It is well–known (due to C. Parsons) that the extension of primitive recursive arithmetic PRA by first–order predicate logic and the rule of Π02–induction Π02–IR is Π 0 2–conservative over PRA. We show that this is no longer true in the presence of function quantifiers and quantifier–free choice for numbers AC– qf. More precisely we show that T :=PRA + Π02–IR+AC –qf proves the totality of the A...

SOFT (‘Second-Order Functions and Theorems’) is a tool to mimic second-order functions and theorems in the first-order logic of ACL2. Second-order functions are mimicked by first-order functions that reference explicitly designated uninterpreted functions that mimic function variables. First-order theorems over these second-order functions mimic second-order theorems universally quantified over...

We present an illative system Is of classical higher-order logic with subtyping and basic inductive types. The system Is allows for direct definitions of partial and general recursive functions, and provides means for handling functions whose termination has not been proven. We give examples of how properties of some recursive functions may be established in our system. In a technical appendix ...

We show new upper and lower bounds for the effective differential Nullstellensatz for differential fields of characteristic zero with several commuting derivations. Seidenberg was the first to address this problem in 1956, without giving a complete solution. The first explicit bounds appeared in 2009 in a paper by Golubitsky, Kondratieva, Szanto, and Ovchinnikov,with the upper bound expressed i...

Let P be a poset in which each point is incomparable to at most ∆ others. Tanenbaum, Trenk, and Fishburn asked in a 2001 paper if the linear discrepancy of such a poset is bounded above by b(3∆−1)/2c. This paper answers their question in the affirmative for two classes of posets. The first class is the interval orders, which are shown to have linear discrepancy at most ∆, with equality precisel...