# Elementary-base cirquent calculus II: Choice quantifiers

## TL;DR

This paper introduces CL17, a first-order extension of cirquent calculus with choice quantifiers, which is sound, complete, and decidable, advancing the logical framework for elementary games and computability logic.

## Contribution

It constructs the first-order version CL17 of the propositional calculus CL16, incorporating choice quantifiers and establishing its soundness, completeness, and decidability.

## Key findings

- CL17 is sound and complete for elementary games.
- CL17 is decidable, unlike classical predicate calculus.
- It extends the previous propositional calculus with first-order features.

## Abstract

Cirquent calculus is a novel proof theory permitting component-sharing between logical expressions. Using it, the predecessor article "Elementary-base cirquent calculus I: Parallel and choice connectives" built the sound and complete axiomatization CL16 of a propositional fragment of computability logic (see http://www.csc.villanova.edu/~japaridz/CL/ ). The atoms of the language of CL16 represent elementary, i.e., moveless, games, and the logical vocabulary consists of negation, parallel connectives and choice connectives. The present paper constructs the first-order version CL17 of CL16, also enjoying soundness and completeness. The language of CL17 augments that of CL18 by including choice quantifiers. Unlike classical predicate calculus, CL17 turns out to be decidable.