# Canonicity results for mu-calculi: an algorithmic approach

@article{Conradie2017CanonicityRF, title={Canonicity results for mu-calculi: an algorithmic approach}, author={Willem Conradie and A. P. K. Craig}, journal={J. Log. Comput.}, year={2017}, volume={27}, pages={705-748} }

We investigate the canonicity of inequalities of the intuitionistic mu-calculus. The notion of canonicity in the presence of fixed point operators is not entirely straightforward. In the algebraic setting of canonical extensions we examine both the usual notion of canonicity and what we will call tame canonicity. This latter concept has previously been investigated for the classical mu-calculus by Bezhanishvili and Hodkinson. Our approach is in the spirit of Sahlqvist theory. That is, we…

## 28 Citations

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### Canonicity and Relativized Canonicity via Pseudo-Correspondence: an Application of ALBA

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This work defines a suitable enhancement of the algorithm ALBA, which is used to prove the canonicity of certain syntactically defined classes of DLE-inequalities (called the meta-inductive inequalities), relative to the structures in which the formulas asserting the additivity of some given terms are valid.

### Constructive Canonicity for Lattice-Based Fixed Point Logics

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This result simultaneously generalizes Conradie and Craig's canonicity for $\mu$-inequalities based on a bi-intuitionistic bi-modal language, and ConradIE and Palmigiano's constructive canonicityFor inductive inequalities (restricted to normal lattice expansions to keep the page limit).

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It is proved the canonicity of inductive inequalities in a constructive meta-theory, for classes of logics algebraically captured by varieties of normal and regular lattice expansions, based on an application of the tools of unified correspondence theory.

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The present paper proves the soundness of the algorithm with respect to both full possibility frames and filter-descriptive possibility frames, and uses the algorithm to give an alternative proof to the one in the work by Holliday (2016, Possibility frames and forcing for modal logic).

### Algorithmic Correspondence and Canonicity for Possibility Semantics (Abstract)

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A uniform and modular theory which subsumes the above results and extends them to logics with a non-classical propositional base has emerged, and has been dubbed unified correspondence.

### Unified Correspondence as a Proof-Theoretic Tool

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- 2018

The present paper aims at establishing formal connections between correspondence phenomena, well known from the area of modal logic, and the theory of display calculi, originated by Belnap, and applies unified correspondence theory, with its tools and insights, to extend Kracht's results and prove his claims in the setting of DLE-logics.

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Unified correspondence theory is extended to Kripke frames with impossible worlds and their associated regular modal logics and it is shown that additivity and multiplicativity turn out to be key to extend Jonsson’s original proof of canonicity to the full Sahlqvist class of certain regular distributives naturally generalizing distributive modal logic.

## References

SHOWING 1-10 OF 35 REFERENCES

### Algorithmic correspondence for intuitionistic modal mu-calculus

- MathematicsTheor. Comput. Sci.
- 2015

### Elementary canonical formulae: extending Sahlqvist's theorem

- MathematicsAnn. Pure Appl. Log.
- 2006

### Semantics for a Class of Intuitionistic Modal Calculi

- Philosophy
- 1980

A general criterion is proposed which enables us to define a rather large number of intuitionistic modal calculi and the idea that lies behind this criterion is that of presenting a uniform rule by means of which the authors can find the ‘intuitionistic analogues’ for some of the most usual classical modal systems.

### Algorithmic correspondence and canonicity for distributive modal logic

- MathematicsAnn. Pure Appl. Log.
- 2012

### Algebraic Canonicity in Non-Classical Logics

- Philosophy
- 2013

This thesis is a study of the notion of canonicity (as is understood e.g. in modal logic) from an algebraic viewpoint. The main conceptual contribution of this thesis is a better understanding of the…

### Minimal predicates, fixed-points, and definability

- Computer Science, MathematicsJournal of Symbolic Logic
- 2005

The main technical result is a preservation theorem showing PIA-conditions to be expressively complete for all those first-order formulas that are preserved under a natural model-theoretic operation of ‘predicate intersection’.

### Modal mu-calculi

- PhilosophyHandbook of Modal Logic
- 2007

In this chapter, least and greatest solutions to recursive modal equations were represented using the fixed point quantifiers μZ and υZ and the connectives to modal logic are added, thereby providing a very rich temporal logic.