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# Logic algebra package for Maxima

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## 1. Introduction

This is a draft version of logic algebra package for Maxima. It is being developed by Alexey Beshenov (al@beshenov.ru). All source code is available uder the terms of GNU GPL 2.1.

List of recognized operators:

Operator | Type | Binding power | Description | Properties
--- | --- | --- | --- | ---
`not` | Prefix | `70` | Logical NOT (negation) |
`and` | N-ary | `65` | Logical AND (conjunction) | Commutative
`nand` | N-ary | `62` | Sheffer stroke (alternative denial, NAND) | Commutative
`nor` | N-ary | `61` | Webb-operation or Peirce arrow (Quine's dagger, NOR) | Commutative
`or` | N-ary | `60` | Logical OR (disjunction) | Commutative
`implies` | Infix | `59` | Implication |
`eq` | N-ary | `58` | Equivalence | Commutative
`xor` | N-ary | `58` | Sum modulo 2 (exclusive or) | Commutative

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## 2. TeX output

logic.mac assigns the following TeX output:

`not`

  `\neg`

`and`

  `\wedge`

`nand`

  `\mid`

`nor`

  `\downarrow`

`or`

  `\vee`

`implies`

  `\rightarrow`

`eq`

  `\sim`

`xor`

  `\oplus`

Examples:

    (%i1) load ("logic.mac")$
    (%i2) tex (a implies b)$
    $$a \rightarrow b$$
    (%i3) tex ((a nor b) nand c)$
    $$\left(a \downarrow b\right) \mid c$$
    (%i4) tex (zhegalkin_form (a or b or c))$
    $$a \wedge b \wedge c \oplus a \wedge b \oplus a \wedge c \oplus b
      \wedge c \oplus a \oplus b \oplus c$$
    (%i5) tex (boolean_form (a implies b implies c));
    $$ \neg \left( \neg a \vee b\right) \vee c$$
    (%i6) tex (a eq b eq c);
    $$a \sim b \sim c$$

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## 3. Functions for logic algebra

Function: logic_simp (expr)

  Returns a simplified version of logical expression expr.

  Examples:

      (%i1) load ("logic.mac")$
      (%i2) logic_simp (a or (b or false or (a or b)));
      (%o2)                               a or b
      (%i3) logic_simp (b eq a eq false eq true);
      (%o3)                           eq a eq b false
      (%i4) logic_simp ((a xor true) xor b xor true);
      (%o4)                               a xor b

  The function applies only basic simplification rules without introducing new functions.

  N.B. It should be merged somehow with the basic Maxima simplifier.

Function: characteristic_vector (expr, var_1, ..., var_n)

  Returns a list of size 2^n with all possible values of expr.

  For example, `characteristic_vector (f(x,y,z), x, y, z)` is equivalent to list

      [
        f (false, false, false),
        f (false, false,  true),
        f (false,  true, false),
        f (false,  true,  true),
        f ( true, false, false),
        f ( true, false,  true),
        f ( true,  true, false),
        f ( true,  true,  true)
      ]

  If `var_1, ..., var_n` is omitted, it is assumed that

      [var_1, ..., var_n] = sort(listofvars(expr))

  Examples:

      (%i1) load ("logic.mac")$
      (%i2) characteristic_vector (true);
      (%o2)                               [true]
      (%i3) characteristic_vector (a xor b);
      (%o3)                     [false, true, true, false]
      (%i4) characteristic_vector (a implies b);
      (%o4)                      [true, true, false, true]
      (%i5) characteristic_vector (a implies b, a, b);
      (%o5)                      [true, true, false, true]
      (%i6) characteristic_vector (a implies b, b, a);
      (%o6)                      [true, false, true, true]

Function: zhegalkin_form (expr)

  Returns the representation of expr in Zhegalkin basis `{xor, and, true}`.

