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I want to model fixed-size arrays that can contain records and other fixed-size arrays. I then want to model store and select accesses to them.
I currently use ArraySorts for the arrays and Datatypes for the records. However, according to this answer (arrays) and this answer (records), these aren't really intended for my usecase. Is there anything else in Z3 I could use instead?
Background: I want to model pointers as they occur in the LLVM IR. For this, each pointer has a data array that represents the memory buffer into which it is pointing and an indices array that represents the indices used in getelementptr calls. Since the memory buffer could contain pointers or structs, I need to be able to nest the arrays (or store the records in arrays).
An example (in z3py) looks like this:
Vec3 = z3.Datatype("Vec3")
Vec3.declare("Vec3__init",
("x", z3.IntSort()),
("y", z3.IntSort()),
("z", z3.IntSort())
)
Vec3 = Vec3.create()
PointerVec3 = z3.Datatype("Pointer__Vec3")
PointerVec3.declare("Pointer__Vec3__init",
("data", z3.ArraySort(z3.BitVecSort(32), Vec3)),
("nindices", z3.IntSort()),
("indices", z3.ArraySort(z3.IntSort(), z3.BitVecSort(32)))
)
PointerVec3 = PointerVec3.create()
Arrays and records are the only way to model these things in z3/SMT-Lib. And they can be nested as you wish.
It is true that SMTLib arrays are not quite like arrays you find in regular programming languages. But it's not true that they are always unbounded. Their size exactly matches the cardinality of their domain. If you want a bounded array, I recommend using an appropriate BitVec type for your source. For an n-bit bit vector, your array will have 2^n elements. You can work with a small enough n, and treat that as your array; which typically matches what you see in practice: Most of such internal arrays will be a power-of-two anyhow. (Moreover, provers usually don't do well with large arrays; so sticking to a small power-of-two is a good idea, at least to start with.)
So, these are your only options. Stack-overflow works the best if you try it out and actually ask about what sort of problems you ran into.
How can I increase the length of a tuple in Erlang? For example, suppose Tup={1,2,3}, and now I want to add another element to it. Is there any way to do this?
Tuple is not suppose to be a flexible data structure. If you are resizing it often, then you should consider using other erlang data structures like lists, maps or sets - depends on your expectation. Here is nice introduction to key-value stores.
But if you really have to extend that tuple, then you can use erlang:append_element/2:
{1,2,3,4} = erlang:append_element({1,2,3}, 4).
Tuples aren't mutable so you can't, strictly speaking, increase the length.
Generally, if you want a variable-number-of-things datatype, a tuple will be very inconvenient. For example, iterating over all elements of a list is highly idiomatic, while iterating over all elements of a tuple whose size is unknown at compile-time is a pain.
However, a common pattern is to get a tuple as a result from some function and return elements of that tuple plus additions.
country_coords(Name) ->
{Lat, Lng} = find_address(Name),
{_Street, _City, _Zip, Country} = geocode(Lat, Lng),
{ok, Lat, Lng, Country}.
erlang:append_element(tuple_whose_length_to_increase, element_to_be).This is the inbuilt function but tuples,lists are not meant to be flexible.So avoid using this function unless there is no other way
I want to construct a string in Java that represents a DICT term and that will be passed to an Erlang process for being reflected back as an erlang term ( string-to-term ).
I can achieve this easily for ORDDICT 's, since they are structured as a simple sorted key / value pair in a list of tuples such as : [ {field1 , "value1"} , {field2 , "value2} ]
But, for DICTS, they are compiled into a specific term that I want to find how to reverse-engineer it. I am aware this structure can change over new releases, but the benefits for performance and ease of integration to Java would overcome this. Unfortunately Erlang's JInterface is based on simple data structures. An efficient DICT type would be of great use.
A simple dict gets defined as follows:
D1 = dict:store("field1","AAA",dict:new()).
{dict,1,16,16,8,80,48,
{[],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[]},
{{[],[],[],[],[],[],[],[],
[["field1",65,65,65]],
[],[],[],[],[],[],[]}}}
As it can be seen above, there are some coordinates which I do not understand what they mean ( the numbers 1,16,16,8,80,48 and a set of empty lists, which likely represent something as well.
Adding two other rows (key-value pairs) causes the data to look like:
D3 = dict:store("field3","CCC",D2).
{dict,3,16,16,8,80,48,
{[],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[]},
{{[],[],
[["field3",67,67,67]],
[],[],[],[],[],
[["field1",65,65,65]],
[],[],[],[],
[["field2",66,66,66]],
[],[]}}}
From the above I can notice that:
the first number (3) reppresets the number of items in the DICT.
the second number (16) shows the number of list slots in the first tuple of lists
the third number (16) shows the number of list slots in the second typle of lists, of which the values ended up being placed on ( in the middle ).
the fourth number (8) appears to be the number of slots in the second row of tuples from where the values are placed ( a sort of index-pointer )
the remaining numbers (80 and 48)... no idea...
adding a key "field0" gets placed not in the end but just after "field1"'s data. This indicates the indexing approach.
