# Algorithmic correspondence and canonicity for non-distributive logics

@article{Conradie2019AlgorithmicCA, title={Algorithmic correspondence and canonicity for non-distributive logics}, author={Willem Conradie and Alessandra Palmigiano}, journal={Ann. Pure Appl. Log.}, year={2019}, volume={170}, pages={923-974} }

## 62 Citations

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

- MathematicsWoLLIC
- 2017

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).

### Constructive Canonicity of Inductive Inequalities

- MathematicsLog. Methods Comput. Sci.
- 2020

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.

### Unified Correspondence as a Proof-Theoretic Tool

- Computer ScienceJ. Log. Comput.
- 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.

### Unified inverse correspondence for DLE-Logics

- Computer ScienceArXiv
- 2022

This work presents an algorithm that makes use of ALBA’s rules and algebraic language to invert its steps in the DLE setting; therefore effectively computing an inductive formula starting from its first order correspondent.

### Algorithmic correspondence and canonicity for possibility semantics

- Computer ScienceJ. Log. Comput.
- 2021

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).

### Categories of Residuated Lattices

- Mathematics
- 2018

We present dual variants of two algebraic constructions of certain classes of residuated lattices: The Galatos-Raftery construction of Sugihara monoids and their bounded expansions, and the…

### Canonical extensions and ultraproducts of polarities

- MathematicsAlgebra universalis
- 2018

Jónsson and Tarski’s notion of the perfect extension of a Boolean algebra with operators has evolved into an extensive theory of canonical extensions of lattice-based algebras. After reviewing this…

### Canonical extensions and ultraproducts of polarities

- MathematicsAlgebra universalis
- 2018

Jónsson and Tarski’s notion of the perfect extension of a Boolean algebra with operators has evolved into an extensive theory of canonical extensions of lattice-based algebras. After reviewing this…

### Slanted Canonicity of Analytic Inductive Inequalities

- MathematicsACM Trans. Comput. Log.
- 2021

We prove an algebraic canonicity theorem for normal LE-logics of arbitrary signature, in a generalized setting in which the non-lattice connectives are interpreted as operations mapping tuples of…

### Sahlqvist theory for impossible worlds

- PhilosophyJ. Log. Comput.
- 2017

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

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- MathematicsArXiv
- 2015

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

- MathematicsWoLLIC
- 2017

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).

### Constructive Canonicity of Inductive Inequalities

- MathematicsLog. Methods Comput. Sci.
- 2020

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.

### Unified Correspondence as a Proof-Theoretic Tool

- Computer ScienceJ. Log. Comput.
- 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.

### Algorithmic correspondence and canonicity for possibility semantics

- Computer ScienceJ. Log. Comput.
- 2021

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).

### A General Algebraic Semantics for Sentential Logics

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The purpose of this monograph is to develop a very general approach to the algebra- ization of sententiallogics, to show its results on a number of particular logics, and to relate it to other…

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This book considers both the algebraic and logical perspective within a common framework, and shows how proof theoretical methods like cut elimination are preferable for small logics/theories, but semantic tools like Rabin's theorem work better for big ones.

### Canonicity results for mu-calculi: an algorithmic approach

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- 2017

The algorithm introduced is closely related to the algorithms ALBA and mu-ALBA studied by Conradie, Palmigiano, et al, and is based on a calculus of rewrite rules, the soundness of which rests upon the way in which algebras embed into their canonical extensions and the order-theoretic properties of the latter.

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