Lambda Calculus via C# (5) List

[FP & LINQ via C# series]

[Lambda Calculus via C# series]

In lambda calculus and Church encoding, there are various ways to represent a list with anonymous functions.

Tuple as list node

With Church pair, it is easy to model Church list as a linked list, where each list node is a a Church pair (2-tuple) of current node’s value and the next node, So that

CreateListNode := CreateTuple = λv.λn.λf.f v n
ListNode := Tuple = λf.f v n

Here variable v is the value of the current node, so it is the first item of the tuple; And variable n is the next node of the current node, so it is the second item of the tuple:

Value := Item1 = λl.l (λv.λn.v)
Next := Item2 = λl.l (λv.λn.n)

Here variable l is the list node. The C# implementation is similar to tuple and signed numeral, except ListNode<T> function type now has 1 type parameter, which is the type of its value:

// ListNode<T> is the alias of Tuple<T, ListNode<T>>.
public delegate dynamic ListNode<out T>(Boolean f);

public static partial class ChurchList<T>
{
    // Create = value => next => (value, next)
    public static readonly Func<T, Func<ListNode<T>, ListNode<T>>>
        Create = value => next => new ListNode<T>(ChurchTuple<T, ListNode<T>>.Create(value)(next));

    // Value = node => node.Item1()
    public static readonly Func<ListNode<T>, T> 
        Value = node => new Tuple<T, ListNode<T>>(node).Item1();

    // Next = node => node.Item2()
    public static readonly Func<ListNode<T>, ListNode<T>> 
        Next = node => new Tuple<T, ListNode<T>>(node).Item2();
}

Usually, when a list ends, its last node’s next node is flagged as a special null node. Here in lambda calculus, since a node is an anonymous function, the null node is also an anonymous function:

Null := λf.λx.x

And IsNull predicate returns a Church Boolean to indicate whether a list node is null:

IsNull := λl.l (λv.λn.λx.False) True

When IsNull is applied with a null node:

  IsNull Null
≡ (λl.l (λv.λn.λx.False) True) (λf.λx.x)
≡ (λf.λx.x) (λv.λn.λx.False) True
≡ (λx.x) True
≡ True

And when IsNull is applied with a non-null node:

  IsNull (CreateListNode 0 Null)
≡ IsNull (λf.f 0 Null)
≡ (λl.l (λv.λn.λx.False) True) (λf.f 0 Null)
≡ (λf.f 0 Null) (λv.λn.λx.False) True
≡ (λv.λn.λx.False) 0 Null True
≡ (λn.λx.False) Null True
≡ (λx.False) True
≡ False

The C# implementation is noisy because a lot of type information has to be provided. This is Null:

using static ChurchBoolean;

public static partial class ChurchList<T>
{
    // Null = False;
    public static readonly ListNode<T>
        Null = new ListNode<T>(False);

    // IsNull = node => node(value => next => _ => False)(True)
    public static readonly Func<ListNode<T>, Boolean> 
        IsNull = node => node(value => next => new Func<Boolean, Boolean>(_ => False))(True);
}

And the indexer for list can be easily defined with as a function accepts a start node and a Church numeral i as the specified index. To return the node at the specified index, just call Next function for i times from the start node:

ListNodeAt := λl.λi.i Next l

C#:

public static readonly Func<ListNode<T>, Func<Numeral, ListNode<T>>>
    ListNodeAt = start => index => index(node => Next(node))(start);

The following are the extension methods wrapping the list operators:

public static class ListNodeExtensions
{
    public static T Value<T>(this ListNode<T> node) => ChurchList<T>.Value(node);

    public static ListNode<T> Next<T>(this ListNode<T> node) => ChurchList<T>.Next(node);

    public static Boolean IsNull<T>(this ListNode<T> node) => ChurchList<T>.IsNull(node);

    public static ListNode<T> ListNodeAt<T>(this ListNode<T> start, Numeral index) => ChurchList<T>.ListNodeAt(start)(index);
}

And the following code demonstrates how the list works:

