a value and what as a sequence (if you remember the order of arguments to index-of()). But it expresses more than we need to: it asks for the index, when all we really want to know is whether X occurs in S. This is somewhat misleading to the reader. An experienced developer will figure out what's intended, with some effort and with understanding of the context. But the more we rely on context to understand the intent of each line, the more understanding the code becomes a circular (spiral) and potentially Sisyphean task! Also, since index-of() is designed to return a list of all the indexes of occurrences of X, it could be more expensive than necessary: a smart processor, in order to evaluate X = S, wouldn't necessarily have to find all the contents of S, nor enumerate them in order; but for index-of(S, X), correct order would have to be determined, and all contents of S must be compared to X. One other drawback of using index-of() is that it's limited to using eq for comparison; you can't, for example, use it to ask whether a node is identical to any node in a given sequence.
Correction: This form, used as a conditional test, can result in a runtime error: Effective boolean value is not defined for a sequence of two or more items starting with a numeric value. (But at least we won't get wrong boolean values, since index-of() can't return a zero.) If S can have multiple instances of X, this is another good reason to prefer form 3 or 6.
exists(index-of(X, S)): makes the intent clearer, and would help the processor eliminate the performance penalty if the processor is smart enough.
some $m in S satisfies $m eq X: This one is very clear, and matches our intent exactly. It seems long-winded compared to 1, and that in itself can reduce readability. But maybe that's an acceptable price for clarity. Keep in mind that X and S could potentially be complex expressions themselves -- they're not necessarily just variable references. An advantage is that since the eq operator is explicit, you can replace it with is or any other comparison operator.
S[. eq X]: clearer than 1, but shares the semantic drawbacks of 2: it computes all members of S that are equal to X. Actually, this could return a false negative (incorrect effective boolean value), if X is falsy. E.g. (0, 1)[. eq 0] returns 0 which is falsy, even though 0 occurs in (0, 1).
exists(S[. eq X]): Clearer than 1, 2, 3, and 5. Not as clear as 4, but shorter. Avoids the drawbacks of 5 (or at least most of them, depending on the processor smarts).
I'm kind of leaning toward the last one, at this point: exists(S[. eq X])
What about you... As a developer coming to a complex, unfamiliar XSLT or XQuery or other program that uses XPath 2.0, and wanting to figure out what that program is doing, which would you find easiest to read?
Apologies for the long question. Thanks for reading this far.
Edit: I changed = to eq wherever possible in the above discussion, to make it easier to see where a "value comparison" (as opposed to a general comparison) was intended.
For what it's worth, if names or context make clear that X is a singleton, I'm happy to use your first form, X = S -- for example when I want to check an attribute value against a set of possible values:
<xsl:when test="#type = ('A', 'A+', 'A-', 'B+')" />
or
<xsl:when test="#type = $magic-types"/>
If I think there is a risk of confusion, then I like your sixth formulation. The less frequently I have to remember the rules for calculating an effective boolean value, the less frequently I make a mistake with them.
I prefer this one:
count(distinct-values($seq)) eq count(distinct-values(($x, $seq)))
When $x is itself a sequence, this expression implements the (value-based) subset of relation between two sets of values, that are represented as sequences. This implementation of subset of has just linear time complexity -- vs many other ways of expressing this, that have O(N^2)) time complexity.
To summarize, the question whether a single value belongs to a set of values is a special case of the question whether one set of values is a subset of another. If we have a good implementation of the latter, we can simply use it for answering the former.
The functx library has a nice implementation of this function, so you can use
functx:is-node-in-sequence($X, $Y)
(this particular function can be found at http://www.xqueryfunctions.com/xq/functx_is-node-in-sequence.html)
The whole functx library is available for both XQuery (http://www.xqueryfunctions.com/) and XSLT (http://www.xsltfunctions.com/)
Marklogic ships the functx library with their core product; other vendors may also.
Another possibility, when you want to know whether node X occurs in sequence S, is
exists((X) intersect S)
I think that's pretty readable, and concise. But it only works when X and the values in S are nodes; if you try to ask
exists(('bob') intersect ('alice', 'bob'))
you'll get a runtime error.
In the program I'm working on now, I need to compare strings, so this isn't an option.
As Dimitri notes, the occurrence of a node in a sequence is a question of identity, not of value comparison.
Is it possible to obtain the maximum difference between two columns (for example starting and ending weights)?
Right now I'm leaning towards no as this would require a new column with the difference between the two columns for each row, then taking the max of that. Doing it the way I orginally intended doesn't work either since arithmetic operations are not allowed in the conditions of select operations (e.g. SIGMA (c1 - c2 < c3 - c4)(Table) is not allowed).
Disclosure: this is part of a homework question.
It can be done, exactly in the way you planned, but you need generalized projection for that. The generalized projection is the operator
Π(E1, E2,..., En)R
where R is a relation, and E1...En are expressions in the form a⊕b, where a and b are attributes of R or constants, and ⊕ is an arbitrary binary operator between them. The result is a relation with attributes E1...En.
This would allow you to project the differences into a new relation (R' := Π(x-y)R), then find the maximum on that, just as you planned.
If we're not allowed to use generalized projection, then I think we have no means to actually subtract an attribute from another, or to actually calculate anything from them, as the definition of projection allow only attribute names, and the definition of selection allow only expressions of the form aθb where a and b are attributes or constants and θ is a binary relational operator (this is logical, in its way, because if we have a relation R(X,Y), then we have no idea about the type of X or Y, making operations on them quite meaningless).
I think generalized projection is a great extension to relational algebra. It's obviously immensely useful in real life, and it can be defended even from a more scientific point of view: if we allow binary conditional operators on the values like "X > 50", then we made assumptions on the type already, rendering that point kind of moot. Your instructor may disagree, though.
If you're looking to do this in the real world, you should be able to do this with a subquery (or a view, which amounts to much the same thing), something like:
select max (diff) from (
select high - low as diff from blah blah blah
)
Whether this applies to the abstract world of relational algebra, I couldn't say. I'm too busy fixing those damn real-world problems :-)
This is a small part of a homework question so I can understand the whole.
SQL query to list car prices that occur more than once:
select car_price from cars
group by car_price
having count (car_price) > 1;
The general form of this in relational algebra is
Y (gl, al) R
where Y is the greek symbol, gl is list of attributes to group, and al is a list of aggregations.
The relational algebra:
Y (count(car_price)) cars
How is the having clause written in that statement? Is there a shorthand? If not, do I just need to select from that relation? Like this?
SELECT (count(car_price) > 1) [Y (count(car_price)) cars]
select count(*) from (select * from cars where price > 1) as cars;
also known as relational closure.
For a more or less precise answer to the actual question asked, "Relational algebra - what is the proper way to represent a ‘having’ clause?", it needs to be stated first that the question itself seems to suggest, or presume, that there exists such a thing as "THE" relational algebra, but that presumption is simply untrue !
An algebra is a set of operators, and anyone can define any set of operators he likes, meaning anyone can define any algebra he likes ! In his most recent publication, Hugh Darwen mentions that RESTRICT is not a fundamental operator of the algebra, though lots of others do consider it as such.
Especially with respect to aggregations and summaries, there is little consensus as to how those should be incorporated in a relational algebra. Defining operators such as COUNT() (that take a relation as an argument value and return an integer) as part of the algebra, might be problematic wrt the closure property of the algebra, precisely because such operators do not return a relation ...
So the sorry, but nevertheless most appropriate, answer here seems to be that a conclusive answer to this question is almost impossible to give ...