Documentation

Complexity.Computability.Circuit.Monotone.Basic

Monotone circuits #

The monotone basis has the Boolean constants, binary conjunction and binary disjunction: the De Morgan basis (Cslib.Circuits.Boolean.signature) without negation. Each of its operations is monotone (interpretation_mono), and so is every function computed by a monotone circuit. The depth of monotone formulas charges AND and OR gates and lets constants be free (binaryCost), as for De Morgan formulas, and the monotone basis maps into the De Morgan basis (toDeMorgan) keeping meanings and these costs (binaryCost_toDeMorgan).

Main definitions #

Main results #

References #

source
inductive Cslib.Circuits.Monotone.Op :
Type

Operations of the monotone basis.

  • const (b : Bool) : Op

    A Boolean constant.

  • and : Op

    Binary conjunction.

  • or : Op

    Binary disjunction.

Instances For
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    @[instance_reducible]
    instance Cslib.Circuits.Monotone.instDecidableEqOp :
    DecidableEq Op
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    def Cslib.Circuits.Monotone.instDecidableEqOp.decEq (x✝ x✝¹ : Op) :
    Decidable (x✝ = x✝¹)
    Instances For
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      @[instance_reducible]
      instance Cslib.Circuits.Monotone.instFintypeOp :
      Fintype Op
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      @[reducible, inline]
      abbrev Cslib.Circuits.Monotone.signature :
      Signature

      The monotone signature: constants, binary conjunction and binary disjunction.

      Instances For
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        def Cslib.Circuits.Monotone.interpretation :
        Interpretation signature Bool

        The Boolean interpretation of the monotone basis.

        Instances For
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          @[simp]
          theorem Cslib.Circuits.Monotone.interpretation_const (b : Bool) (x : Fin 0 → Bool) :
          interpretation (Op.const b) x = b
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          @[simp]
          theorem Cslib.Circuits.Monotone.interpretation_and (x : Fin 2 → Bool) :
          interpretation Op.and x = (x 0 && x 1)
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          @[simp]
          theorem Cslib.Circuits.Monotone.interpretation_or (x : Fin 2 → Bool) :
          interpretation Op.or x = (x 0 || x 1)
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          theorem Cslib.Circuits.Monotone.interpretation_mono (op : Op) {x y : Fin (signature.Arity op) → Bool} (h : ∀ (i : Fin (signature.Arity op)), x i ≤ y i) :
          interpretation op x ≤ interpretation op y

          The operations of the monotone basis are monotone.

          source
          def Cslib.Circuits.Monotone.binaryCost :
          Op → ℕ

          The cost function charging one for each AND and OR gate and nothing for constants, the monotone counterpart of Boolean.binaryCost (binaryCost_toDeMorgan).

          Instances For
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            @[simp]
            theorem Cslib.Circuits.Monotone.binaryCost_and :
            binaryCost Op.and = 1

            An AND gate costs one level.

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            @[simp]
            theorem Cslib.Circuits.Monotone.binaryCost_or :
            binaryCost Op.or = 1

            An OR gate costs one level.

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            @[simp]
            theorem Cslib.Circuits.Monotone.binaryCost_const (b : Bool) :
            binaryCost (Op.const b) = 0

            Constants are free.

            source
            def Cslib.Circuits.Monotone.andCost :
            Op → ℕ

            Count AND gates, with constants and OR gates free.

            Instances For
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              @[simp]
              theorem Cslib.Circuits.Monotone.andCost_and :
              andCost Op.and = 1
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              @[simp]
              theorem Cslib.Circuits.Monotone.andCost_or :
              andCost Op.or = 0
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              @[simp]
              theorem Cslib.Circuits.Monotone.andCost_const (b : Bool) :
              andCost (Op.const b) = 0

              The monotone basis inside the De Morgan basis #

              source
              def Cslib.Circuits.Monotone.toDeMorgan :
              signature.Hom Boolean.signature

              The monotone basis is the De Morgan basis without negation.

              Instances For
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                @[simp]
                theorem Cslib.Circuits.Monotone.toDeMorgan_const (b : Bool) :
                toDeMorgan (Op.const b) = Boolean.Op.const b
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                @[simp]
                theorem Cslib.Circuits.Monotone.toDeMorgan_and :
                toDeMorgan Op.and = Boolean.Op.and
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                @[simp]
                theorem Cslib.Circuits.Monotone.toDeMorgan_or :
                toDeMorgan Op.or = Boolean.Op.or
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                instance Cslib.Circuits.Monotone.instPreservesBoolToDeMorganInterpretationInterpretation :
                toDeMorgan.Preserves interpretation Boolean.interpretation

                The monotone operations mean what their De Morgan images mean.

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                @[simp]
                theorem Cslib.Circuits.Monotone.binaryCost_toDeMorgan (op : Op) :
                Boolean.binaryCost (toDeMorgan op) = binaryCost op

                The De Morgan image of an operation costs what it costs in the monotone basis.

                source
                theorem Cslib.Circuits.Monotone.toDeMorgan_injective :
                Function.Injective ⇑toDeMorgan

                The monotone operations embed injectively into the De Morgan basis.