[TestClass]
public class ChurchListTests
{
    [TestMethod]
    public void CreateValueNextTest()
    {
        ListNode<int> node1 = ChurchList<int>.Create(1)(ChurchList<int>.Null);
        ListNode<int> node2 = ChurchList<int>.Create(2)(node1);
        ListNode<int> node3 = ChurchList<int>.Create(3)(node2);
        Assert.AreEqual(1, node1.Value());
        Assert.AreEqual(ChurchList<int>.Null, node1.Next());
        Assert.AreEqual(2, node2.Value());
        Assert.AreEqual(node1, node2.Next());
        Assert.AreEqual(3, node3.Value());
        Assert.AreEqual(node2, node3.Next());
        Assert.AreEqual(node2.Value(), node3.Next().Value());
        Assert.AreEqual(node1.Value(), node3.Next().Next().Value());
        Assert.AreEqual(ChurchList<int>.Null, node3.Next().Next().Next());
        try
        {
            ChurchList<object>.Null.Next();
            Assert.Fail();
        }
        catch (InvalidCastException exception)
        {
            exception.WriteLine();
        }
    }

    [TestMethod]
    public void IsNullTest()
    {
        ListNode<int> node1 = ChurchList<int>.Create(1)(ChurchList<int>.Null);
        ListNode<int> node2 = ChurchList<int>.Create(2)(node1);
        ListNode<int> node3 = ChurchList<int>.Create(3)(node2);
        Assert.IsTrue(ChurchList<object>.Null.IsNull().Unchurch());
        Assert.IsFalse(node1.IsNull().Unchurch());
        Assert.IsFalse(node2.IsNull().Unchurch());
        Assert.IsFalse(node3.IsNull().Unchurch());
        Assert.IsTrue(node1.Next().IsNull().Unchurch());
        Assert.IsFalse(node2.Next().IsNull().Unchurch());
        Assert.IsFalse(node3.Next().IsNull().Unchurch());
    }

    [TestMethod]
    public void IndexTest()
    {
        ListNode<int> node1 = ChurchList<int>.Create(1)(ChurchList<int>.Null);
        ListNode<int> node2 = ChurchList<int>.Create(2)(node1);
        ListNode<int> node3 = ChurchList<int>.Create(3)(node2);
        Assert.AreEqual(node3, node3.NodeAt(0U.Church()));
        Assert.AreEqual(node2, node3.NodeAt(1U.Church()));
        Assert.AreEqual(node1, node3.NodeAt(2U.Church()));
        Assert.IsTrue(node3.NodeAt(3U.Church()).IsNull().Unchurch());
        try
        {
            node3.NodeAt(4U.Church());
            Assert.Fail();
        }
        catch (InvalidCastException exception)
        {
            exception.WriteLine();
        }
    }
}

Aggregate function as list node

Remember the LINQ Aggregate query method accepting a seed and a accumulator function:

TAccumulate Aggregate<TSource, TAccumulate>(this IEnumerable<TSource> source, TAccumulate seed, Func<TAccumulate, TSource, TAccumulate> func);

Assume seed is x, and accumulator function is f:

  • When source is empty, the aggregation result is x
  • When source is { 0 }, the aggregation result is f(x, 0)
  • When source is { 1, 0 }, the aggregation result is f(f(x, 1), 0)
  • When source is { 2, 1, 0 }, the aggregation result is f(f(f(x, 2), 1), 0)

Church list can also be encoded with a similar Aggregate function with seed and accumulator function:

dynamic AggregateListNode<T>(dynamic x, Func<dynamic, T, dynamic> f);

Its type parameter T is the type of node value. And since the seed can be anything, just leave it as dynamic as usual. So the list node is of above aggregate function type (dynamic, (dynamic , T) -> dynamic) -> dynamic. After currying the aggregate function and the accumulator function, it becomes dynamic -> (dynamic –> T -> dynamic) -> dynamic. So this is the function type of list node, and an alias can be defined as:

// Curried from: (dynamic, dynamic -> T -> dynamic) -> dynamic.
// AggregateListNode is the alias of: dynamic -> (dynamic -> T -> dynamic) -> dynamic.
public delegate Func<Func<dynamic, Func<T, dynamic>>, dynamic> AggregateListNode<out T>(dynamic x);

And this is the creation and definition of list node:

