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Step-indexed kripke models over recursive worlds

Lars Birkedal, Bernhard Reus, Jan Schwinghammer, Kristian Støvring, Jacob Thamsborg, Hongseok Yang
2011 Proceedings of the 38th annual ACM SIGPLAN-SIGACT symposium on Principles of programming languages - POPL '11  
A recent line of work has identified a common solution to some of these challenges: Kripke models over worlds that are recursively defined in a category of metric spaces.  ...  Moreover, the method provides a high-level understanding of the essence of recent approaches based on step indexing.  ...  at is the same as . 2 Observe that the natural model of types here is a Kripke model over a recursively-defined set of worlds.  ... 
doi:10.1145/1926385.1926401 dblp:conf/popl/BirkedalRSSTY11 fatcat:lmkx6nkutjhtfbpw6xcwksp5ai

Step-indexed kripke models over recursive worlds

Lars Birkedal, Bernhard Reus, Jan Schwinghammer, Kristian Støvring, Jacob Thamsborg, Hongseok Yang
2011 SIGPLAN notices  
A recent line of work has identified a common solution to some of these challenges: Kripke models over worlds that are recursively defined in a category of metric spaces.  ...  Moreover, the method provides a high-level understanding of the essence of recent approaches based on step indexing.  ...  at is the same as . 2 Observe that the natural model of types here is a Kripke model over a recursively-defined set of worlds.  ... 
doi:10.1145/1925844.1926401 fatcat:2j33cl3unnbx5pxjrrwdqn5nci

OUP accepted manuscript

2019 Journal of Logic and Computation  
This raises the question of whether every Kripke complete, recursively enumerable predicate modal logic can be characterized by an elementary class of Kripke frames.  ...  formula, is recursively enumerable.  ...  ., w = w k , for some k > 1 (notice that the case w = w * is identical to k being the maximal index of a world in M), we define the sought model as follows.  ... 
doi:10.1093/logcom/exz028 fatcat:starp6pbvfhjvemvfmt36m7qya

Sequent Calculus in the Topos of Trees [article]

Ranald Clouston, Rajeev Goré
2015 arXiv   pre-print
Birkedal et al modelled this modality via the internal logic of the topos of trees.  ...  Nakano's "later" modality, inspired by Gödel-Löb provability logic, has been applied in type systems and program logics to capture guarded recursion.  ...  We regard the positive integers as worlds and functions x : 1 → Ω as truth values over these worlds, by considering x true at j iff x j = j.  ... 
arXiv:1501.03293v2 fatcat:gjav5oyn3vbbnbo43xzx46o6yi

Sequent Calculus in the Topos of Trees [chapter]

Ranald Clouston, Rajeev Goré
2015 Lecture Notes in Computer Science  
As a logical connective, ⊲ was married to separation logic in [18] , then to higher-order separation logic in [2] , and to step-indexed logical relations for reasoning about programming languages with  ...  Nakano's "later" modality, inspired by Gödel-Löb provability logic, has been applied in type systems and program logics to capture guarded recursion.  ...  We regard the positive integers as worlds and functions x : 1 → Ω as truth values over these worlds, by considering x true at j iff x j = j.  ... 
doi:10.1007/978-3-662-46678-0_9 fatcat:hxn3oyxyf5dvzij4abuepbdrke

Logic of subtyping

Pavel Naumov
2006 Theoretical Computer Science  
We also consider logics S and S that incorporate into S mutually recursive types over arbitrary and well-founded universes correspondingly.  ...  We introduce new modal logical calculi that describe subtyping properties of Cartesian product and disjoint union type constructors as well as mutually recursive types defined using those type constructors  ...  Lemma 23 . 23 For any positive recursion , every -sound Kripke model can be extended to a -sound grounded Kripke model preserving relation on the existing worlds.  ... 
doi:10.1016/j.tcs.2006.03.017 fatcat:whfco7nxdfctlch4npdqs66qrq

A very modal model of a modern, major, general type system

Andrew W. Appel, Paul-André Melliès, Christopher D. Richards, Jérôme Vouillon
2007 SIGPLAN notices  
We present a model of recursive and impredicatively quantified types with mutable references.  ...  It is also simple, and reduces mainly to defining a Kripke semantics of the Gödel-Löb logic of provability.  ...  program safety, models induction over future steps, models type preservation, !  ... 
doi:10.1145/1190215.1190235 fatcat:xipuoxv6crg3vaa2l6prx4ahai

Bisimulation-invariant PTIME and higher-dimensional μ-calculus

Martin Otto
1999 Theoretical Computer Science  
Consider the class of all those properties of worlds in finite Kripke structures (or of states in finite transition systems), that are l recognizable in polynomial time, and l closed under bisimulation  ...  It is also shown that higher-dimensional p-calculus is undecidable for satisfiability in finite structures, and even Et-hard over general structures.  ...  For the standard semantics of ML, ML, and L,, one usually deals with Kripke structures ('%,a) in which one element is designated, one also speaks of model-world pairs: -a E A the distinguished world of  ... 
doi:10.1016/s0304-3975(98)00314-4 fatcat:klwctsikb5fhpm45567db6vn6y