So the question, is there a way (algorithm) to reliably directly create a DICT string from outside of Erlang ?
The comprehensive specification how dict is implemented can be found simply in the dict.erl sourcecode.
But I'm not sure replicating dict.erl's implementation in Java is worthwhile. This would only make sense if you want a fast dict like data structure that you need to pass often between Java and Erlang code. It might make more sense to use a Key-Value store both from Erlang and Java without passing it directly around. Depending on your application this could be e.g. riak or maybe even connect your different language worlds with RabbitMQ. Both examples are implemented in Erlang and are easily accessible from both worlds.
Or phrased another way, what kind of benefits do you get from having a basic, singly linked list with only a head pointer? The benefits of a tail pointer that I can see are:
O(1) list concatenation
O(1) Appending stuff to the right side of the list
Both of which are rather convenient things to have, as opposed to O(n) list concatenation (where n is the length of the left-side list?). What advantages does dropping the tail pointer have?
F#, like many other functional[-ish] languages, has a cons-list (the terminology originally comes from LISP, but the concept is the same). In F# the :: operator (or List.Cons) is used for cons'ing: note the signature is ‘a –> ‘a list –> ‘a list (see Mastering F# Lists).
Do not confuse a cons-list with an opaque Linked List implementation which contains a discrete first[/last] node - every cell in a cons-list is the start of a [different] list! That is, a "list" is simply the chain of cells that starts at a given cons-cell.
This offers some advantages when used in a functional-like manner: one is that all the "tail" cells are shared and since each cons-cell is immutable (the "data" might be mutable, but that's a different issue) there is no way to make a change to a "tail" cell and flub up all the other lists which contain said cell.
Because of this property, [new] lists can be efficiently built - that is, they do not require a copy - simply by cons'ing to the front. In addition, it is also very efficient to deconstruct a list to head :: tail - once again, no copy - which is often very useful in recursive functions.
This immutable property generally does not exist in a [standard mutable] Double Linked List implementation in that appending would add side-effects: the internal 'last' node (the type is now opaque) and one of the "tail" cells are changed. (There are immutable sequence types that allow an "effectively constant time" append/update such as immutable.Vector in Scala -- however, these are heavy-weight objects compared to a cons-list that is nothing more than a series of cells cons'ed together.)
As mentioned, there are also disadvantages a cons-list is not appropriate for all tasks - in particular, creating a new list except by cons'ing to the head is an O(n) operation, fsvo n, and for better (or worse) the list is immutable.
I would recommend creating your own version of concat to see how this operation is really done. (The article Why I love F#: Lists - The Basics covers this.)
Happy coding.
Also see related post: Why can you only prepend to lists in functional languages?
F# lists are immutable, there's no such thing as "append/concat", rather there's just creating new lists (that may reuse some suffixes of old lists). Immutability has many advantages, outside the scope of this question. (All pure languages, and most functional languages have this data structure, it is not an F#-ism.)
See also
http://diditwith.net/2008/03/03/WhyILoveFListsTheBasics.aspx
which has good picture diagrams to explain things (e.g. why consing on the front is cheaper than at the end for an immutable list).
In addition to what the others said: if you need efficient, but yet immutable data structures (which should be an idiomatic F# way), you have to consider reading Chris Okasaki, Purely Functional Data Structures. There is also a thesis available (on which the book is based).
In addition to what has been already said, the Introducing Functional Programming section on MSDN has an article about Working with Functional Lists that explains how lists work and also implements them in C#, so it may be a good way to understand how they work (and why adding reference to the last element would not allow efficient implementation of append).
If you need to append things to the end of the list, as well as to the front, then you need a different data structure. For example, Norman Ramsey posted source code for DList which has these properties here (The implementation is not idiomatic F#, but it should be easy to fix).
If you find you want a list with better performance for append operations, have a look at the QueueList in the F# PowerPack and the JoinList in the FSharpx extension libraries.
QueueList encapsulates two lists. When you prepend using the cons, it prepends an element to the first list, just as a cons-list. However, if you want to append a single element, it can be pushed to the top of the second list. When the first list runs out of elements, List.rev is run on the second list, and the two are swapped putting your list back in order and freeing the second list to append new elements.
JoinList uses a discriminated union to more efficiently append whole lists and is a bit more involved.
Both are obviously less performant for standard cons-list operations but offer better performance for other scenarios.
You can read more about these structures in the article Refactoring Pattern Matching.
