Hierarchic Superposition Revisited

Abstract

Many applications of automated deduction require reasoning in first-order logic modulo background theories, in particular some form of integer arithmetic. A major unsolved research challenge is to design theorem provers that are "reasonably complete" even in the presence of free function symbols ranging into a background theory sort. The hierarchic superposition calculus of Bachmair, Ganzinger, and Waldmann already supports such symbols, but, as we demonstrate, not optimally. This paper aims to rectify the situation by introducing a novel form of clause abstraction, a core component in the hierarchic superposition calculus for transforming clauses into a form needed for internal operation. We argue for the benefits of the resulting calculus and provide two new completeness results: one for the fragment where all background-sorted terms are ground and another one for a special case of linear (integer or rational) arithmetic as a background theory. ⋆ The final authenticated version is available online at https://doi.org/TBD.

Symbol Elimination for Parametric Second-Order Entailment Problems (with Applications to Problems in Wireless Network Theory)

“…Our goal is to compute, if possible, a first-order Π-formula G such that G ≡ T ∃P F . We adapt the hierarchical superposition calculus proposed in [8,9] to this case.…”

“…, where each support of interpreted sort is considered to be fixed. Following the terminology used in [8,9], we will refer to elements in the fixed domain of sort s ∈ S i as domain elements of sort s. Let F be a universal first-order formula over signature Π ′ = (S, Σ, Pred ∪ {P }). We can assume, without loss of generality, that F is a set of clauses of the form ∀x D(x) ∨ C(x), where D(x) is a clause over the signature Π and C(x) is a clause containing literals of the form (¬)P (x 1 , .…”

“…Let ≻ be a strict, well-founded ordering on terms that is compatible with contexts and stable under substitutions. As in [9] we assume that ≻ has the following properties: 4 (i) ≻ is total on ground terms, (ii) t ≻ d for every domain element d of interpreted sort s and every ground term t that is not a domain element.…”

“…As discussed before, the value of C in A w.r.t. β is the same as the value of C ′ = Cσ in A A , where σ is the substitution that associates with every variable x the constant Proof: First, note that for constrained P -clauses the hierarchical superposition calculus specializes to HRes P ≻ : With the terminology used in [5,8,9], the background signature is Π; the only foreground symbol is P . Since there are no "background"-sorted terms starting with a "foreground" function symbol, sets of P -clauses are sufficiently complete.…”

“…In [5], Bachmair et al mention that hierarchical superposition (cf. [8,9] for further refinements) can be used for second-order quantifier elimination modulo a theory. In [34,24], Hoder et al study possibilities of symbol elimination in inference systems (e.g.…”