Abstract This paper presents a formal theory of Krivine’s classical realizability interpretation for first-order Peano arithmetic ($\mathsf{PA}$). To formulate the theory as an extension of $\mathsf{PA}$, we first modify Krivine’s original definition to the form of number realizability, similar to Kleene’s intuitionistic realizability for Heyting arithmetic. By axiomatizing our realizability with additional predicate symbols, we obtain a first-order theory of compositional realizability ($\mathsf{CR}$), which can formally realize every theorem of $\mathsf{PA}$. Although $\mathsf{CR}$ itself is conservative over $\mathsf{PA}$, adding a type of reflection principle that roughly states that ‘realizability implies truth’ results in $\mathsf{CR}$ being essentially equivalent to the Tarskian compositional truth theory ($\mathsf{CT}$) of typed compositional truth, which is known to be proof-theoretically stronger than $\mathsf{PA}$. We also prove that a weaker reflection principle, which preserves the distinction between realizability and truth, is sufficient for $\mathsf{CR}$ to achieve the same strength as $\mathsf{CT}$. Furthermore, we formulate transfinite iterations of $\mathsf{CR}$ and its variants, and then we determine their proof-theoretic strength.

Video games are utilized in some studies for evaluation experiments and demonstrations. However, because commercial video games cannot be edited, optimizing them for specific experiments is challenging. Thus, researchers may have to develop their own video games individually, which requires effort and impedes the comparison of their study with other similar studies. To overcome these limitations, in this study, we propose a user-friendly "Open Video Game Library" that can be used by researchers for experiments. We conducted a survey of previous studies and designed video games to meet researcher needs. The library features characteristics such as GUI-based parameter editing, play log output, open-source availability, and support for operation on multiple devices, with the aim of becoming the de facto standard for video games used for research purposes.