Programmatically checking equivalence of statements

Question

So as part of a theorem-prover/checker, I'm using Prolog to try to determine the equivalence of statements that have been parsed into tree form, e.g. $x=2$ is represented as eq(x,2), or $x=2 \land y\leq z$ if and(eq(x,2),leq(y,z)), and I want to determine if these statements are equal.

The way I'm currently (attempting) to do this is by trying to convert statements to some normal form, e.g. converting all greater-than statements to less-than statements, or eq(x,x) simplifies to true, and then statements would be compared syntactically to determine equivalence. However, this feels like it might be a tad tricky/complicated, and to some extent I'm a bit uncomfortable simply using syntax rather than semantics to determine whether they are the same.

Is there some standard way of solving determining these equivalences algorithmically, or perhaps can anyone think of a better alternative?

Much appreciated.

Answer 1

So, there are two possibilities, depending on how complex your statements are.

If your statements are relatively simple, then there's a good chance they fit in what is decidable with Satisfiability Modulo Theories. It's basically a way of taking a SAT solver and integrating it with a solver for a specific theory, for example, theory of linear arithmetic. Here is a list of common theories.

If your statements are too complex, then reducing these equations to a normal form becomes undecidable. This is because there are ways to encode logic and computation in arithmetic, so determining the equivalence of some formulas can be equivalent to solving arbitrary logical equations.

The decidability of your problems depends heavily on whether you're using integers or reals, linear or non-linear arithmetic, etc. As was mentioned in the comments, if you can encode arbitrary Diophantine equations, then you're definitely undecidable.

Regardless of decidability, your problem is definitely $NP$-hard. Suppose we could efficiently check if the normal form of problems were syntactically the same. Then we could efficiently solve 3SAT by checking if a boolean formula is equivalent to False.