Detecting problematic divisions under SMT-LIB’s uninterpreted division semantics

Determine whether there exists an effective procedure that, given a first-order formula over the reals where division is interpreted as an uninterpreted total function per SMT-LIB (i.e., the value of div(x,0) is unconstrained), can decide whether occurrences of division in the formula cause the kinds of problematic behaviors identified by the authors—specifically, making problems jump in complexity, switch from decidable to undecidable, or cease to be well-defined for tasks such as computing dimension or connectivity of semi-algebraic sets.

Background

The paper analyzes the SMT-LIB approach of treating division by zero as an unconstrained value in a total function. Under this semantics, divisions can simulate fresh variables, dramatically increasing complexity, and, in the presence of quantifiers, can render satisfiability equivalent to undecidable problems (e.g., integer solutions to polynomial equations).

The authors also show that geometric properties like dimension or connectivity of sets defined by formulas with divisions may depend on the chosen interpretation of div(x,0), making those tasks ill-defined in the SMT-LIB framework.

Because these behaviors can be subtle and context-dependent, the authors remark that it is unclear whether one can algorithmically detect when a given formula’s divisions will lead to such issues, motivating their alternative fair-SAT semantics and translation algorithm for computer algebra contexts.