# Provenance Analysis for Logic and Games

## TL;DR

This paper introduces a novel provenance analysis method for logics with negation, using dual-indeterminate semirings, enhancing understanding of logical computations and game strategies.

## Contribution

It develops a new provenance framework for logics with negation, extending existing methods to include negative information and strategies in game analysis.

## Key findings

- Proposes dual-indeterminate semirings for provenance analysis.
- Provides detailed insights into strategies in logic games.
- Extends provenance analysis to logics with negation.

## Abstract

A model checking computation checks whether a given logical sentence is true in a given finite structure. Provenance analysis abstracts from such a computation mathematical information on how the result depends on the atomic data that describe the structure. In database theory, provenance analysis by interpretations in commutative semirings has been rather succesful for positive query languages (such a unions of conjunctive queries, positive relational algebra, or datalog). However, it did not really offer an adequate treatment of negation or missing information. Here we propose a new approach for the provenance analysis of logics with negation, such as first-order logic and fixed-point logics. It is closely related to a provenance analysis of the associated model-checking games, and based on new semirings of dual-indeterminate polynomials or dual-indeterminate formal power series. These are obtained by taking quotients of traditional provenance semirings by congruences that are generated by products of positive and negative provenance tokens. Beyond the use for model-checking problems in logics, provenance analysis of games is of independent interest. Provenance values in games provide detailed information about the number and properties of the strategies of the players, far beyond the question whether or not a player has a winning strategy from a given position.