## Many-valued Coalgebraic Logic: From Boolean Algebras to Primal Varieties
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###### CALCO 2023, June 19
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###### *Alexander Kurz*, Wolfgang Poiger
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### Introduction
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- Coalgebraic Logic
- Many-Valued Logic
- Universal Algebra
- Modal Logic
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### Coalgebraic Logic
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| State Machines | Logics |
| :---: | :---: |
| state space $X\in \cal X$ | algebra of propositions $A\in \cal A$ |
| type of transitions $\sf T:\cal X\to X$ | type of modalities $\sf L:\cal A\to A$ |
| coalgebra $X\to \mathsf TX$ | algebra $\mathsf LA\to A$ |
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(Goldblatt, Scott, Johnstone, Smyth, Abramsky, Vickers, ...)
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### Stone Duality
*Two-valued* dualising object: $\mathsf PX={\sf Set}(X,2)$ and $\mathsf SA = {\sf BA}(A,2)$.
*Idea:* Starting from $\sf T$, define $\sf L = PTS$ on finitely generated free BAs and extend to BA via sifted colimits.
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### Abstract Coalgebraic Logics
From $\sf L = PTS$ one obtains the **one-step semantics** as a natural transformation
$$\delta:\sf LP\to PT$$
We call $$(\sf L,\delta)$$
an *abstract coalgebraic logic*. This gives a "semantics functor"
$$\sf Coalg(T) \to Alg(L)$$
Invariance under bisimulation, expressiveness, completeness can be established for abstract coalgebric logics.
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### Example
Let $A$ be a Boolean algebra.
Define $LA$ as the free $\sf BA$ over the meet-semi lattice $A$.
It is well-known that $LP\cong P\cal P$ on finite sets.
Explicitely: $LA$ is the free $\sf BA$
- generated by $\Box a$, $a\in A$
- modulo $\Box\top=\top$ and $\Box(a\wedge a')=\Box a \wedge \Box a'$.
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### Presenting Functors
**Theorem (K-Rosicky 2012):** A functor $L$ on a variety has a presentation by operations and equations iff it is determined by its action on finitely generated free algebras.
... iff it is the left-Kan extension of its restriction to finitely generated free algebras
... iff $L$ preserves filtered colimits
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### Concrete Coalgebraic Logics
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### Many-Valued Coalgebraic Logic
$\sf BA$ is the variety generated by $2$.
$\mathcal V = \mathbb {HSP}(D)$ is generated by a *finite lattice $D$ of truth degrees*.
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### Universal Algebra
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### Primal Algebras (Foster 1953)
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### Semi-Primal Algebras (Foster 1967)
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Originally, we wanted to write this paper for semi-primal algebras, continuing *K-P-Teheux (2023) [New perspectives on semi-primal varieties](https://arxiv.org/pdf/2301.13406.pdf)* but it turned out that the interesting question already arises in the primal case.
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For every variety $\cal A$ generated by a semi-primal algebra $\bf L$ one has the following diagram.
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### Hu's Theorem (Hu 1969)
In the primal case, the previous diagram collapses:
From a logical point of view, adding constants for each truth value is sufficient for the collapse.
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### Lifting Abstract Coalgebraic Logics
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$\cal A$ is a variety generated by a finite primal algebra.
$(\frak P, \frak S)$ is the equivalence of Hu's theorem, lifting $(\sf P,S)$ to $(\sf P',S')$.
$\sf L' = \frak P \sf L\frak S$ is the abstract lifting of $\sf L$.
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### Lifting Concrete Coalgebraic Logics
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$D$ is a finite primal algebra with a lattice reduct.
The order on $D$ is term-definable:
We can consider $\tau_d$ as a modal operator:
\begin{gather}
\tau_d(x\wedge y)=\tau_d(x)\wedge \tau_d(y)\\
\tau_d1 = 1
\end{gather}
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Compare this with
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This example is a special case of part (b) of the theorem. We will see later that neighbourhood frames give rise to an example that falls under (a) but not under (b).
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As a corollary we obtain a generalization of the example:
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### Neighborhood Frames
Instance of Theorem 15 (a):
To present the lifting $\sf L'$ of $\sf L$ we need a modal operator $\Box_d$ for each $d\in D\setminus\{0\}$, satisfying
where $T_1(d)=1$ if $d=1$ and $T_1(d)=0$ if $d\not=1$.
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### Conclusion
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Using an equivalence of categories $\mathbb {HSP}(2) \simeq\mathbb {HSP}(D)$ we investigated how to lift a 2-valued modal logic to a $D$-valued modal logic in a principled way.
Future Work:
- From a universal algebra point of view: generalizing from primal to semi-primal (and beyond) algebras.
- From a coalgebraic point of view: generalizing to many-valued semantics via enriching over a quantale of truth values (or distances).
- Combining both approaches using eniched universal algebra.
- Approximation techniques for reasoning with infinitely many truth-values.
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