CreateListNode := λv.λn.λx.λf.f (n x f) v
ListNode := λx.λf.f (n x f) v

In C#:

public static partial class ChurchAggregateList<T>
{
    public static readonly Func<T, Func<AggregateListNode<T>, AggregateListNode<T>>>
        Create = value => next => x => f => f(next(x)(f))(value);
}

Similarly, here variable v is the value of current node, variable n is the next node of the current node. And variable x is the seed for aggregation, variable f is the accumulator function. The list is still modeled as a linked list, so Null is also needed to represent the end of list:

Null := λx.λf.x

Null is defined to call f for 0 times. For example, to create a linked list { 2, 1, 0 }, first create the last list node, with value 2 and Null as its next node:

  CreateListNode 0 Null
≡ (λv.λn.λx.λf.f (n x f) v) 0 (λx.λf.x)
≡ (λn.λx.λf.f (n x f) 0) (λx.λf.x)
≡ λx.λf.f ((λx.λf.x) x f) 0
≡ λx.λf.f x 0

Then the previous node can be created with value 1 and the above node:

  CreateListNode 1 (CreateListNode 0 Null)
≡ CreateListNode 1 (λx.λf.f x 0)
≡ (λv.λn.λx.λf.f (n x f) v) 1 (λx.λf.f x 0)
≡ (λn.λx.λf.f (n x f) 1) (λx.λf.f x 0)
≡ λx.λf.f ((λx.λf.f x 0) x f) 1
≡ λx.λf.f (f x 0) 1

And the first node has value 0:

  CreateListNode 2 (CreateListNode 1 (CreateListNode 0 Null))
≡ CreateListNode 2 (λx.λf.f (f x 0) 1)
≡ (λv.λn.λx.λf.f (n x f) v) 2 (λx.λf.f (f x 0) 1)
≡ (λn.λx.λf.f (n x f) 2) (λx.λf.f (f x 0) 1)
≡ λx.λf.f (λx.λf.f (f x 0) 1) x f) 2
≡ λx.λf.f (f (f x 0) 1) 2

So the list nodes are represented in the same pattern as LINQ aggregation.

The IsNull predicate can be defined as following:

IsNull := λl.l True (λx.λv.False)

The variable l is the list node, which is an aggregate function, and is applied with seed True and accumulate function λv.λx.False. When IsNull is applied with a null node, the accumulate function is not applied, and seed True is directly returned:

  IsNull Null
≡ (λl.l True (λx.λv.False)) (λx.λf.x)
≡ (λx.λf.x) True (λx.λv.False)
≡ (λf.True) (λx.λv.False)
≡ True

And when IsNull is applied with a non null node, the accumulator function is applied and constantly returns False, so IsNull returns False:

  IsNull (CreateListNode 2 Null)
≡ IsNull (λx.λf.f x 2)
≡ (λl.l True (λx.λv.False)) (λx.λf.f x 2)
≡ (λx.λf.f x 2) True (λx.λv.False)
≡ (λf.f True 2) (λx.λv.False)
≡ (λx.λv.False) True 2
≡ False

In C#:

using static ChurchBoolean;

public static partial class ChurchAggregateList<T>
{
    public static readonly AggregateListNode<T>
        Null = x => f => x;

    public static readonly Func<AggregateListNode<T>, Boolean>
        IsNull = node => node(True)(x => value => False);
}

The following function returns the value from the specified node:

Value := λl.l Id (λx.λv.v)

When Value is applied with a node, which has value v and next node n:

  Value (CreateListNode v n)
≡ Value (λx.λf.f (n x f) v)
≡ (λl.l Id (λx.λv.v)) (λx.λf.f (n x f) v)
≡ (λx.λf.f (n x f) v) Id (λx.λv.v)
≡ (λf.f (n Id f) v) (λx.λv.v)
≡ (λx.λv.v) (n Id f) v
≡ (λv.v) v
≡ v

In C#:

// Value = node => node(Id)(x => value => value)
public static readonly Func<AggregateListNode<T>, T>
    Value = node => node(Functions<T>.Id)(x => value => value);

It is not very intuitive to get a node’s next node:

Next := λl.λx.λf.l (λf.x) (λx.λv.λg.g (x f) v) (λx.λv.v)

In C#:

