Cut-Elimination for Provability Logic by Terminating Proof-Search: Formalised and Deconstructed Using Coq

Cut elimination and complexity bounds for intuitionistic epistemic logic

Journal of Logic and Computation ◽

10.1093/logcom/exaa012 ◽

2020 ◽

Vol 30 (1) ◽

pp. 281-294

Author(s):

Vladimir N Krupski

Keyword(s):

Epistemic Logic ◽

Sequent Calculus ◽

Formal System ◽

Point Of View ◽

Cut Elimination ◽

Polynomial Space ◽

Complexity Bounds ◽

Proof Search

Abstract The formal system of intuitionistic epistemic logic (IEL) was proposed by S. Artemov and T. Protopopescu. It provides the formal foundation for the study of knowledge from an intuitionistic point of view based on Brouwer–Heyting–Kolmogorov semantics of intuitionism. We construct a cut-free sequent calculus for IEL and establish that polynomial space is sufficient for the proof search in it. We prove that IEL is PSPACE-complete.

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Proof Search Tree and Cut Elimination

Pillars of Computer Science - Lecture Notes in Computer Science ◽

10.1007/978-3-540-78127-1_28 ◽

2008 ◽

pp. 521-536

Author(s):

Grigori Mints

Keyword(s):

Search Tree ◽

Cut Elimination ◽

Proof Search

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Syntactic Cut-Elimination and Backward Proof-Search for Tense Logic via Linear Nested Sequents

Lecture Notes in Computer Science - Automated Reasoning with Analytic Tableaux and Related Methods ◽

10.1007/978-3-030-29026-9_11 ◽

2019 ◽

pp. 185-202

Author(s):

Rajeev Goré ◽

Björn Lellmann

Keyword(s):

Tense Logic ◽

Cut Elimination ◽

Proof Search ◽

Nested Sequents

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Hypersequent calculi for non-normal modal and deontic logics: countermodels and optimal complexity

Journal of Logic and Computation ◽

10.1093/logcom/exaa072 ◽

2020 ◽

Author(s):

Tiziano Dalmonte ◽

Björn Lellmann ◽

Nicola Olivetti ◽

Elaine Pimentel

Keyword(s):

Search Strategy ◽

Cut Elimination ◽

Relational Semantics ◽

Optimal Complexity ◽

Proof Search ◽

Polynomial Size ◽

Hypersequent Calculi

Abstract We present some hypersequent calculi for all systems of the classical cube and their extensions with axioms ${T}$, ${P}$ and ${D}$ and for every $n \geq 1$, rule ${RD}_n^+$. The calculi are internal as they only employ the language of the logic, plus additional structural connectives. We show that the calculi are complete with respect to the corresponding axiomatization by a syntactic proof of cut elimination. Then, we define a terminating proof search strategy in the hypersequent calculi and show that it is optimal for coNP-complete logics. Moreover, we show that from every failed proof of a formula or hypersequent it is possible to directly extract a countermodel of it in the bi-neighbourhood semantics of polynomial size for coNP logics, and for regular logics also in the relational semantics. We finish the paper by giving a translation between hypersequent rule applications and derivations in a labelled system for the classical cube.

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Focused Proof-search in the Logic of Bunched Implications

Lecture Notes in Computer Science - Foundations of Software Science and Computation Structures ◽

10.1007/978-3-030-71995-1_13 ◽

2021 ◽

pp. 247-267

Author(s):

Alexander Gheorghiu ◽

Sonia Marin

Keyword(s):

Data Structure ◽

Proof Theory ◽

Sequent Calculus ◽

Operational Semantics ◽

Search Space ◽

Difficult Problem ◽

Cut Elimination ◽

Proof Search ◽

Nested Sequents

AbstractThe logic of Bunched Implications (BI) freely combines additive and multiplicative connectives, including implications; however, despite its well-studied proof theory, proof-search in BI has always been a difficult problem. The focusing principle is a restriction of the proof-search space that can capture various goal-directed proof-search procedures. In this paper we show that focused proof-search is complete for BI by first reformulating the traditional bunched sequent calculus using the simpler data-structure of nested sequents, following with a polarised and focused variant that we show is sound and complete via a cut-elimination argument. This establishes an operational semantics for focused proof-search in the logic of Bunched Implications.

