Modal Logics with Hard Diamond-Free Fragments

2015

Multi-Agent Systems

Self Cite

The Logic of Proofs, LP, and its successor, Justification Logic, is a refinement of the modal logic approach to epistemology in which proofs/justifications are taken into account. In 2000 Kuznets showed that satisfiability for LP is in the second level of the polynomial hierarchy, a result which has been successfully repeated for all other one-agent justification logics whose complexity is known. We introduce a family of multi-agent justification logics with interactions between the agents' justifications, by extending and generalizing the twoagent versions of the Logic of Proofs introduced by Yavorskaya in 2008. Known concepts and tools from the single-agent justification setting are adjusted for this multiple agent case. We present tableau rules and some preliminary complexity results. In several cases the satisfiability problem for these logics remains in the second level of the polynomial hierarchy, while for others it is PSPACE or EXP-hard. Furthermore, this problem becomes PSPACE-hard even for certain two-agent logics, while there are EXP-hard logics of three agents.

Section: 3mentioning

confidence: 99%

Tableaux and Complexity Bounds for a Multiagent Justification Logic with Interacting Justifications

2015

Multi-Agent Systems

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“…In the multi-agent setting we have already examined, this is still the case: many justification logics that so far demonstrate a complexity jump (to PSPACE-or EXP-completeness) have corresponding modal logics with an EXP-complete satisfiability problem (c.f. [20,9,1,2]). It is notable that, assuming EXP = NEXP, this is the first time we have a justification logic with a higher complexity than its corresponding modal logic, and, in fact, the reduction we use makes heavy use of the effects of the way a justification term is constructed.…”

Section: Introductionmentioning

confidence: 99%

“…This raised the question of whether the condition that the constant specification is schematically injective is a necessary one, which is answered in this paper. 2 We present a general lower bound, which applies to all logics from [4]. This includes all the single-agent logics whose complexity was studied in [12,13,8,1].…”

Section: Introductionmentioning

confidence: 99%

NEXP-Completeness and Universal Hardness Results for Justification Logic

Lecture Notes in Computer Science

2015

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Abstract. We provide a lower complexity bound for the satisfiability problem of a multi-agent justification logic, establishing that the general NEXP upper bound from our previous work is tight. We then use a simple modification of the corresponding reduction to prove that satisfiability for all multi-agent justification logics from there is Σ p 2 -hard -given certain reasonable conditions. Our methods improve on these required conditions for the same lower bound for the single-agent justification logics, proven by Buss and Kuznets in 2009, thus answering one of their open questions. IntroductionJustification Logic is the logic of justifications. Where in Modal Epistemic Logic we use formulas of the form φ to denote that φ is known (or believed, etc), in Justification Logic, we use t : φ to denote that φ is known for reason t (i.e. t is a justification for φ). Artemov introduced LP, the first justification logic, in 1995 [5], originally as a link between Intuitionistic Logic and Peano Arithmetic. Since then the field has expanded significantly, both in the variety of logical systems and in the fields it interacts with and is applied to (see [6,7] for an overview).In [21] Yavorskaya introduced two-agent LP with agents whose justifications may interact. We studied the complexity of a generalization in [3] and [4], discovering that unlike the case with single-agent Justification Logic as studied in [12,13,15,8,1], the complexity of satisfiability jumps to PSPACE-and EXPcompleteness when two or three agents are involved respectively, given appropriate interactions. In fact, the upper bound we proved was that all logics in this family have their satisfiability problem in NEXP -under reasonable assumptions.The NEXP upper complexity bound was not met with the introduction of a NEXP-hard logic in [4]. The main contribution of this paper is that we present a NEXP-hard justification logic from the family that was introduced in [4], thus establishing that the general upper bound is tight.In general, the complexity of the satisfiability problem for a justification logic tends to be lower than the complexity of its corresponding modal logic 1 (given 1 That is, the modal logic that is the result of substituting all justification terms in the axioms with boxes and adding the Necessitation rule.the usual complexity-theoretic assumptions). For example, while satisfiability for K, D, K4, D4, T, and S4 is PSPACE-complete, the complexity of the corresponding justification logics (J, JD, J4, JD4, JT, and LP respectively) is in the second level of the polynomial hierarchy (in Σ p 2 , specifically). In the multi-agent setting we have already examined, this is still the case: many justification logics that so far demonstrate a complexity jump (to PSPACE-or EXP-completeness) have corresponding modal logics with an EXP-complete satisfiability problem (c.f.[20, 9, 1, 2]). It is notable that, assuming EXP = NEXP, this is the first time we have a justification logic with a higher complexity than its corresponding modal logic, and, in fact, ...

“…Satisfiability for HML (and therefore for HML as well) is known to be in PSPACE [Vardi 1988]. That satisfiability for HML is PSPACE-hard results from the observation that HML with at least two actions can encode the 1-variable, diamond-free fragment of D ⊕ ⊆ D4, which is PSPACE-complete [Achilleos 2016]. The reduction can be found in Appendix B.2.…”

mentioning

confidence: 99%

Adventures in monitorability: from branching to linear time and back again

Aceto

Proc. ACM Program. Lang.

Francalanza

et al. 2019

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This paper establishes a comprehensive theory of runtime monitorability for Hennessy-Milner logic with recursion, a very expressive variant of the modal µ-calculus. It investigates the monitorability of that logic with a linear-time semantics and then compares the obtained results with ones that were previously presented in the literature for a branching-time setting. Our work establishes an expressiveness hierarchy of monitorable fragments of Hennessy-Milner logic with recursion in a linear-time setting and exactly identifies what kinds of guarantees can be given using runtime monitors for each fragment in the hierarchy. Each fragment is shown to be complete, in the sense that it can express all properties that can be monitored under the corresponding guarantees. The study is carried out using a principled approach to monitoring that connects the semantics of the logic and the operational semantics of monitors. The proposed framework supports the automatic, compositional synthesis of correct monitors from monitorable properties.