// Next = node => x => f => node(_ => x)(accumulate => value => (g => g(accumulate(f))(value)))(accumulate => value => accumulate);
public static readonly Func<AggregateListNode<T>, AggregateListNode<T>>
    Next = node => x => f => node(new Func<Func<dynamic, Func<T, dynamic>>, dynamic>(_ => x))(accumulate => value => new Func<Func<dynamic, Func<T, dynamic>>, dynamic>(g => g(accumulate(f))(value)))(new Func<dynamic, Func<T, dynamic>>(accumulate => value => accumulate));

The above definition is similar to the the pattern of initial version Subtract function for Church numeral. So it can be defined by shifting tuple too. Again, list node with value v and next node n is a aggregate function, it can be applied with a tuple of Null nodes as seed, and a accumulator function to swap the tuple:

  (CreateListNode v n) (Null, Null) (λt.λv.Shift (CreateListNode v) t)
≡ (λx.λf.f (n x f) v) (Null, Null) (λt.λv.Shift (CreateListNode v) t)
≡ (λf.f (n (Null, Null) f) v) (λt.λv.Shift (CreateListNode v) t)
≡ (λt.λv.Shift (CreateListNode v) t) (n (Null, Null) (λt.λv.Shift (CreateListNode v)) t) v
≡ (λv.Shift (CreateListNode v) (n (Null, Null) (λt.λv.Shift (CreateListNode v)) t)) v
≡ Shift (CreateListNode v) (n (Null, Null) (λt.λv.Shift (CreateListNode v)) t)

Take list { n, n – 1, …, 2, 1, 0 } as example, assume its nodes are ListNoden, ListNoden - 1, …, ListNode2, ListNode1, ListNode0:

  • the last node  is: CreateListNode 0 Null
  • the second last node is: CreateListNode 1 (CreateListNode 0 Null)
  • the third last node is: CreateListNode 2 (CreateListNode 1 (CreateListNode 0 Null))

Now apply these nodes with above tuple seed and tuple shifting accumulator function:

  ListNode0 (Null, Null) (λt.λv.Shift (CreateListNode v) t)
≡ (CreateListNode 0 Null) (Null, Null) (λt.λv.Shift (CreateListNode v) t)
≡ Shift (CreateListNode 0) (Null (Null, Null) (λt.λv.Shift (CreateListNode v)) t)
≡ Shift (CreateListNode 0) ((λx.λf.λx) (Null, Null) (λt.λv.Shift (CreateListNode v)) t)
≡ Shift (CreateListNode 0) (Null, Null)
≡ (Null, CreateListNode 0 Null)
≡ (Null, ListNode0)

  ListNode1 (Null, Null) (λt.λv.Shift (CreateListNode v) t)
≡ (CreateListNode 1 (CreateListNode 0 Null)) (Null, Null) (λt.λv.Shift (CreateListNode v) t)
≡ Shift (CreateListNode 1) ((CreateListNode 0 Null) (Null, Null) (λt.λv.Shift (CreateListNode v)) t)
≡ Shift (CreateListNode 1) (Null, Create ListNode 0 Null)
≡ (CreateListNode 0 Null, (CreateListNode 1 (CreateListNode 0 Null))
≡ (ListNode0, ListNode1)

  ListNode2 (Null, Null) (λt.λv.Shift (CreateListNode v) t)
≡ (CreateListNode 2 (CreateListNode 1 (CreateListNode 0 Null))) (Null, Null) (λt.λv.Shift (CreateListNode v) t)
≡ Shift (CreateListNode 2) ((CreateListNode 1 (CreateListNode 0 Null)) (Null, Null) (λt.λv.Shift (CreateListNode v)) t)
≡ Shift (CreateListNode 2) (CreateListNode 0 Null, (CreateListNode 1 (CreateListNode 0 Null))
≡ ((CreateListNode 1 (CreateListNode 0 Null), CreateListNode 2 (CreateListNode 1 (CreateListNode 0 Null)))
≡ (ListNode1, ListNode2)

...