  Examples:

      (%i1) load ("logic.mac")$
      (%i2) zhegalkin_form (a or b or c);
      (%o2) (a and b and c) xor (a and b) xor (a and c)
                                     xor (b and c) xor a xor b xor c
      (%i3) zhegalkin_form ((a implies b) or c);
      (%o3) (a and b and c) xor (a and b) xor (a and c) xor a
                                                            xor true

Function: logic_equiv (expr_1, expr_2)

  Returns `true` if expr_1 is equivalent to expr_2 and `false` otherwise.

  Examples:

      (%i1) load ("logic.mac")$
      (%i2) e : ((a or b) xor c) and d$
      (%i3) zhegalkin_form (e);
      (%o3) (a and b and d) xor (a and d) xor (b and d)
                                                       xor (c and d)
      (%i4) logic_equiv (%i2, %o3);
      (%o4)                                true
      (%i5) is (characteristic_vector(%i2) = characteristic_vector(%o3));
      (%o5)                                true
      (%i6) logic_equiv (x and y eq x, x implies y);
      (%o6)                                true

Function: dual_function (expr)

      dual_function (f (x_1, ..., x_n)) := not f (not x_1, ..., not x_n).

  Example:

      (%i1) load ("logic.mac")$
      (%i2) dual_function (x or y);
      (%o2)                     not ((not x) or (not y))
      (%i3) demorgan (%);
      (%o3)                               x and y

Function: self_dual (expr)

  Returns `true` if expr is equivalent to `dual_function (expr)` and `false` otherwise.

  Examples:

      (%i1) load ("logic.mac")$
      (%i2) self_dual (a);
      (%o2)                               true
      (%i3) self_dual (not a);
      (%o3)                               true
      (%i4) self_dual (a eq b);
      (%o4)                               false

Function: closed_under_f (expr)

  `closed_under_f (f (x_1, ..., x_n)` returns `true` if `f (false, ..., false) = false` and `false` otherwise.

  Examples:

      (%i1) load ("logic.mac")$
      (%i2) closed_under_f (x and y);
      (%o2)                                true
      (%i3) closed_under_f (x or y);
      (%o3)                                true

Function: closed_under_t (expr)

  `closed_under_t (f (x_1, ..., x_n)` returns `true` if `f (true, ..., true) = true` and `false` otherwise.

  Examples:

      (%i1) load ("logic.mac")$
      (%i2) closed_under_t (x and y);
      (%o2)                                true
      (%i3) closed_under_t (x or y);
      (%o3)                                true

Function: monotonic (expr)

  Returns `true` if characteristic vector of expr is monotonic, i.e.

      charvec : characteristic_vector(expr)
      charvec[i] <= charvec[i+1],   i = 1, ..., n-1

  where `a<=b := (a=b or (a=false and b=true))`.

  Examples:

      (%i1) load ("logic.mac")$
      (%i2) monotonic (a or b);
      (%o2)                                true
      (%i3) monotonic (a and b);
      (%o3)                                true
      (%i4) monotonic (a implies b);
      (%o4)                                false
      (%i5) monotonic (a xor b);
      (%o5)                                false
      (%i6) characteristic_vector (a or b);
      (%o6)                     [false, true, true, true]
      (%i7) characteristic_vector (a and b);
      (%o7)                    [false, false, false, true]
      (%i8) characteristic_vector (a implies b);
      (%o8)                     [true, true, false, true]
      (%i9) characteristic_vector (a xor b);
      (%o9)                    [false, true, true, false]

Function: linear (expr)

  Returns `true` if `zhegalkin_form(expr)` is linear and `false` otherwise.

  Examples:

      (%i1) load ("logic.mac")$
      (%i2) linear (a or b);
      (%o2)                                false
      (%i3) linear (a eq b);
      (%o3)                                true
      (%i4) zhegalkin_form (a or b);
      (%o4)              (a and b) xor a xor b
      (%i5) zhegalkin_form (a eq b);
      (%o5)                 a xor b xor true

  Linear functions are also known as counting or alternating functions.

