Interactive Learning Based Realizability and 1-Backtracking Games

This paper describes axiomatic theories SA and SAR, which are versions of second order arithmetic with countably many sorts for sets of natural numbers. The theories are intended to be applied in reverse mathematics because their multi-sorted language allows to express some mathematical statements in more natural form than in the standard second order arithmetic. We study metamathematical properties of the theories SA, SAR and their fragments. We show that SA is mutually interpretable with the theory of arithmetical truth PATr obtained from the Peano arithmetic by adding infinitely many truth predicates. Corresponding fragments of SA and PATr are also mutually interpretable. We compare the proof-theoretical strengths of the fragments; in particular, we show that each fragment SAs with sorts <=s is weaker than next fragment SAs+1.

This paper is a contribution to the project of determining which set existence axioms are needed to prove various theorems of analysis. For more on this project and its history we refer the reader to [1] and [2].We work in a weak subsystem of second order arithmetic. The language of second order arithmetic includes the symbols 0, 1, =, <, +, ·, and ∈, together with number variables x, y, z, … (which are intended to stand for natural numbers), set variables X, Y, Z, … (which are intended to stand for sets of natural numbers), and the usual quantifiers (which can be applied to both kinds of variables) and logical connectives. We write ∀x < t φ and ∃x < t φ as abbreviations for ∀x(x < t → φ) and ∃x{x < t ∧ φ) respectively; these are called bounded quantifiers. A formula is said to be if it has no quantifiers applied to set variables, and all quantifiers applied to number variables are bounded. It is if it has the form ∃xθ and it is if it has the form ∀xθ, where in both cases θ is .The theory RCA0 has as axioms the usual Peano axioms, with the induction scheme restricted to formulas, and in addition the comprehension scheme, which consists of all formulas of the formwhere φ is , ψ is , and X does not occur free in φ(n). (“RCA” stands for “Recursive Comprehension Axiom.” The reason for the name is that the comprehension scheme is only strong enough to prove the existence of recursive sets.) It is known that this theory is strong enough to allow the development of many of the basic properties of the real numbers, but that certain theorems of elementary analysis are not provable in this theory. Most relevant for our purposes is the fact that it is impossible to prove in RCA0 that every continuous function on the closed interval [0, 1] attains maximum and minimum values (see [1]).Since the most common proof of the Mean Value Theorem makes use of this theorem, it might be thought that the Mean Value Theorem would also not be provable in RCA0. However, we show in this paper that the Mean Value Theorem can be proven in RCA0. All theorems stated in this paper are theorems of RCA0, and all of our reasoning will take place in RCA0.

We prove that the modal mu-calculus model-checking problem for (ranked and ordered) node-labelled trees that are generated by order- recursion schemes (whether safe or not, and whether homogeneously typed or not) is - EXPTIME complete, for every n \geqslant 0. It follows that the monadic second-order theories of these trees are decidable. There are three major ingredients. The first is a certain transference principle from the tree generated by the scheme - the value tree - to an auxiliary computation tree, which is itself a tree generated by a related order-0 recursion scheme (equivalently, a regular tree). Using innocent game semantics in the sense of Hyland and Ong, we establish a strong correspondence between paths in the value tree and traversals in the computation tree. This allows us to prove that a given alternating parity tree automaton (APT) has an (accepting) run-tree over the value tree iff it has an (accepting) traversal-tree over the computation tree. The second ingredient is the simulation of an (accepting) traversal-tree by a certain set of annotated paths over the computation tree; we introduce traversal-simulating APT as a recognising device for the latter. Finally, for the complexity result, we prove that traversal-simulating APT enjoy a succinctness property: for deciding acceptance, it is enough to consider run-trees that have a reduced branching factor. The desired bound is then obtained by analysing the complexity of solving an associated (finite) acceptance parity game.

Higher-order model checking - the model checking of trees generated by higher-order recursion schemes (HORS) - is a natural generalisation of finite-state and pushdown model checking. Recent work has shown that it can serve as a basis for software model checking for functional languages such as ML and Haskell. In this paper, we introduce higher-order recursion schemes with cases (HORSC), which extend HORS with a definition-by-cases construct (to express program branching based on data) and non-determinism (to express abstractions of behaviours). This paper is a study of the universal HORSC model checking problem for deterministic trivial automata : does the automaton accept every tree in the tree language generated by the given HORSC? We first characterise the model checking problem by an intersection type system extended with a carefully restricted form of union types. We then present an algorithm for deciding the model checking problem, which is based on the notion of traversals induced by the fully abstract game semantics of these schemes, but presented as a goal-directed construction of derivations in the intersection and union type system. We view HORSC model checking as a suitable backend engine for an approach to verifying functional programs. We have implemented the algorithm in a tool called TravMC , and demonstrated its effectiveness on a test suite of programs, including abstract models of functional programs obtained via an abstraction-refinement procedure from pattern-matching recursion schemes.

The theory of classical realizability was introduced by Krivine [Kri09] in the middle of the 90’s in order to analyze the computational contents of classical proofs, following the connection between classical reasoning and control operators discovered by Griffin [Gri90]. More than an extension of the theory of intuitionistic realizability, classical realizability is a complete reformulation of the very principles of realizability based on a combination [OS08, Miq10] of Kleene’s realizability [Kle45] with Friedman’s A-translation [Fri78].

We use the technique of “classical realizability” to build new models of ZF + DC in which R is not well ordered. This gives new relative consistency results, which are not obtainable by forcing. This gives also a new method to get programs from proofs of arithmetical formulas with dependent choice.

Learning based realizability for HA + EM1 and 1-Backtracking games: Soundness and completeness