  ListNoden (Null, Null) (λt.λv.Shift (CreateListNode v) t)
≡ (ListNoden - 1, ListNoden)

Generally, there is:

  (CreateListNode v n) (Null, Null) (λt.λv.Shift (CreateListNode v) t)
≡ (n, Create v n)

So Next can be defined as:

Next := λl.Item2 (l (CreateTuple Null Null) (λt.λv.Shift (CreateListNode v) t))

In C#:

// Next = node => node((Null, Null))(tuple => value => tuple.Shift(ChurchTuple.Create(value))).Item1()
public static readonly Func<AggregateListNode<T>, AggregateListNode<T>>
    Next = node =>
        ((Tuple<AggregateListNode<T>, AggregateListNode<T>>)node
            (ChurchTuple<AggregateListNode<T>, AggregateListNode<T>>.Create(Null)(Null))
            (tuple => value => ((Tuple<AggregateListNode<T>, AggregateListNode<T>>)tuple).Shift(Create(value))))
        .Item1();

The indexer can be defined the same as above:

ListNodeAt := λl.λi.i Next l

In C#;

public static readonly Func<AggregateListNode<T>, Func<Numeral, AggregateListNode<T>>>
    ListNodeAt = start => index => index(node => Next(node))(start);

The following are the extension methods wrapping the list operators:

public static class AggregateListNodeExtensions
{
    public static Boolean IsNull<T>(this AggregateListNode<T> node) => ChurchAggregateList<T>.IsNull(node);

    public static T Value<T>(this AggregateListNode<T> node) => ChurchAggregateList<T>.Value(node);

    public static AggregateListNode<T> Next<T>(this AggregateListNode<T> node) => 
        ChurchAggregateList<T>.Next(node);

    public static AggregateListNode<T> ListNodeAt<T>(this AggregateListNode<T> start, Numeral index) => 
        ChurchAggregateList<T>.ListNodeAt(start)(index);
}

And the following code demonstrate how the list works:

[TestClass]
public class ChurchAggregateListTests
{
    [TestMethod]
    public void CreateValueNextTest()
    {
        AggregateListNode<int> node1 = ChurchAggregateList<int>.Create(1)(ChurchAggregateList<int>.Null);
        AggregateListNode<int> node2 = ChurchAggregateList<int>.Create(2)(node1);
        AggregateListNode<int> node3 = ChurchAggregateList<int>.Create(3)(node2);
        Assert.AreEqual(1, node1.Value());
        Assert.IsTrue(node1.Next().IsNull().Unchurch());
        Assert.AreEqual(2, node2.Value());
        Assert.AreEqual(node1.Value(), node2.Next().Value());
        Assert.AreEqual(3, node3.Value());
        Assert.AreEqual(node2.Value(), node3.Next().Value());
        Assert.AreEqual(node1.Value(), node3.Next().Next().Value());
        Assert.IsTrue(node3.Next().Next().Next().IsNull().Unchurch());
    }

    [TestMethod]
    public void IsNullTest()
    {
        AggregateListNode<int> node1 = ChurchAggregateList<int>.Create(1)(ChurchAggregateList<int>.Null);
        AggregateListNode<int> node2 = ChurchAggregateList<int>.Create(2)(node1);
        AggregateListNode<int> node3 = ChurchAggregateList<int>.Create(3)(node2);
        Assert.IsTrue(ChurchAggregateList<int>.Null.IsNull().Unchurch());
        Assert.IsFalse(node1.IsNull().Unchurch());
        Assert.IsFalse(node2.IsNull().Unchurch());
        Assert.IsFalse(node3.IsNull().Unchurch());
        Assert.IsTrue(node1.Next().IsNull().Unchurch());
        Assert.IsFalse(node2.Next().IsNull().Unchurch());
        Assert.IsFalse(node3.Next().IsNull().Unchurch());
    }

    [TestMethod]
    public void IndexTest()
    {
        AggregateListNode<int> node1 = ChurchAggregateList<int>.Create(1)(ChurchAggregateList<int>.Null);
        AggregateListNode<int> node2 = ChurchAggregateList<int>.Create(2)(node1);
        AggregateListNode<int> node3 = ChurchAggregateList<int>.Create(3)(node2);
        Assert.AreEqual(node3.Value(), node3.NodeAt(0U.Church()).Value());
        Assert.AreEqual(node2.Value(), node3.NodeAt(1U.Church()).Value());
        Assert.AreEqual(node1.Value(), node3.NodeAt(2U.Church()).Value());
        Assert.IsTrue(node3.NodeAt(3U.Church()).IsNull().Unchurch());
    }
}

