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Proofs and Countermodels in Non-Classical Logics

Sara Negri
2014 Logica Universalis  
The method is a synthesis of a generation of calculi with internalized relational semantics, a Tait-Schütte-Takeuti style completeness proof, and procedures to finitize the countermodel construction.  ...  A method is presented that makes it possible to establish completeness in a direct way: For any given sequent either a proof in the given logical system or a countermodel in the corresponding frame class  ...  The methodology of generation of complete analytic countermodel-producing calculi will be also illustrated for the following (families of) logics and extensions: Intermediate logics and their modal companions  ... 
doi:10.1007/s11787-014-0097-1 fatcat:yzxgrtllgrga5j3nsc4qm5hgoy

MOIN: A Nested Sequent Theorem Prover for Intuitionistic Modal Logics (System Description) [chapter]

Marianna Girlando, Lutz Straßburger
2020 Lecture Notes in Computer Science  
We have implemented single-conclusion systems (Gentzen-style) and multi-conclusion systems (Maehara-style) for all logics in the intuitionistic modal IS5-cube.  ...  To our knowledge this is the first automated theorem prover for intuitionistic modal logics. For wider usability, we also implemented classical normal modal logics in the S5cube.  ...  The proof theory of intuitionistic modal logics has seen many advances in the last decade, and we felt the need for a tool that makes these advances accessible to a wider audience.  ... 
doi:10.1007/978-3-030-51054-1_25 fatcat:4fi3kmvkljgzja7jokg7s4nrym

Countermodels from Sequent Calculi in Multi-Modal Logics

Deepak Garg, Valerio Genovese, Sara Negri
2012 2012 27th Annual IEEE Symposium on Logic in Computer Science  
Based on backwards search in labeled sequent calculi, the procedure employs a novel termination condition and countermodel construction.  ...  A novel countermodel-producing decision procedure that applies to several multi-modal logics, both intuitionistic and classical, is presented.  ...  Negri [16] and Viganó [24] provide sound, complete and terminating labeled sequent calculi for uni-modal classical logics K, T, K4 and S4.  ... 
doi:10.1109/lics.2012.42 dblp:conf/lics/GargGN12 fatcat:h2etn5ezibey3buidgqcicrcwi

Hypersequent calculi for non-normal modal and deontic logics: Countermodels and optimal complexity [article]

Tiziano Dalmonte, Björn Lellmann, Nicola Olivetti, Elaine Pimentel
2020 arXiv   pre-print
Then we define a terminating root-first proof search strategy based on the hypersequent calculi and show that it is optimal for coNP-complete logics.  ...  Moreover, we obtain that from every saturated leaf of a failed proof it is possible to define a countermodel of the root hypersequent in the bi-neighbourhood semantics, and for regular logics also in the  ...  In order to facilitate the countermodel construction for underivable hypersequents in the next section, we show termination for all considered logics, even those containing axiom C: Proof.  ... 
arXiv:2006.05436v1 fatcat:aq5snfalffcvtigqtcgj57wklq

Forward proof-search and Countermodel Construction in Intuitionistic Propositional Logic

Camillo Fiorentini, Mauro Ferrari
2020 Italian Conference on Theoretical Computer Science  
A significant investigation is presented in [4, 5] , where focused calculi and polarization of formulas are exploited to reduce the search space in forward proof-search for Intuitionistic Logic.  ...  The method has been popularized by Degtyarev and Voronkov [7], who provide the general recipe to design forward calculi, with applications to Classical Predicate Logic and some non-classical logics.  ...  As a future work we plan to investigate the applicability of our method to other logics, in particular to modal logics (see [13] for a preliminary work) and intermediate logics such as Gödel-Dummett  ... 
dblp:conf/ictcs/Fiorentini020 fatcat:giuwlagjm5ealicgzsh6httxgi

A theorem prover and countermodel constructor for provability logic in HOL Light [article]

Marco Maggesi, Cosimo Perini Brogi
2022 arXiv   pre-print
We introduce our implementation in HOL Light of a prototype of a general theorem prover for normal modal logics. In the present work, we start by considering G\"odel-L\"ob provability logic (GL).  ...  We discuss our code for the formal proof of modal completeness, and the design of our proof search algorithm. Furthermore some examples of both interactive and automated use of the latter are shown.  ...  Most relevantly, proving termination of the proof search in these calculi provides a neat proof of decidability for the corresponding logics, which allows the direct construction of a countermodel for  ... 
arXiv:2205.03659v1 fatcat:5ik6aviqlfgafb5rwkh75bhsu4

Hypersequent Calculi for Lewis' Conditional Logics with Uniformity and Reflexivity [chapter]

Marianna Girlando, Björn Lellmann, Nicola Olivetti, Gian Luca Pozzato
2017 Lecture Notes in Computer Science  
These calculi provide both a decision procedure for the respective logics and constructive countermodel extraction from a failed proof search attempt.  ...  We present the first internal calculi for Lewis' conditional logics characterized by uniformity and reflexivity, including non-standard internal hypersequent calculi for a number of extensions of the logic  ...  Connections to modal logic The constructed hypersequent calculi provide purely syntactical proofs of results from [13] connecting the conditional logics to, e.g., modal logic S5.  ... 
doi:10.1007/978-3-319-66902-1_8 fatcat:ren4zmqudjf2zkrxxdd3pa4oue

Labeled Sequent Calculi for Access Control Logics: Countermodels, Saturation and Abduction