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VALENTINI’S CUT-ELIMINATION FOR PROVABILITY LOGIC RESOLVED

The Review of Symbolic Logic ◽

10.1017/s1755020311000323 ◽

2012 ◽

Vol 5 (2) ◽

pp. 212-238 ◽

Cited By ~ 9

Author(s):

RAJEEV GORÉ ◽

REVANTHA RAMANAYAKE

Keyword(s):

Sequent Calculus ◽

Point Of View ◽

Cut Elimination ◽

Provability Logic ◽

Sequent Calculi ◽

Original Proof ◽

Underlying Assumption ◽

Induction Argument ◽

Formal Point ◽

Explicit Rule

Valentini (1983) has presented a proof of cut-elimination for provability logic GL for a sequent calculus using sequents built from sets as opposed to multisets, thus avoiding an explicit contraction rule. From a formal point of view, it is more syntactic and satisfying to explicitly identify the applications of the contraction rule that are ‘hidden’ in proofs of cut-elimination for such sequent calculi. There is often an underlying assumption that the move to a proof of cut-elimination for sequents built from multisets is straightforward. Recently, however, it has been claimed that Valentini’s arguments to eliminate cut do not terminate when applied to a multiset formulation of the calculus with an explicit rule of contraction. The claim has led to much confusion and various authors have sought new proofs of cut-elimination for GL in a multiset setting.Here we refute this claim by placing Valentini’s arguments in a formal setting and proving cut-elimination for sequents built from multisets. The use of sequents built from multisets enables us to accurately account for the interplay between the weakening and contraction rules. Furthermore, Valentini’s original proof relies on a novel induction parameter called “width” which is computed ‘globally’. It is difficult to verify the correctness of his induction argument based on “width.” In our formulation however, verification of the induction argument is straightforward. Finally, the multiset setting also introduces a new complication in the case of contractions above cut when the cut-formula is boxed. We deal with this using a new transformation based on Valentini’s original arguments.Finally, we discuss the possibility of adapting this cut-elimination procedure to other logics axiomatizable by formulae of a syntactically similar form to the GL axiom.

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Risperidone-treated Children and Adolescents with Behavioral Disorders: Do Drug Dose and Patients’ Gender and Age Relate to Drug and Metabolite Plasma Levels?

European Psychiatry ◽

10.1016/j.eurpsy.2017.01.322 ◽

2017 ◽

Vol 41 (S1) ◽

pp. S104-S104

Author(s):

D. Piacentino ◽

M. Grözinger ◽

A. Saria ◽

F. Scolati ◽

D. Arcangeli ◽

...

Keyword(s):

Regression Analysis ◽

Children And Adolescents ◽

Behavioral Disorders ◽

Plasma Levels ◽

Linear Models ◽

Parent Drug ◽

Drug Dose ◽

Cut Elimination ◽

Younger Patients ◽

Gender And Age

IntroductionBehavioral disorders, such as conduct disorder, influence choice of treatment and its outcome. Less is known about other variables that may have an influence.Objectives/AimsWe aimed to measure the parent drug and metabolite plasma levels in risperidone-treated children and adolescents with behavioral disorders and investigate the role of drug dose and patients’ gender and age.MethodsWe recruited 115 children/adolescents with DSM-5 behavioral disorders (females = 24; age range: 5–18 years) at the Departments of Psychiatry of the Hospitals of Bolzano, Italy, and Innsbruck, Austria. We measured risperidone and its metabolite 9-hydroxyrisperidone plasma levels and the parent drug-to-metabolite ratio in the plasma of all patients by using LC-MS/MS. A subsample of 15 patients had their risperidone doses measured daily. We compared risperidone and 9-hydroxyrisperidone plasma levels, as well as risperidone/9-hydroxyrisperidone ratio, in males vs. females and in younger (≤ 14 years) vs. older (15–18 years) patients by using Mann-Whitney U test. We fitted linear models for the variables “age” and “daily risperidone dose” by using log-transformation, regression analysis and applying the R2 statistic.ResultsFemales had significantly higher median 9-hydroxyrisperidone plasma levels (P = 0.000). Younger patients had a slightly lower median risperidone/9-hydroxyrisperidone ratio (P = 0.052). At the regression analysis, daily risperidone doses and metabolite, rather than parent drug–plasma levels were correlated (R2 = 0.35).ConclusionsGender is significantly associated with plasma levels, with females being slower metabolizers than males. Concerning age, younger patients seem to be rapid metabolizers, possibly due to a higher activity of CYP2D6. R2 suggests a clear-cut elimination of the metabolite.Disclosure of interestThe authors have not supplied their declaration of competing interest.

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