Undecidability and non-axiomatizability of modal many-valued logics [article]

Amanda Vidal
2022 arXiv   pre-print
This family includes the global modal logics arising from Kripke frames evaluated over the standard Lukasiewicz and Product algebras.  ...  In this work we study the decidability of a class of global modal logics arising from Kripke frames evaluated over certain residuated lattices, known in the literature as modal many-valued logics.  ...  The logics studied in this work are defined over valued Kripke models: Kripke frames W, R enriched with a world-wise evaluation 1 of the formulas e : W × F m → A into some algebra A, and where , ✸ are  ... 
arXiv:2101.08767v2 fatcat:effdf4mpqfehlmrqcctjbutv2q

Logics of Kripke meta-models

S. Babenyshev, V. Rybakov
2009 Logic Journal of the IGPL  
In particular we show that refinement of logics preserves the effective finite model property and decidability for a wide class of constituent logics. 2 Logics of Kripke meta-models of frames from K indexed  ...  by clusters of F, where each F C is based on the cluster C ⊆ F (i.e., Thus, in a refinement, the Kripke structure of frames from the indexed family X is imposed on the clusters of the frame F, in addition  ...  Kripke frame of the Kripke signature R I , I ∩J =∅. The refinement of F by the indexed family of frames {F C DEFINITION 2.2.  ... 
doi:10.1093/jigpal/jzp047 fatcat:q7zj246tb5blzfrsadzsaogmju

On Kripke-style Semantics for the Provability Logic of Gödel's Proof Predicate with Quantifiers on Proofs

Rostislav Yavorskiy
2005 Journal of Logic and Computation  
Kripke-style semantics is suggested for the provability logic with quantifiers on proofs corresponding to the standard Gödel proof predicate. It is proved that the set of valid formulas is decidable.  ...  Suppose, for any σ ⊂ σ there is an algorithm to transform any sentence S over σ (in the language LPqC ) into a propositional modal formula µ(S) such that for any world w of any Kripke model one has w S  ...  Lemma 5.10 (Inductive step for σ A ) Suppose that for all σ B such that σ B ⊂ σ A the set of true formulas over σ B is decidable in every world w.  ... 
doi:10.1093/logcom/exi037 fatcat:dd22qypiknhudfc4cr2avhadvy

Model Checking Social Network Models

Raúl Pardo, Gerardo Schneider
2017 Electronic Proceedings in Theoretical Computer Science  
Kripke models are also sound w.r.t.  ...  SNMs; iii) We give a satisfaction-preserving encoding of SNMs into canonical Kripke models, and we also characterise which Kripke models may be translated into SNMs; iv) We show that, for SNMs, the model  ...  PPF has been specifically designed for privacy policies for real social networks, and that is why the language PPL and the underlying logic K BL are interpreted over SNMs and not over Kripke models (possible-worlds  ... 
doi:10.4204/eptcs.256.17 fatcat:wjtkkcpq2neyvlkcc7wlhnzozq

Model Checking BDI Logics over Finite-state Worlds

Salvatore La Torre, Gennaro Parlato
2019 International Conference of the Italian Association for Artificial Intelligence  
In this paper, we introduce the model-checking problem of Ctl * bdi over finite-state models that are described by tuples of Kripke structures (one for each world) and where the BDI relations are captured  ...  We consider the logic Ctl * bdi that augments the branching-time logic Ctl * with the BDI modalities and adopt the possible-world semantics by Rao and Georgeff.  ...  For the model-checking problem we consider (1) a finite number of possible worlds each corresponding to the unrolling of a Kripke structure, and (2) each bdi relation defined by a finite automaton over  ... 
dblp:conf/aiia/TorreP19 fatcat:agryolx2rvgbnnakv4nbgf6eny

Multimodal Separation Logic for Reasoning About Operational Semantics

Robert Dockins, Andrew W. Appel, Aquinas Hobor
2008 Electronical Notes in Theoretical Computer Science  
The final component of a traditional Kripke model is a binary relation on worlds, R, called the accessibility relation.  ...  Kripke semantics in particular are a standard way to to model about these logics [19, 17] .  ... 
doi:10.1016/j.entcs.2008.10.002 fatcat:vi5buoqlavcojmj3kp6toxq3vu

The impact of higher-order state and control effects on local relational reasoning

Derek Dreyer, Georg Neis, Lars Birkedal
2010 Proceedings of the 15th ACM SIGPLAN international conference on Functional programming - ICFP '10  
In this paper, we marry the aspirations of the semantic cube to the powerful proof method of step-indexed Kripke logical relations.  ...  Central to our story is the use of state transition systems to model the way in which properties of local state evolve over time.  ...  Worlds and Auxiliary Definitions HOS As described in Section 3, we employ a step-indexed Kripke logical relation, which is a kind of possible-worlds model.  ... 
doi:10.1145/1863543.1863566 dblp:conf/icfp/DreyerNB10 fatcat:6tx5tvqr7fc4xcmsd4ojwhiksi
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