This paper presents a sound and complete five-sided sequent calculus for first-order weak Kleene valuations which permits not only elegant representations of four logics definable on first-order we

https://www.tandfonline.com/doi/pdf/10.1080/11663081.2020.1782593

Proof Theory

Structural proof theory for first-order weak Kleene logics

We provide a constructive direct semantic proof of the completeness of the cut-free part of the hypersequent calculus HIF for the standard first-order Godel logic (thereby proving both completeness of the calculus for its standard semantics, and the admissibility of the cut rule in the full calculus). The results also apply to derivations from assumptions (or ‘non-logical axioms’), showing in particular that when the set of assumptions is closed under substitutions, then cuts can be confined to formulas occurring in the assumptions. The methods and results are then extended to handle the (Baaz) Delta connective as well.

https://www.cs.tau.ac.il/~orilahav/papers/godel-jlc2.pdf

A semantic proof of strong cut-admissibility for first-order Gödel logic

We investigate a new paradigm in the context of learning in the limit: learning correction grammars for classes of r.e. languages. Knowing a language may feature a representation of the target language in terms of two sets of rules (two grammars). The second grammar is used to make corrections to the first grammar. Such a pair of grammars can be seen as a single description of (or grammar for) the language. We call such grammars correction grammars. Correction grammars capture the observable fact that people do correct their linguistic utterances during their usual linguistic activities.

We show that learning correction grammars for classes of r.e. languages in the TxtEx-model (i.e., converging to a single correct correction grammar in the limit) is sometimes more powerful than learning ordinary grammars even in the TxtBc-model (where the learner is allowed to converge to infinitely many syntactically distinct but correct conjectures in the limit). For each n =≥ 0, there is a similar learning advantage, where we compare learning correction grammars that make n + 1 corrections to those that make n corrections.

The concept of a correction grammar can be extended into the constructive transfinite, using the idea of counting-down from notations for transfinite constructive ordinals. For u a notation in Kleene's general system (O, <o) of ordinal notations, we introduce the concept of an u-correction grammar, where u is used to bound the number of corrections that the grammar is allowed to make.We prove a general hierarchy result: if u and v are notations for constructive ordinals such that u <o v, then there are classes of r.e. languages that can be TxtEx-learned by conjecturing v-correction grammars but not by conjecturing u-correction grammars.

Surprisingly, we show that -- above "ω-many" corrections -- it is not possible to strengthen the hierarchy: TxtEx-learning u-correction grammars of classes of r.e. languages, where u is a notation in O for any ordinal, can be simulated by TxtBc-learning w-correction grammars, where w is any notation for the smallest infinite ordinal ω.

/pdf/learning-correction-grammars-5fqg3zhgvy.pdf

Learning correction grammars

The minimal extension of intuitionistic propositional language is characterized, where propositional quantifiers are eliminable w.r.t. Kripke frames of type ω.

Quantifier Elimination for Quantified Propositional Logics on Kripke Frames of Type ω

We survey a selection of results about majorization hierarchies. The main focus is on classical and recent results about the comparison between the slow and fast growing hierarchies.

Proof Theory

Citations

Structural proof theory for first-order weak Kleene logics

A semantic proof of strong cut-admissibility for first-order Gödel logic

Learning correction grammars

Quantifier Elimination for Quantified Propositional Logics on Kripke Frames of Type ω

Slow versus fast growing

Related Papers (5)

Untersuchungen über das logische Schließen. II

Introduction to Metamathematics

The collected papers of Gerhard Gentzen

Basic proof theory

Linear logic