Model everything

Once again, in lambda calculus the only primitive is anonymous function. So far many data types and operations are modeled by anonymous functions, including Boolean, unsigned and signed numeral, tuple, list, logic, arithmetic (except division, which will be implemented later), predicate, etc. With these facilities, many other data types and operations can be modeled too. For example:

  • Floating point number can be represented in the form of significand * baseexponent. In IEEE 754 (aka IEC 60559), floating point numbers are represented as binary format (sign) significand * 2exponent (System.Single and System.Double in .NET), and decimal format (sign) significand * 10exponent (System.Decimal). So either representation can be modeled with a 3-tuple of (Boolean, unsigned numeral, signed numeral).
  • Character (System.Char in .NET) can be represented by unsigned numeral.
  • String (System.String in .NET) can be modeled by a list of characters.
  • Tuple and list can represent other data structures, like tree, stack, queue, etc.

And eventually everything can be modeled with anonymous function represented by lambda expression. Actually, lambda calculus is a classic example of Turing completeness. Lambda calculus is introduced by Alonzo Church before Turing machine was introduced by Alan Turing, and they are equivalent. Lambda calculus, as a universal model of computation, is the rationale and foundations of functional programming. Functional languages (or the functional subset of languages) can be viewed as lambda calculus with more specific syntax, and the execution of functional program can be viewed as reduction of lambda calculus expression.

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  • خرید گیم تایم 60 روزه با توجه به اینکه گیم تایم60 روزه در طی این 2 سال تبدیل به محبوب ترین گیم تایم حال حاضر وارکرافت شده است. یکی از دلایل محبوبیت گیم تایم 2 ماهه قیمت مناسب آن است. شما می توانید برای خرید گیم تایم 60 روزه به سایت جت گیم مراجعه کنید. گیم تایم 2 ماهه به شکل 60 روزه است و بعد از گذشت مدت زمان می بایست دوباره گیم تایم را خریداری کنید. یکی از ویژگی های که سایت جت گیم دارد برای ارائه گیم تایم دو ماهه مناسب و تحویل دادن آنی آن به کاربران عزیز می باشد.

  • خرید گیم تایم 60 روزه با توجه به اینکه گیم تایم60 روزه در طی این 2 سال تبدیل به محبوب ترین گیم تایم حال حاضر وارکرافت شده است. یکی از دلایل محبوبیت گیم تایم 2 ماهه قیمت مناسب آن است. شما می توانید برای خرید گیم تایم 60 روزه به سایت جت گیم مراجعه کنید. گیم تایم 2 ماهه به شکل 60 روزه است و بعد از گذشت مدت زمان می بایست دوباره گیم تایم را خریداری کنید. یکی از ویژگی های که سایت جت گیم دارد برای ارائه گیم تایم دو ماهه مناسب و تحویل دادن آنی آن به کاربران عزیز می باشد.

  • Call of DutyUltimate Edition کاملا اوریجینال
    کیفیت تضمین شده
    به تازگی بازی Call of Duty: Vanguard که توسط استودیو Sledgehammer توسعه یافته است، با انتشار یک تریلر معرفی شد و همچنین در مراسم گیمزکام شاهد نشر یک کلیپ ویدیو ۱۰ دقیقه‌ای با محوریت قسمت داستانی آن بودیم. در این کلیپ ویدیو می‌توانیم آغاز روند داستان کاراکتر پالینا با بازی لارا بیلی را که براساس کاراکتری واقعی به نام لیودمیلا پاولیچنکو خلق شده به تماشا بنشینیم . این کلیپ ویدیو با به تصویر کشیدن قسمت آغازین روند داستانی این شخصیت ویژگی‌های جدید گان ‌پلی و بعضی محیط ‌های این بازی را نشان می‌دهد.