Valerio Genovese, Deepak Garg, Daniele Rispoli
2012 2012 IEEE 25th Computer Security Foundations Symposium  
We show that Kripke semantics of modal logic, manifest in the syntactic proof formalism of labeled sequent calculi, can be used to solve three central problems in access control: Generating evidence for  ...  At the core of our work is a single, non-trivial, countermodel producing decision procedure for a specific access control logic.  ...  earlier work on uni-modal tableaux calculi [18] .  ... 
doi:10.1109/csf.2012.11 dblp:conf/csfw/GenoveseGR12 fatcat:ww3roswbpzdfdhihsnb7bhiqg4

Proof Theory for Intuitionistic Strong Löb Logic [article]

Iris van der Giessen, Rosalie Iemhoff
2020 arXiv   pre-print
This paper introduces two sequent calculi for intuitionistic strong Löb logic iSL_: a terminating sequent calculus G4iSL_ based on the terminating sequent calculus G4ip for intuitionistic propositional  ...  Here a constructive proof of this fact is obtained by using a countermodel construction based on a variant of G4iSL_.  ...  We thank Tadeusz Litak and Albert Visser for several enjoyable conversations about Intuitionistic Strong Löb Logic and related matters.  ... 
arXiv:2011.10383v1 fatcat:zqt23b5mive3je3ukq5a3zv6dy

Non-Normal Modal Logics: Bi-Neighbourhood Semantics and Its Labelled Calculi

Tiziano Dalmonte, Nicola Olivetti, Sara Negri
2018 Advances in Modal Logic  
On the basis of this alternative semantics, labelled sequent calculi are developed for all the logics of the classical cube.  ...  It is shown that this semantics characterises the cube of non-normal modal logics and that there is a mutual correspondence between models in the standard and in the bi-neighbourhood semantics.  ...  We show that these calculi are terminating and thereby provide a decision procedure for the respective logics, and we prove semantic completeness of the calculi with respect to bi-neighbourhood semantics  ... 
dblp:conf/aiml/DalmonteON18 fatcat:w3tu2l2f6jav7kpcmmwvdjcb6e

Page 1693 of Mathematical Reviews Vol. , Issue 87d [page]

1987 Mathematical Reviews  
A criterion for admissibility of rules in the modal system S4 and intuitionistic logic. (Russian) Algebra i Logika 23 (1984), no. 5, 546-572, 600.  ...  Topo-Boolean and pseudo-Boolean algebras arose as an algebraic tool for the study of modal and intuitionistic logics. (The term “topo-Boolean algebra” was proposed by A. V.  ... 

Forward Countermodel Construction in Modal Logic K

Mauro Ferrari, Camillo Fiorentini, Guido Fiorino
2018 Italian Conference on Computational Logic  
Here we apply this method to derive the unprovability of a formula in the modal logic K. To this aim, we design a forward calculus to check the K-satisfiability of a set of modal formulas.  ...  The inverse method is a saturation based theorem proving technique; it relies on a forward proof-search strategy and can be applied to cut-free calculi enjoying the subformula property.  ...  In this paper we only focus on K; we plan to extend the techniques here introduced to other modal logics.  ... 
dblp:conf/cilc/FerrariFF18 fatcat:c6pwrq53uvhmtpaq3nqukoapam

A forward internal calculus for model generation in S4

Camillo Fiorentini, Mauro Ferrari
2021 Journal of Logic and Computation  
Our calculus directly supports model extraction and is designed so to implement a forward proof-search strategy that can be understood as a top-down construction of a model.  ...  Starting from these motivations, in [12, 15, 16] , we addressed the problem of countermodel construction for IPL and for the modal logic K by devising refutation calculi directly supporting model extraction  ...  Also in this case it is not trivial to adapt the strategies defined for IPL and the modal logic K to the case of S4.  ... 
doi:10.1093/logcom/exab014 fatcat:j2cslmzcjbbkleonvsibbqvdfu

Deciding intuitionistic propositional logic via translation into classical logic [chapter]

Daniel S. Korn, Christoph Kreitz
1997 Lecture Notes in Computer Science  
The translation is based on the constructive description of a nite countermodel for any intuitionistic non-theorem.  ...  A formula A is called forced (otherwise unforced) at w | denoted as w A (and w 6 A respectively) | i one of the following conditions holds: 1. A is atomic, and A 2 V(w ) for any w 2.  ...  as its source code is short and cryptic.  ... 
doi:10.1007/3-540-63104-6_15 fatcat:dod3yikixjhk5diqrcro3euqbq

A cut-free sequent system for Grzegorczyk logic, with an application to the Gödel–McKinsey–Tarski embedding

Roy Dyckhoff, Sara Negri
2013 Journal of Logic and Computation  
The completeness proof, being constructive, yields a constructive decision procedure, i.e. both a proof procedure for derivable sequents and a countermodel construction for underivable sequents.  ...  It is well-known that intuitionistic propositional logic Int may be faithfully embedded not just into the modal logic S4 but also into the provability logics GL and Grz of Gödel-Löb and Grzegorczyk, and  ...  The following rules (in the context of labelled sequent calculi such as G3K for the modal logic K, as presented in [26] , in which and are multisets of labelled formulae x : A, with possibly also some  ... 
doi:10.1093/logcom/ext036 fatcat:wxy26qn65reubasipfkzy2lmq4
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