Hierarchic Superposition: Completeness without Compactness

Many applications of automated deduction and verification require reasoning in combinations of theories, such as, on the one hand (some fragment of) first-order logic, and on, the other hand a background theory, such as some form of arithmetic. Unfortunately, due to the high expressivity of the full logic, complete reasoning is impossible in general. It is a realistic goal, however, to devise theorem provers that are “reasonably complete” in practice, and the hierarchic superposition calculus has been designed as a theoretical basis for that. In a recent paper we introduced an extension of hierarchic superposition and proved its completeness for the fragment where every term of the background sort is ground. In this paper, we extend this result and obtain completeness for a larger fragment that admits variables in certain places. 1 Hierarchic Superposition Many applications of automated deduction and verification require reasoning in combinations of theories, such as, on the one hand (some fragment of) first-order logic and on the other hand some form of arithmetic. In hierarchic superposition [2, 3] we consider the following scenario: We assume that we have a background (“BG”) prover that accepts as input a set of clauses over a BG signature ΣB = (ΞB, ΩB), where ΞB is a set of BG sorts and ΩB is a set of BG operators. Terms/clauses over ΣB and BG-sorted variables are called BG terms/clauses. For instance, ΞB might be {int, boolB} and ΩB might contain the integer numbers, +, −, <, ≤, true<, true≤, and additional parameters α, β, . . . that may be interpreted freely over the int-domain. The BG prover decides the satisfiability of ΣB-clause sets w. r. t. a BG specification, say linear integer arithmetic (LIA). For technical reasons, we assume that equality is the only predicate symbol in our language and that any non-equational atom p(t1, . . . , tn) is encoded as an equation p(t1, . . . , tn) ≈ truep. We refer to the terms that result from this encoding of atoms as atom terms; all other terms are called proper terms. When we simply write, say, x ≤ y, this should always be taken as a shorthand for an equation as above. ⋆ NICTA is funded by the Australian Government through the Department of Communications and the Australian Research Council through the ICT Centre of Excellence Program. The foreground (“FG”) theorem prover accepts as inputs clauses over a signature Σ = (Ξ,Ω), where ΞB ⊆ Ξ and ΩB ⊆ Ω. The sorts in ΞF = Ξ \ ΞB and the operator symbols in ΩF = Ω\ΩB are called FG sorts and FG operators. For instance, ΞF might be {list, boolF} and ΩF might contain operators cons : int × list → list, length : list → int, isempty : list → boolF, and trueisempty :→ boolF, among others. Σ-terms that are not BG terms are called FG terms. Notice that FG terms such as length(x) can have BG sorts. After abstracting out certain BG terms that occur as subterms of FG terms,3 the FG prover saturates the set of Σ-clauses using the inference rules of hierarchic superposition, such as, e. g., Negative superposition l ≈ r ∨ C s[u] 0 t ∨ D abstr((s[r] 0 t ∨ C ∨ D)σ) if (i) neither l nor u is a BG term, (ii) u is not a variable, (iii) σ is a simple mgu of l and u, (iv) rσ lσ, (v) (l ≈ r)σ is strictly maximal in (l ≈ r ∨ C)σ, (vi) the first premise does not have selected literals, (vii) tσ sσ, and (viii) if the second premise has selected literals, then s 0 t is selected in the second premise, otherwise (s 0 t)σ is maximal in (s 0 t ∨ D)σ. These differ from the standard superposition inference rules [1] mainly in that only the FG parts of clauses are overlapped and that any BG clauses derived during the saturation are instead passed to the BG prover. The BG prover implements an inference rule