Abstract
In the present paper, we prove canonicity results for lattice-based fixed point logics in a constructive meta-theory. Specifically, we prove two types of canonicity results, depending on how the fixed-point binders are interpreted. These results smoothly unify the constructive canonicity results for inductive inequalities, proved in a general lattice setting, with the canonicity results for fixed point logics on a bi-intuitionistic base, proven in a non-constructive setting.
The research of the first author has been funded by the National Research Foundation of South Africa, Grant number 81309. The research of the third and fourth author has been funded by the NWO Vidi grant 016.138.314, the NWO Aspasia grant 015.008.054, and a Delft Technology Fellowship awarded in 2013.
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Notes
- 1.
Namely, the tame and proper canonicity, cf. Sect. 3.
- 2.
The same methodology can be used also to define Gentzen calculi, as is witnessed by [25] in the setting of strict implication logics.
- 3.
We say that \(g:\mathbb {A}^n\rightarrow \mathbb {A}\) is the right residual of \(f:\mathbb {A}^n\rightarrow \mathbb {A}\) in its i-th coordinate if for any \(a_1, \ldots , a_{n}, b\in \mathbb {A}\), \(f(a_1, \ldots , a_i, \ldots , a_{n})\le b\) iff \(a_i\le g(a_1, \ldots , b, \ldots , a_{n})\). We also say that f is the left residual of g in its j-th coordinate.
- 4.
We say that \(g:\mathbb {A}^n\rightarrow \mathbb {A}\) is the left Galois adjoint of \(f:\mathbb {A}^n\rightarrow \mathbb {A}\) in its i-th coordinate if for any \(a_1, \ldots , a_{n}, b\in \mathbb {A}\), \(f(a_1, \ldots , a_i, \ldots , a_{n})\le b\) iff \(g(a_1, \ldots , b, \ldots , a_{n})\le a_i\). We say that g is the right Galois adjoint of f in its i-th coordinate if for any \(a_1, \ldots , a_{n}, b\in \mathbb {A}\), \(b\le f(a_1, \ldots , a_i, \ldots , a_{n})\) iff \(a_i\le g(a_1, \ldots , b, \ldots , a_{n})\).
- 5.
Although that argument is given in the distributive setting, it can be easily verified that it does not make use of any specific feature of that setting.
- 6.
In other settings, nominals are interpreted as completely join-irreducible elements in \(\mathbb {A}^{\delta }\), while in the constructive setting, the constructive canonical extensions might not be perfect, therefore there might not be enough completely join-irreducible elements, therefore nominals are interpreted as closed elements instead in the constructive setting.
- 7.
Indeed, the U-shaped argument is the algorithmic version of the Sambin–Vaccaro canonicity proof [29]. In [26], this method has been unified with Jónsson’s canonicity proof [24]; in [12], it has been unified with constructive canonicity introduced in Ghilardi and Meloni [22]; in [16], the canonicity via pseudo-correspondence has been presented as an instance of this method.
- 8.
A term \(\varphi \) is positive (negative) in a variable p if in the signed generation tree \(+\varphi \) all p-nodes are signed \(+\) (−). An inequality \(\varphi \le \psi \) is positive (negative) in p if \(\varphi \) is negative (positive) in p and \(\psi \) is positive (negative) in p.
- 9.
The purpose of this restriction is to enforce preservation of non-empty joins by the term function \(\varphi '^{\mathbb {C}}\). The soundness of the rule is founded upon this and approximation of the argument \(\gamma \) as the join of all closed elements below it. In the non-constructive setting of [14] the same strategy is followed, except that the approximation is done by means of completely join-irreducibles. Since this can give rise to empty sets of approximants and hence empty joins, \(+\vee \) is excluded in the analogous approximation rule in [14], as the join does not preserve empty joins coordinate-wise. In the present setting, the set of closed approximants is never empty, and hence this restriction may be dropped. Similar considerations apply to \(-\wedge \).
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Conradie, W., Craig, A., Palmigiano, A., Zhao, Z. (2017). Constructive Canonicity for Lattice-Based Fixed Point Logics. In: Kennedy, J., de Queiroz, R. (eds) Logic, Language, Information, and Computation. WoLLIC 2017. Lecture Notes in Computer Science(), vol 10388. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-55386-2_7
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