As others have pointed out, an F# list could be represented by a data structure:
List<T> { T Value; List<T> Tail; }
From here, the convention is that a list goes from the List you have a reference to until Tail is null. Based on that definition, the benefits/features/limitations in the other answers come naturally.
However, your original question seems to be why the list is not defined more like:
List<T> { Node<T> Head; Node<T> Tail; }
Node<T> { T Value; Node<T> Next; }
Such a structure would allow both appending and prepending to the list without any visible effects to the a reference to the original list, since it still only sees a "window" of the now expanded list. Although this would (sometimes) allow O(1) concatenation, there are several issues such a feature would face:
The concatenation only works once. This can lead to unexpected performance behavior where one concatenation is O(1), but the next is O(n). Say for example:
listA = makeList1 ()
listB = makeList2 ()
listC = makeList3 ()
listD = listA + listB //modified Node at tail of A for O(1)
listE = listA + listC //must now make copy of A to concat with C
You could argue that the time savings for the cases where possible are worth it, but the surprise of not knowing when it will be O(1) and when O(n) are strong arguments against the feature.
All lists now take up twice as much space, even if you never plan to concatenate them.
You now have a separate List and Node type. In the current implementation, I believe F# only uses a single type like the beginning of my answer. There may be a way to do what you are suggesting with only one type, but it is not obvious to me.
The concatenation requires mutating the original "tail" node instance. While this shouldn't affect programs, it is a point of mutation, which most functional languages tend to avoid.
Or phrased another way, what kind of benefits do you get from having a basic, singly linked list with only a head pointer? The benefits of a tail pointer that I can see are:
O(1) list concatenation
O(1) Appending stuff to the right side of the list
Both of which are rather convenient things to have, as opposed to O(n) list concatenation (where n is the length of the left-side list?).
If by "tail pointer" you mean a pointer from every list to the last element in the list, that alone cannot be used to provide either of the benefits you cite. Although you could then get at the last element in the list quickly, you cannot do anything with it because it is immutable.
You could write a mutable doubly-linked list as you say but the mutability would make programs using it significantly harder to reason about because every function you call with one might change it.
As Brian said, there are purely functional catenable lists. However, they are many times slower at common operations than the simple singly-linked list that F# uses.
What advantages does dropping the tail pointer have?
30% less space usage and better performance on virtually all list operations.
I'm trying to learn how to think in a functional programming way, for this, I'm trying to learn Erlang and solving easy problems from codingbat. I came with the common problem of comparing elements inside a list. For example, compare a value of the i-th position element with the value of the i+1-th position of the list. So, I have been thinking and searching how to do this in a functional way in Erlang (or any functional language).
Please, be gentle with me, I'm very newb in this functional world, but I want to learn
Thanks in advance
Define a list:
L = [1,2,3,4,4,5,6]
Define a function f, which takes a list
If it matches a list of one element or an empty list, return the empty list
If it matches the first element and the second element then take the first element and construct a new list by calling the rest of the list recursivly
Otherwise skip the first element of the list.
In Erlang code
f ([]) -> [];
f ([_]) -> [];
f ([X, X|Rest]) -> [X | f(Rest)];
f ([_|Rest]) -> f(Rest).
Apply function
f(L)
This should work... haven't compiled and run it but it should get you started. Also in case you need to do modifications to it to behave differently.
Welcome to Erlang ;)
I try to be gentle ;-) So main thing in functional approach is thinking in terms: What is input? What should be output? There is nothing like comparing the i-th element with the i+1-th element alone. There have to be always purpose of it which will lead to data transformation. Even Mazen Harake's example doing it. In this example there is function which returns only elements which are followed by same value i.e. filters given list. Typically there are very different ways how do similar thing which depends of purpose of it. List is basic functional structure and you can do amazing things with it as Lisp shows us but you have to remember it is not array.
Each time you need access i-th element repeatable it indicates you are using wrong data structure. You can build up different data structures form lists and tuples in Erlang which can serve your purposes better. So when you face problem to compare i-th with i+1-th element you should stop and think. What is purpose of it? Do you need perform some stream data transformation as Mazen Harake does or You need random access? If second you should use different data structure (array for example). Even then you should think about your task characteristics. If you will be mostly read and almost never write then you can use list_to_tuple(L) and then read using element/2. When you need write occasionally you will start thinking about partition it to several tuples and as your write ratio will grow you will end up with array implementation.
So you can use lists:nth/2 if you will do it only once or several times but on short list and you are not performance freak as I'm. You can improve it using [X1,X2|_] = lists:nthtail(I-1, L) (L = lists:nthtail(0,L) works as expected). If you are facing bigger lists and you want call it many times you have to rethink your approach.
P.S.: There are many other fascinating data structures except lists and trees. Zippers for example.