  • ارتباط گیم تایم به شدولند
    از همان روز اولی که شدولند به دنیای world of warcraft آمد گیم تایم نیز ارائه شد. می توان گفت که اصلی ترین هدف ارتباط گیم تایم به شدولند جلوگیری از چیت زدن است. چرا که برای اینکه شما بتوانید گیم تایم را بازی کنید باید هزینه زیادی را پرداخت کنید. از طرفی دیگر قوی کردن سرور ها است. بعد از به وجود آمدن سرور های گیم تایم سرور های بازی خود وارکرافت نیز قوی تر شده است.




    سخن آخر خرید گیم تایم 60 روزه
    جمع بندی که می توان از این مطلب داشته باشیم این است که شما می توانید برای خرید گیم تایم 60 روزه از فروشگاه جت گیم آن را خریداری کنید. گیم تایم 60 روزه دارای سرور اروپا و آمریکا است که بهتر است سرور گیم تایم شما با شدولند شما یکی باشد تا از لحاظ پینگی مشکلی را به وجود نیاورد. امیدوارم مطالب برای علاقمندان این گیم جذاب مفید قرار گرفته باشه با تشکر.

  • خرید گیم تایم 60 روزه از جت گیم

    اگر به دنبال این هستید که یک گیم تایم 60 روزه را خریداری کنید برای بازی world of warcraft خود می توانید به فروشگاه جت گیم مراجعه کنید. یکی از ویژگی های این فروشگاه آنی بودن آن است. پس از پرداخت قیمت کد محصول به شما در سریع ترین مدت زمان تحویل داده می شود. در حال حاضر مزیت فروشگاه جت گیم همین است که نسبت به فروشگاه های دیگر سریع تر است. و با کادری مجرب و با پشتیبانی محصولات ارائه شده به کاربران با مناسب ترین قیمت در حال فعالیت می باشد.

    بهترین راه برای اکتیو کردن گیم تایم 60 روزه
    راحت ترین راه و بهترین راه برای فعال کردن گیم تایم ارائه به کلاینت بتل نت است. بعد از اینکه شما گیم تایم 60 روزه را از جت گیم خریداری کنید به شما یک کد ارسال می شود. شما باید این کد را در کلاینت بتل نت بخش Rededm a Code وارد کنید تا گیم تایم 60 روزه برای شما فعال شود. اما راه دیگر شما برای اکتیو کردن گیم تایم مراجعه به سایت بتل نت است.

    ارتباط گیم تایم به شدولند
    از همان روز اولی که شدولند به دنیای world of warcraft آمد گیم تایم نیز ارائه شد. می توان گفت که اصلی ترین هدف ارتباط گیم تایم به شدولند جلوگیری از چیت زدن است. چرا که برای اینکه شما بتوانید گیم تایم را بازی کنید باید هزینه زیادی را پرداخت کنید. از طرفی دیگر قوی کردن سرور ها است. بعد از به وجود آمدن سرور های گیم تایم سرور های بازی خود وارکرافت نیز قوی تر شده است.




    سخن آخر خرید گیم تایم 60 روزه
    جمع بندی که می توان از این مطلب داشته باشیم این است که شما می توانید برای خرید گیم تایم 60 روزه از فروشگاه جت گیم آن را خریداری کنید. گیم تایم 60 روزه دارای سرور اروپا و آمریکا است که بهتر است سرور گیم تایم شما با شدولند شما یکی باشد تا از لحاظ پینگی مشکلی را به وجود نیاورد. امیدوارم مطالب برای علاقمندان این گیم جذاب مفید قرار گرفته باشه با تشکر.

  • Two-month gametime popularity:
    As mentioned above, 60-day gametime has been more popular than other gametime for a few months. This is because it has both the right time and the right price. The reason world of warcraft players use this type of game time is time. Because 60 days of game time is an average game time and most people use the days of this game time. One of the advantages of this game time over other game times is its length of time.

