# Local Basic Strategy Logic

## Valentin Goranko

Basic Strategy Logic (**BSL**) introduced in [1] is a minimal system of strategy logic of the type studied in [2]. **BSL** builds on
a fixed set of agents \(\mathsf{Agt}\) and on some (usually temporalized) language for expressing agents goals, such as **LTL**, that defines a set of *goal formulae* \(\Gamma\) which are evaluated on plays in concurrent game models.

The language of **BSL** extends \(\Gamma\) with standard Boolean connectives and associates with each agent \(\mathsf{a} \in \mathsf{Agt}\) a *strategy variable*, denoted by \(\mathsf{x}_{\mathsf{a}}\). These variables range over strategies for the respective agents and can be quantified over within the formulae of **BSL**.
Thus, **BSL** can be used for formalising the reasoning about strategic abilities of agents (players) and coalitions in concurrent multi-player games. It is shown in [1] that **BSL** is sufficiently expressive to capture many naturally defined recently studied operators and logics for strategic abilities.

In the present work I study its local version **LBSL**, which only involves in the language of \(\Gamma\) the Nexttime temporal operator, referring to the immediate outcomes states from playing single-round concurrent games at the states of the model. I explore and characterise the expressiveness of **LBSL**, study its validities, present an axiomatic system for them, and discuss the problems of its completeness and of the decidability of **LBSL**.

## Bibliography

- V. Goranko Logics for Strategic Reasoning of Socially Interacting Rational Agents: An Overview and Perspectives.
**Logics**, vol. 1(1), 2023. Online publication: https://www.mdpi.com/2813-0405/1/1/3. - Fabio Mogavero, Aniello Murano, and Moshe Y. Vardi. Reasoning about strategies. In
**Proc. of {FSTTCS} 2010**, volume 8 of**LIPIcs**, pages 133–144.