Computational reverse mathematics and foundational analysis

Reverse mathematics studies which subsystems of second order arithmetic are equivalent to key theorems of ordinary, non-set-theoretic mathematics. The main philosophical application of reverse mathematics proposed thus far is foundational analysis, which explores the limits of different foundations for mathematics in a formally precise manner. This paper gives a detailed account of the motivations and methodology of foundational analysis, which have heretofore been largely left implicit in the practice. It then shows how this account can be fruitfully applied in the evaluation of major foundational approaches by a careful examination of two case studies: a partial realization of Hilbert's program due to Simpson [1988], and predicativism in the extended form due to Feferman and Sch\"{u}tte. Shore [2010, 2013] proposes that equivalences in reverse mathematics be proved in the same way as inequivalences, namely by considering only $\omega$-models of the systems in question. Shore refers to this approach as computational reverse mathematics. This paper shows that despite some attractive features, computational reverse mathematics is inappropriate for foundational analysis, for two major reasons. Firstly, the computable entailment relation employed in computational reverse mathematics does not preserve justification for the foundational programs above. Secondly, computable entailment is a $\Pi^1_1$ complete relation, and hence employing it commits one to theoretical resources which outstrip those available within any foundational approach that is proof-theoretically weaker than $\Pi^1_1\text{-}\mathsf{CA}_0$.Comment: Submitted. 41 page

Computational Complexity for Physicists

These lecture notes are an informal introduction to the theory of computational complexity and its links to quantum computing and statistical mechanics.Comment: references updated, reprint available from http://itp.nat.uni-magdeburg.de/~mertens/papers/complexity.shtm

Testing for the ground (co-)reducibility property in term-rewriting systems

AbstractGiven a term-rewriting system R, a term t is ground-reducible by R if every ground instance tσ of it is R-reducible. A pair (t, s) of terms is ground-co-reducible by R if every ground instance (tσ, sσ] of it for which tσ and sσ are distinct is R-reducible. Ground (co-)reducibility has been proved to be the fundamental tool for mechanizing inductive proofs, together with the Knuth-Bendix completion procedure presented by Jouannaud and Kounalis (1986, 1989).Jouannaud and Kounalis (1986, 1989) also presented an algorithm for testing ground reducibility which is tractable in practical cases but restricted to left-linear term-rewriting systems. The solution of the ground (co-)reducibility problem, for the general case, turned out to be surprisingly complicated. Decidability of ground reducibility for arbitrary term-rewriting systems has been first proved by Plaisted (1985) and independently by Kapur (1987). However, the algorithms of Plaisted and Kapur amount to intractable computation, even in very simple cases.We present here a new algorithm for the general case which outperforms the algorithms of Plaisted and Kapur and even our previous algorithm in case of left-linear term-rewriting systems. We then show how to adapt it to check for ground co-reducibility

The intuitionistic fragment of computability logic at the propositional level

This paper presents a soundness and completeness proof for propositional intuitionistic calculus with respect to the semantics of computability logic. The latter interprets formulas as interactive computational problems, formalized as games between a machine and its environment. Intuitionistic implication is understood as algorithmic reduction in the weakest possible -- and hence most natural -- sense, disjunction and conjunction as deterministic-choice combinations of problems (disjunction = machine's choice, conjunction = environment's choice), and "absurd" as a computational problem of universal strength. See http://www.cis.upenn.edu/~giorgi/cl.html for a comprehensive online source on computability logic

The computational complexity of density functional theory

Density functional theory is a successful branch of numerical simulations of quantum systems. While the foundations are rigorously defined, the universal functional must be approximated resulting in a `semi'-ab initio approach. The search for improved functionals has resulted in hundreds of functionals and remains an active research area. This chapter is concerned with understanding fundamental limitations of any algorithmic approach to approximating the universal functional. The results based on Hamiltonian complexity presented here are largely based on \cite{Schuch09}. In this chapter, we explain the computational complexity of DFT and any other approach to solving electronic structure Hamiltonians. The proof relies on perturbative gadgets widely used in Hamiltonian complexity and we provide an introduction to these techniques using the Schrieffer-Wolff method. Since the difficulty of this problem has been well appreciated before this formalization, practitioners have turned to a host approximate Hamiltonians. By extending the results of \cite{Schuch09}, we show in DFT, although the introduction of an approximate potential leads to a non-interacting Hamiltonian, it remains, in the worst case, an NP-complete problem.Comment: Contributed chapter to "Many-Electron Approaches in Physics, Chemistry and Mathematics: A Multidisciplinary View