NEXP-Completeness and Universal Hardness Results for Justification Logic

Abstract

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, ...

The complexity of satisfiability in non-iterated and iterated probabilistic logics

“…Let α ∈ L, let v 1 , v 2 be two evaluations and assume that, for every basic formula β that appears in α v 1…”

The Complexity of Non-Iterated Probabilistic Justification Logic

“…The upper complexity bound we established is tight. By a result from [20] which was later strengthened in [8] and [1] we have that for a decidable, schematic and axiomatically appropriate constant specification CS the J CS -satisfiability problem is Σ p 2 -hard. For any α ∈ L J it is not difficult to prove that:…”

“…PJ is a probabilistic logic defined over the basic justification logic J. 1 The language of PJ contains formulas of the form P ≥s α meaning the probability of truthfulness of the justification formula α is at least s. So, in the logic PJ, statements like "evidence t serves as a justification for α with probability at least 30%" can be expressed. PJ does not allow iterations of the probability operator.…”

“…The results of [17,20,8,1] showed that, under some reasonable assumptions, the derivability problem for the justification logic J is Π p 2 -complete, i.e. it is complete in the second level of the polynomial hierarchy.…”

“…PJ is a probabilistic logic defined over the basic justification logic J. 1 The language of PJ contains formulas of the form P ≥s α meaning the probability of truthfulness of the justification formula α is at least s.…”