The Merits of Compositional Abstraction: A Case Study in Propositional Logic

Abstract

We revisit a well-established and old topic in computational logic: algorithms – such as the one by Quine-McCluskey – that convert a formula of propositional logic into a semantically equivalent disjunctive normal form whose clauses are all prime implicants of that formula. This exercise in education is meant to honor Bernhard Steffen, who made important contributions in formal verification and its use of compositional abstraction, and who is a role model in transferring research insights into teaching addressed at students with varying skill levels. The algorithm we propose here is indeed compositional and can teach students about the value of compositional abstractions – making use of simple lattice-theoretic and semantic concepts.

A Exercises

• 1. Let (X, Y) be such that ${\mathsf {DNF}({X})}\equiv \phi$ and ${\mathsf {DNF}({Y})} \equiv \lnot \phi$. Show for all z in ${\mathsf {\mathbf{3}}}^{n}_\top \setminus \{\top \}$ that

  • (a) ${\mathsf {Clause}({z})}$ is an implicant of $\phi$ iff z is inconsistent with all y in Y

  • (b) ${\mathsf {Clause}({z})}$ is an implicant of $\lnot \phi$ iff z is inconsistent with all x in X

• 2. Reconsider the algorithm ${\mathsf {PI}({\phi })}$ and the definition of ${\mathsf {Reduce}({X},{Y})}$. The latter chose one element of a non-empty set Z to make a reduction. Discuss whether the algorithm ${\mathsf {PI}({\phi })}$ can be amended to make reductions for more than one or even all elements of set Z, concurrently or sequentially.

• 3. Consider a different implementation of ${\mathsf {Reduce}({X},{Y})}$ shown in Fig. 3 and let ${\mathsf {PI}'({\phi })}$ be the implementation of ${\mathsf {PI}({\phi })}$ that uses this new version ${\mathsf {Reduce}'({X},{Y})}$ of ${\mathsf {Reduce}({X},{Y})}$ instead.

  • (a) Prove that ${\mathsf {PI}'({\phi })}$ terminates for all formulas $\phi$ that are legitimate input to ${\mathsf {PI}({\phi })}$.

  • (b) Prove that ${\mathsf {PI}'({\phi })}$ satisfies the following:

    “(PI’) The call ${\mathsf {PI}({\phi })}$ returns a pair (X, Y) of finite setsets of ${\mathsf {\mathbf{3}}}^{n}_\top \setminus \{\top \}$ such that ${\mathsf {DNF}({X})}\equiv \phi$ and ${\mathsf {DNF}({Y})}\equiv \lnot \phi$.”

  • (c) Show that (PI’) cannot be strengthened to the statement (PI) on page 8, i.e. the DNFs may not only contain prime implicants.

  • (d) Consider a variant of ${\mathsf {PI}({\phi })}$ that uses both ${\mathsf {Reduce}({X},{Y})}$ and ${\mathsf {Reduce}'({X},{Y})}$ so that (PI) on page 8 will be satisfied of this variant. Discuss in which order these two types of reduction may have to be performed to guarantee (PI).

  • (e) Let (X, Y) be the output of ${\mathsf {PI}({\phi })}$ for some formula $\phi$. Show that the call to ${\mathsf {Reduce}'({X},{Y})}$ will compute an empty set Z.

• 4. Consider another implementation of function ${\mathsf {Conj}({(X,Y)},{(U,V)})}$, in which P and N are still computed as in that function but where there is no repeat statement for function ${\mathsf {Reduce}({X},{Y})}$. Instead, function ${\mathsf {Reduce}({X},{Y})}$ calls another function ${\mathsf {makePrime}({\emptyset },{X},{Y})}$ with header ${\mathsf {makePrime}({A},{X},{Y})}$. The idea is that ${\mathsf {makePrime}({A},{X},{Y})}$ is a tail-recursive function that iteratively goes through each element of X, converts it into a prime implicant of ${\mathsf {DNF}({X})}$ that is below that x, adds that prime implicant to set A, and removes any elements from X that are above that new prime implicant. The desired output of ${\mathsf {Reduce}({X},{Y})}$ is then the final value of set A.

  • (a) Propose such an amended implementation of ${\mathsf {Conj}({X},{Y})}$ and the methods it calls.

  • (b) Prove that this still realizes an implementation that satisfies requirement (PI) on page 8.