    Types of game time regions
    In general, the two-month game time is made up of 2 regions, Europe and America. But an important argument is that it is recommended to get a gametime regimen that is compatible with your Shodland region. If you are looking for our advice, we recommend that you buy the European region. Because it is close to Middle East servers and you usually get a better ping. More products from Jet Game site

  • گیم تایم 60 روزه در حال حاضر تنها گیم تایمی است که از طرف کمپانی blizzard برای بازیکنان گیم ، ورد اف وارکرافت ارائه شده است. در گذشته گیم تایم هایی مانند 30 روزه و 180 روزه هم موجود بود اما به دلیل سیاست های جدید این کمپانی و خط مشی که در نظر گرفته است، تنها گیم تایمی که در حال حاضر امکان فراهم کردن آن برای گیمر های عزیز، گیم تایم 60 روزه می باشد. در ادامه توضیحات جالبی در مورد گیم تایم برای شما جمع آوری کرده ایم که خواندنشان خالی از لطف نیست.

    کاربرد گیم تایم دو ماهه

    در حال حاضر گیم تایم 2 ماهه در تمامی زمینه های world of warcraft کاربرد دارد. اما اگر می خواهید که یک سری تجربه های جذاب و جدید را تجربه کنید باید این گیم تایم را خریداری کنید. این تجربه ها عبارتند از:
    استفاده از اکسپنشن های جدید
    بازی در مپ های جدید
    لول آپ به سبک جدید
    تغییر در شکل بازی
    تهیه از سایت جت گیم
    حمایت کنید لطفا

  • Popularity of Gametime for two months:
    As mentioned above, the 60-day game time has been more popular than other game times for several months. This is because it has both the right time and the right price. The reason why World of Warcraft players use this type of game time is the duration. Because the game time of 60 days is an average game time and most people use the days of this game time. One advantage that this game time has over other game times is its duration.

    All kinds of game time regions
    In general, the two-month game time is made from 2 regions, Europe and America. But an important point is that it is recommended to get a region of Gametime that is compatible with your Shadowland region. If you are looking for our advice, we recommend you to buy Region Europe. Because it is close to Middle East servers and usually you get better ping.
    Prepared from the Jet Game website

  • گیم تایم 60 روزه در حال حاضر تنها گیم تایمی است که از طرف کمپانی blizzard برای بازیکنان گیم ، ورد اف وارکرافت ارائه شده است. در گذشته گیم تایم هایی مانند 30 روزه و 180 روزه هم موجود بود اما به دلیل سیاست های جدید این کمپانی و خط مشی که در نظر گرفته است، تنها گیم تایمی که در حال حاضر امکان فراهم کردن آن برای گیمر های عزیز، گیم تایم 60 روزه می باشد. در ادامه توضیحات جالبی در مورد گیم تایم برای شما جمع آوری کرده ایم که خواندنشان خالی از لطف نیست.

    کاربرد گیم تایم دو ماهه

    در حال حاضر گیم تایم 2 ماهه در تمامی زمینه های world of warcraft کاربرد دارد. اما اگر می خواهید که یک سری تجربه های جذاب و جدید را تجربه کنید باید این گیم تایم را خریداری کنید. این تجربه ها عبارتند از:
    استفاده از اکسپنشن های جدید
    بازی در مپ های جدید
    لول آپ به سبک جدید
    تغییر در شکل بازی

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  • قهرمانان فوق العاده لحظات حماسی. فقط Overwatch.
    به بیش از 50 میلیون بازیکن بپیوندید و جای خود را در دنیای Overwatch بگیرید. قهرمان خود را ازبین گروه های گوناگون سربازان، دانشمندان، ماجراجویان و چیزهای عجیب و غریب انتخاب کنید. زمان را خم کنید، از فیزیک سرپیچی کنید و مجموعه ای فوق العاده از قدرت ها و سلاح های خارق العاده را آزاد کنید. دشمنان خود را در یک بازی اکشن نهایی تیمی درگیر کنید. شلیک خود را از دست ندهید
    قهرمانان فوق العاده لحظات حماسی. فقط Overwatch.
    به بیش از 50 میلیون بازیکن بپیوندید و جای خود را در دنیای Overwatch بگیرید. قهرمان خود را ازبین گروه های گوناگون سربازان، دانشمندان، ماجراجویان و چیزهای عجیب و غریب انتخاب کنید. زمان را خم کنید، از فیزیک سرپیچی کنید و مجموعه ای فوق العاده از قدرت ها و سلاح های خارق العاده را آزاد کنید. دشمنان خود را در یک بازی اکشن نهایی تیمی درگیر کنید. شلیک خود را از دست ندهید
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