Function: functionally_complete (expr_1, ..., expr_n)

  Returns `true` if expr_1, ..., expr_n is a functionally complete system and `false` otherwise. The constants are essential (see the example below).

  Examples:

      (%i1) load ("logic.mac")$
      (%i2) functionally_complete (x and y, x xor y);
      (%o2)                                false
      (%i3) functionally_complete (x and y, x xor y, true);
      (%o3)                                true
      (%i4) functionally_complete (x and y, x or y, not x);
      (%o4)                                true

Function: logic_basis (expr_1, ..., expr_n)

  Returns `true` if expr_1, ..., expr_n is a functionally complete system without redundant elements and `false` otherwise.

  Examples:

      (%i1) load ("logic.mac")$
      (%i2) logic_basis (x and y, x or y);
      (%o2)                       false
      (%i3) logic_basis (x and y, x or y, not x);
      (%o3)                       false
      (%i4) logic_basis (x and y, not x);
      (%o4)                       true
      (%i5) logic_basis (x or y, not x);
      (%o5)                       true
      (%i8) logic_basis (x and y, x xor y, true);
      (%o8)                       true

  All possible bases:

      (%i1) load ("logic.mac")$
      (%i2) logic_functions : { not x, x nand y, x nor y,
                                x implies y, x and y, x or y,
                                x eq y, x xor y, true, false }$
      (%i3) subset (powerset(logic_functions),
                    lambda ([s], apply ('logic_basis, listify(s))));
      (%o3) {{false, x eq y, x and y}, {false, x eq y, x or y},
      {false, x implies y}, {true, x xor y, x and y},
      {true, x xor y, x or y}, {not x, x implies y},
      {not x, x and y}, {not x, x or y},
      {x eq y, x xor y, x and y}, {x eq y, x xor y, x or y},
      {x implies y, x xor y}, {x nand y}, {x nor y}}

Function: logic_diff (f, x)

  Returns the logic derivative df/dx of f wrt x.

      logic_diff (f (x_1, ..., x_k, ..., x_n),  x_k) :=
          f (x_1, ..., true, ..., x_n)  xor
              f (x_1, ..., false, ..., x_n)

  Examples:

      (%i1) load ("logic.mac")$
      (%i2) logic_diff (a or b or c, a);
      (%o2)          (b and c) xor b xor c xor true
      (%i3) logic_diff (a and b and c, a);
      (%o3)                      b and c
      (%i4) logic_diff (a or (not a), a);
      (%o4)                       false

Function: boolean_form (expr)

  Returns the representation of expr in Boolean basis `{and, or, not}`.

  Examples:

      (%i1) load ("logic.mac")$
      (%i2) boolean_form (a implies b implies c);
      (%o2)             (not ((not a) or b)) or c
      (%i3) demorgan (%);
      (%o3)               ((not b) and a) or c
      (%i4) logic_equiv (boolean_form (a implies b implies c),
                         zhegalkin_form (a implies b implies c));
      (%o4)                       true

Function: demorgan (expr)

  Applies De Morgan's rules to expr:

      not (x_1 and ... and x_n) => (not x_1  or ...  or not x_n)
      not (x_1  or ...  or x_n) => (not x_1 and ... and not x_n)

  Example:

      (%i1) load ("logic.mac")$
      (%i2) demorgan (boolean_form (a nor b nor c));
      (%o2)          (not a) and (not b) and (not c)

Function: pdnf (expr)

  Returns the perfect disjunctive normal form of expr.

  Example:

      (%i1) load ("logic.mac")$
      (%i2) pdnf (x implies y);
      (%o2) (x and y) or ((not x) and y) or ((not x) and (not y))

Function: pcnf (expr)

  Returns the perfect conjunctive normal form of expr.

  Example:

      (%i1) load ("logic.mac")$
      (%i2) pcnf (x implies y);
      (%o2)                   (not x) or y

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# About This Document

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