Let F be a field with charF 6= 2. We show that there are two groups of order 32, respectively 64, such that a field F with char F 6= 2 is nonrigid if and only if at least one of the two groups is realizable as a Galois group over F . The realizability of those groups turns out to be equivalent to the realizability of certain quotients (of order 16, respectively 32). Using known results on conne...

We settle the equivalence between the problem of hitting a polyhedral set by the orbit of a linear map and the intersection of a regular language and a language of permutations of binary words (PB-realizability problem). The decidability of the both problems is presently unknown and the first one is a straightforward generalization of the famous Skolem problem and the nonnegativity problem in t...

We investigate fundamental properties of threevalued semantics for abstract dialectical frameworks (ADFs). In particular, we deal with realizability, i.e., the question whether there exists an ADF that has a given set of interpretations as its semantics. We provide necessary and sufficient conditions that hold for a set of three-valued interpretations whenever there is an ADF realizing it under...

We explain how recent developments in the fields of realizability models for linear logic [Sei14e] – or geometry of interaction – and implicit computational complexity [AS14, AS15] can lead to a new approach of implicit computational complexity. This semantic-based approach should apply uniformly to various computational paradigms, and enable the use of new mathematical methods and tools to att...

Constructive mathematics is mathematics without the use of the principle of the excluded middle. There exists a wide array of models of constructive logic. One particular interpretation of constructive mathematics is the realizability interpretation. It is utilized as a metamathematical tool in order to derive admissible rules of deduction for systems of constructive logic or to demonstrate the...

This article investigates Krivine’s realizability interpretation of classical second-order arithmetic and its recent extension handling countable choice. We will start by presenting a two-step interpretation which first eliminates classical logic via a negative translation and then applies standard realizability interpretation. We then argue that a slight variant of Krivine’s interpretation cor...

We prove that interactive learning based classical realizability (introduced by Aschieri and Berardi for first order arithmetic [1]) is sound with respect to Coquand game semantics. In particular, any realizer of an implication-and-negation-free arithmetical formula embodies a winning recursive strategy for the 1-Backtracking version of Tarski games. We also give examples of realizer and winnin...

In scenarios where a set of independent business partners engage in complex conversations, interaction models are a means to specify the allowed interaction behavior from a global perspective. Atomic interactions serve as basic building blocks and behavioral dependencies are defined between them. The notion of realizability centers around the question whether there exist a set of roles that col...

The theory of classical realizability is a framework in which we can develop the proof-program correspondence. Using this framework, we show how to transform into programs the proofs in classical analysis with dependent choice and the existence of a well ordering of the real line. The principal tools are: The notion of realizability algebra, which is a three-sorted variant of the well known com...

The field of implicit complexity has recently produced several bounded-complexity programming languages. This kind of language allows to implement exactly the functions belonging to a certain complexity class. We here present a realizability semantics for a higher-order functional language based on a fragment of linear logic called LAL which characterizes the complexity class PTIME. This langua...