An Application of the Feferman-Vaught Theorem to Automata and Logics for<br> Words over an Infinite Alphabet

Feferman-Vaught Decompositions for Prefix Classes of First Order Logic

The Feferman-Vaught theorem provides a way of evaluating a first order sentence $φ$ on a disjoint union of structures by producing a decomposition of $φ$ into sentences which can be evaluated on the individual structures and the results of these evaluations combined using a propositional formula. This decomposition can in general be non-elementarily larger than $φ$. We show that for first order sentences in prenex normal form with a fixed number of quantifier alternations, such a decomposition, further with the same number of quantifier alternations, can be obtained in time elementary in the s…

A game comonadic account of Courcelle and Feferman-Vaught-Mostowski theorems

Game comonads, introduced by Abramsky, Dawar and Wang, and developed by Abramsky and Shah, give a categorical semantics for model comparison games. We present an axiomatic account of Feferman-Vaught-Mostowski (FVM) composition theorems within the game comonad framework, parameterized by the model comparison game. In a uniform way, we produce compositionality results for the logic in question, and its positive existential and counting quantifier variants. Secondly, we extend game comonads to the second order setting, specifically in the case of Monadic Second Order (MSO) logic. We then generali…

Connection Matrices and the Definability of Graph Parameters

In this paper we extend and prove in detail the Finite Rank Theorem for connection matrices of graph parameters definable in Monadic Second Order Logic with counting (CMSOL) from B. Godlin, T. Kotek and J.A. Makowsky (2008) and J.A. Makowsky (2009). We demonstrate its vast applicability in simplifying known and new non-definability results of graph properties and finding new non-definability results for graph parameters. We also prove a Feferman-Vaught Theorem for the logic CFOL, First Order Logic with the modular counting quantifiers.

逻辑中复合方法的范畴论阐述

A categorical account of composition methods in logic

本文扩展博弈余单子语义，建立有限模型理论复合方法的统一范畴论框架，参数化导出 FVM 型定理并恢复、改进多项经典结果。

逻辑中组合方法的范畴论阐述（扩展版）

A categorical account of composition methods in logic (extended version)

本文利用博弈余单子语义建立有限模型理论中组合方法的统一范畴论框架，参数化地得到涵盖多种逻辑片段的 FVM 型定理，并恢复和改进经典结果。