Overview of →

There are three kinds of → filter: the binary form, the ternary form, and the n-ary form for n>3.

The ASCII form of → is ->.

The binary form of the → filter denotes an attack by a piece on a square: 

      ♖→d4 //♖ attacks d4

The ternary form deals with pins and simple interferences: 

      ♖→♞→♚ //♞ pinned by ♖
      ♕→♙→h8 //♙ is between ♕ and h8

The n-ary form, of which the ternary form is a special case, is less commonly used: 

      ♕→♙→♙→h8
means that there are two distinct white pawns preventing an attack of ♕ on h8.

Each of these forms are discussed in more detail below.

binary →

If x and y are set filters, then 
    x→y
matches the current position if a piece on x attacks a piece on y.

A piece u is said to attack a square v if a king of different color than u would be in check from u if that king was on v.

For example, suppose that the current position is the chess start position. The following filters are all true: 

    d1→e2
    ♕→e2
    e2→d3
    not ♕→d4
    ♘g1→f3

Note that the check filter can be written in terms of →: 

    check
    ≡
    △→♚ or ▲→♔
    ≡
    ⬓△→♚

Now suppose we remove all the pawns from the start position. Then the following filters are all true: 

    ♕→d8
    ♕→♛
    ♗→g5

Value returned by →

The value of 
    X→Y
is the set of squares in Y that attack a square in Y.

This is equivalent to 

    x∊X x→Y

For example, suppose the current position is the start position in chess. Then 

    ♙→f3
    ≡
    [g2,e2]
and 
    △→f3
    ≡
    [g2,e2,g1]

←, binary case

The filter 
    x←y
is true exactly when 
    y→x
is true.

However, the value of 

    y←x
is the set of squares in y that are attacked by a piece in x.

For example, if the current position is the start position, then 

    ♙←♘
    ≡
    [e2,d2]

ternary →

Suppose that x, y and z are set filters each representing a single square in the current position. We will identify the set filter with the set of squares it represents as usual.

One can think of 

x→y→z
as being a sort of generalized pin. In fact, this matches a position exactly when 
    pin from x through y to z
matches the position.

The ternary → filter 

x→y→z

matches a position if each of the following four conditions is true:

  1. There is a sliding piece on x which attacks y
  2. y is between x and z
  3. Any squares between x and y are empty
  4. Any squares between y and z are empty

The above discussion was limited to the case where x, y, and z denoted a single square.

In general there can be multiple squares denoted by each of these set filters. In the case, we say that 

x→y→z
matches a position if there are squares X, Y, Z, in x, y, z respectively such that 
X→Y→Z

Examples of ternary →

    ♖→▲→♚
matches positions where a black piece is pinned by a white rook. 
♕→△→♚
matches positions where there is a single white piece between a ♕ and the ♚.

If the current position is the start position, then following are true: 

    d1→d2→d7
    ♗→♙→g5

The value of the ternary →

The value of 
x→y→z
is the set of squares X in x such that 
X→y→z
.

This can be written 

    X∊x X→y→z
.

ternary → versus nested binary →

You can use parentheses to prevent ternary evaluation of →. For example, 
    ♝→♖→♔
is true if a white rook is pinned by a black bishop. But 
    ♝→(♖→♔)
is true if a black bishop is attacking a white rook which is itself attacking the white king.

n-ary →

Chains with more than two → symbols are much rarer but are treated similarly (this section can be safely skipped unless necessary). More formally, consider a chain 
    x1→x2→x3→...→xn
for some integer n≥4.

First, suppose each xi has only one square. The the chain matches a position in that case that, if empty squares were placed on each square 

x2,x3,...,xn-1
then x1 would attack xn. But if any one of those squares were occupied, then x1 would not attack xn.

As before, in the general case where each xi could hold more than one square, we say that 

x1→x2→x3→...→xn
if there are squares 
    X1 in x1,
    X2 in x2,
     ... ,
    Xn in xn
such that 
X1→X2→X3→...→Xn
For example, suppose the current position is the chess starting position. Then the following filters are all true: 
    ♕→d2→d7→d8
    h1→h2→h4→h7→h8
    h1→g1→f1→e1
However: 
    not ♕→d2→d7→e7
    not h1→h3→h7→h8

The value returned by → chains

The value of a filter returned by a chain 
x1→x2→x3→...→xn
is the set of squares X in x1 such that 

X1→x2→x3→...→xn

For example, the set of white pieces pinning some black piece is: 

    △→▲→♚

← chains

z←y←z
matches a position if and only if 
x→y→z
matches the position.

The value returned by 

z←y←z
is the set of squares Z in z such that 
Z←y←x
matches the position.

Likewise, 

x1→x2→x3→...→xn
matches a position if and only if 
      xn←xn-1←...←x1
matches the position.

As before, the value of 

          xn←xn-1←...←x1
is the set of X in xn such that 
          X←xn-1←...←x1

Examples

There are many examples of simple → denoting attacks in the examples, for example, in turton.cql.

There are also a number of examples of length-3 → chains 

x→y→z
.

One use is to detect moves crossing a particular square.

The filter 

  ⊢ ―― from→d4→to
matches positions where the next move is a sliding move that crosses the square d4. That is because, from here is the departing square and to is the arriving square of the move matching ―― .

In many studies derived from problem themes, such as Bristol and Turton, a critical square must be crossed by a particular move. This usage occurs in turton.cql and bristol-universal.cql.

There are only two themes in our corpus where length-4 → chains arise naturally: in indian.cql and likeinterference.cql. We have yet to see a true length-5 → chain arise naturally however.