The liberalized ?-rule in free variable semantic tableaux

Differential Dynamic Logic for Hybrid Systems

“…Yet, unlike in the approaches of Dowek et al [23] and Tinelli [57], the information given to the background prover is not restricted to ground formulas [57] or atomic formulas [23]. Further, real quantifier elimination is quite different from uninterpreted logic [33,26,30] in that the resulting formulas are not obtained by instantiation but by intricate arithmetic recombination. The F-rules can use any theory that admits quantifier elimination (see Def.…”

“…For handling quantifiers, we cannot use the standard rules [33,26,27], because these are for uninterpreted first-order logic and (ultimately) work by instantiating quantifiers, either eagerly as in ground tableaux or lazily by unification as in free variable tableaux [33,26,27]. The basis of dL, instead, is first-order logic interpreted over the reals or in the theory of real-closed fields [55].…”

“…The quantifier rules F1 and F2 correspond to the liberalised δ + -rule of Hähnle and Schmitt [33]. F4 and F5 resemble the usual γ-rule but, unlike in [26,27,33,30], they cannot be applied twice because the original formula is removed (∃x φ(x) in F4).…”

“…For continuous evolutions, we have to prove formulas like ∀t [α]x≥0 expressing that, for all durations t of some evolution in α, x ≥ 0 holds after all executions of system α. Standard first-order quantifier rules [33,26,27] are incomplete for handling these situations, because they are based on instantiation or unification, which is already insufficient for proving the tautology ∀z (z 2 ≥ 0). Unfortunately, decision procedures for real arithmetic like real quantifier elimination [55,16] cannot handle ∀t either, because of the modality [α].…”

A Semantic Completeness Proof for TaMeD

“…Moreover, if an existential quantifier is inside a universal one (as in the formula ∀x ∃y P (x, y)), we generate a new Skolem symbol each time we use δ after a new γ although we could simply use the same one over and over. This inefficiency could be solved either by removing the rule and preskolemizing the input formula or by using one of the improved δ-rules: δ + ( [14]) (or better δ + + ( [1]) or one of the other ones surveyed in [4]. Our completeness proof should not be changed by the use of one the two δ + rules.…”