Logical Design: Constructing an Electronic System with a Minimum Number of Components

Constraints

The following constraints are used in this example:

  • bounds on variables

  • for $\text {gate} \in \text {GATES}$,

    \[ \Variable{AssignAGate[gate]} \le \Variable{UseGate[gate]} \]

  • for $\text {gate} \in \text {GATES}$,

    \[ \Variable{AssignBGate[gate]} \le \Variable{UseGate[gate]} \]

  • for $\text {gate} \in \text {GATES}$,

    \[ \sum _{(\text {pred},\text {gate}) \in \text {ARCS}} \Variable{UseGate[pred]} + \Variable{AssignAGate[gate]} + \Variable{AssignBGate[gate]} \le 2 \]

  • for $\text {row} \in \text {ROWS}$,

    \[ \Variable{Output[1,row]} = \Argument{target\_ output[row]} \]

  • for $\text {gate} \in \text {GATES}$ and $\text {row} \in \text {ROWS}$,

    \[ \Variable{Output[gate,row]} \le \Variable{UseGate[gate]} \]

  • for $\text {gate} \in \text {GATES}$ and $\text {row} \in \text {ROWS}$,

    \[ \Argument{inputA[row]} \cdot \Variable{AssignAGate[gate]} \le 1 - \Variable{Output[gate,row]} \]

  • for $\text {gate} \in \text {GATES}$ and $\text {row} \in \text {ROWS}$,

    \[ \Argument{inputB[row]} \cdot \Variable{AssignBGate[gate]} \le 1 - \Variable{Output[gate,row]} \]

  • for $(\text {pred},\text {gate}) \in \text {ARCS}$ and $\text {row} \in \text {ROWS}$,

    \[ \Variable{Output[pred,row]} \le 1 - \Variable{Output[gate,row]} \]

  • for $\text {gate} \in \text {GATES}$ and $\text {row} \in \text {ROWS}$,

    \begin{align*} & \quad \Argument{inputA[row]} \cdot \Variable{AssignAGate[gate]} + \Argument{inputB[row]} \cdot \Variable{AssignBGate[gate]} \\ & + \sum _{(\text {pred},\text {gate}) \in \text {ARCS}} \Variable{Output[pred,row]} \\ & \ge \Variable{UseGate[gate]} - \Variable{Output[gate,row]} \end{align*}