1. Boolean Unification EE219B Presented by: Jason Shamberger March 1, 2000

2. Outline What is Boolean unification? Boolean unification algorithms Boole’s method Löwenheim’s method Applications

3. Boolean Unification Boolean unification is a procedure to obtain a general solution of a Boolean equation Use parametric variables to represent don’t cares

4. Boolean Unification Given a Boolean equation of the form Where x i ’s are inputs and q i ’s are constants We want to find the most general substitution for x i ’s, that satisfies the above equation

5. Boolean Unification Where p i ’s can be any variable or function

6. Boolean Unification: Example Consider the following example: The solutions for x 1, x 2, x 3 are:

7. Algorithms for Boolean Unification There are two main algorithms for Boolean unification Boole’s method Löwenheim’s method Both algorithms can be implemented using BDD operations

8. Boole’s Method Based on successive elimination of variables Define conjunctive eliminant (ECON) as:

9. Boole’s Method If X, G, and P represent the vectors of inputs, parameters, and parametric functions, then the parametric general solution of the equation: Is a system: Such that the following conditions are satisfied:

10. Theorem It can be shown that for a Boolean function f, for which a solution to f(x)=0 exists: Where:

11. Proof: A sketch of the proof: To show S=P:

12. Proof: Show for all a in B, there is a p in B: Use the property that: Simplifies to: Which is satisfied for all p=a

13. Boole’s Method Using this result, we can generate parametric equations for each of the functions g i Consider a 3 variable function: We first calculate the eliminants:

14. Boole’s Method We can now generate functions for each input x: And the parametric general solution is:

15. Boole’s Method: Example As an example, consider the RST flip-flop, which is defined by the equations: y is the present state, Y is the next state, R,S,T are the reset, set, and toggle inputs. The first equation describes how the next state is determined, and the second equation is the constraint that only one input may be a 1 at any time

16. Boole’s Method: Example We first need to solve the system for R,S, and T which we consider variables, in terms of y and Y which we regard as elements of B. Where f is defined by:

17. Boole’s Method: Example The eliminants are:

18. Boole’s Method: Example We can now derive the parametric expressions for the general solution: Any values can be chosen for p i ’s. For example, let p 1 =1, p 2 =y :

19. Boole’s Method: The method of successive elimination demonstrates that we need at most n parameters, but possibly less as in the previous example. Most unification algorithms are implementations of Boole’s method

20. Löwenheim’s Method: Allows us to form a parametric general solution from any particular solution Useful when it is trivial to find a particular solution, and constructing a general solution is inconvenient

21. Löwenheim’s Method: Theorem: Let U be a particular solution of a consistent Boolean equation f(x)=0, and let P be an n-tuple of arbitrary parameters. Then a parametric general solution is given by: Provided that:

22. Löwenheim’s Method: Proof: Using the previous definition, the following conditions must be satisfied:

23. Löwenheim’s Method: Example Consider the RST flip-flop example from before. An easily obtained particular solution of the RST equation is:

24. Löwenheim’s Method: Example There are three inputs, thus there are three parameters, we will call them p,q,r. Using the previous theorem, the general solution is given by:

25. Löwenheim’s Method: Example After some calculations we arrive at: Notice that we generate n new parameters for an n variable equation.

26. Applications Applications include: Design rectification Boolean matching Minimizing Boolean relations

27. Application: Design Rectification Problem: A circuit has already been laid out, but small modifications are desired (design errors, change in specs) Can only add extra circuitry before input pins, or after output pins

28. Application: Design Rectification To guarantee that the rectification is correct, the following must hold: This is equivalent to computing: Which is in the necessary form for Boolean unification

29. Conclusion Boolean unification allows us to solve a Boolean equation of the form f(X)=0 Generate parametric equations for each of the inputs x i Is feasible to implement using BDD representations

30. References Boolean Reasoning: The Logic of Boolean Equations. Brown, Frank Markham Application of Boolean Unification to Logic Synthesis. Poncino, Massimo Application of Boolean Unification to Combinational Logic Synthesis. Fujita, Masahiro etal.