Expressibility and inexpressibility in propositional team logics

TL;DR

It is shown that in propositional logic and in several important cases, a team theoretical atom can be expressed in terms of atoms of lower arity, and the `price' of such a reduction of arity is estimated, i.e. how much more complicated the new expression is.

Abstract

We develop dimension theoretic methods for propositional team based logics. Such quantitative methods were defined for team based first-order logic in a recent paper by Hella, Luosto and the third author and were used to obtain strong hierarchy results in the first-order logic context. We show that in propositional logic and in several important cases, a team theoretical atom can be expressed in terms of atoms of lower arity. We estimate the `price' of such a reduction of arity, i.e. how much more complicated the new expression is. Our estimates involve as parameters the arity of the atoms involved, as well as the number of times the atom occurs in a formula. We also consider new variants of atoms and propositional operations, inspired by our work. We believe that our quantitative analysis leads to a deeper understanding of the scope and limits of propositional team based logic.

Figures (3)

• Figure 1: The double-edged arrows represent maximal intervals, the white circles critical sets, and the filled-in circles dual critical sets. Each dimension is concerned with the smallest number of some particular convex sets that cover a family; the upper dimension $\mathrm{D}$ with critical sets, the dual upper dimension $\mathrm{D}^d$ with dual critical sets, and the cylindrical dimension $\mathrm{D}^c$ with maximal intervals.
• Figure 2: Let $\mathcal{X}=\{a,b,c,d\}$ be a base set. The four families are illustrated with their maximal intervals shown by a double-edged arrow. The family $\mathcal{A}:=\{\emptyset,\{a\},\{b\}, \{c\}, \{a,b\}, \{b,c\}\}$ is downward closed with $\mathrm{D}(\mathcal{A})=2$ and $\mathrm{D}^d({A})=1$. The family $\mathcal{B}:=\{\emptyset,\{c\}, \{a,b\},\{a,b,c\}, \{a,b,c,d\}\}$ is union closed with $\mathrm{D}(\mathcal{B})=3=\mathrm{D}^d(\mathcal{B})$. The family $\mathcal{C}:=\{\emptyset,\{a,b\},\{b,c\},\{a,b,c\}, \{a,b,d\},\{b,c,d\}, \{a,b,c,d\}\}$ is quasi upward closed with $\mathrm{D}(\mathcal{C})=2$ and $\mathrm{D}^d(\mathcal{C})=3$. Lastly, the family $\mathcal{D}$ is from \ref{['example1']}. What is its dual upper dimension $\mathrm{D}^d(\mathcal{D})$?
• Figure 3: The dark gray teams in the picture belong to $\lVert=\!\!(p_1;q)\rVert^{p_1q}$, where $10(p_1)=1$ and $10(q)=0$ etc. There are four maximal teams, hence $\mathrm{D}(\lVert=\!\!(p_1;q)\rVert^{p_1q})=4$.

Theorems & Definitions (61)

• Definition 2.1: Hella_Luosto_Vaananen_2024
• Definition 2.2: Hella_Luosto_Vaananen_2024
• Lemma 2.3
• proof
• Example 2.4
• Proposition 2.5
• proof
• Lemma 2.6
• proof
• Proposition 2.7