NEXP-completeness and Universal Hardness Results for Justification Logic

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 hard for the Sigma 2 p class of the second level of the polynomial hierarchy - 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.Comment: Shorter version has been accepted for publication by CSR 201

Uncertain Reasoning in Justification Logic

This thesis studies the combination of two well known formal systems for knowledge representation: probabilistic logic and justification logic. Our aim is to design a formal framework that allows the analysis of epistemic situations with incomplete information. In order to achieve this we introduce two probabilistic justification logics, which are defined by adding probability operators to the minimal justification logic J. We prove soundness and completeness theorems for our logics and establish decidability procedures. Both our logics rely on an infinitary rule so that strong completeness can be achieved. One of the most interesting mathematical results for our logics is the fact that adding only one iteration of the probability operator to the justification logic J does not increase the computational complexity of the logic

Interactions and Complexity in Multi-Agent Justification Logic

Justification cation Logic is the logic which introduces justifications to the epistemic setting. In contrast to Modal Logic, when an agent believes (or knows) a certain claim, in Justification Logic we assume the agent believes the claim because of a certain justification. Therefore, instead of having formulas that represent the belief of a claim (ex. □ø or Kø), we have formulas that represent that the belief of a claim follows from a provided justification (ex. t : ø). The original Justification Logic is LP, the Logic of Proofs, and was introduced by Artemov in 1995 as a link between Intuitionistic Truth and Gödel proofs in Peano Arithmetic. The complexity of Justification Logic was first studied by Kuznets in 2000. He demonstrated that for many justification logics, their derivability problem (and thus their satisfiability problem) is in the second level of the Polynomial Hierarchy, a result which was shown to be tight and which was later extended to more justification logics. In fact, so far, given reasonable assumptions, every single-agent justification logic whose complexity has been settled has its satisfiability problem in the second level of the Polynomial Hierarchy. This result is nicely contrasted to Modal Logic, as the corresponding modal systems are PSPACE-complete. We investigate the complexity of Justification Logic and Modal Logic when we allow multiple agents whose justifications affect each other -- by including some combination of the axioms t :iø → t :jø and t :iø → !t :j t :iø (modal cases: □iø→ □jø). We discover complexity jumps new for the field of Justification Logic: in addition to logics with their satisfiability problem in the second level of the polynomial hierarchy (as is the usual case until now), there are logics that have PSPACE-complete, EXP-complete and even NEXP-complete satisfiability problems. It is notable how the behavior of several of these justification logics mirrors the behavior of the corresponding multi-modal logics when we restrict modal formulas (in negation normal form) to use no diamonds. Thus we first study the complexity of such diamond-free modal logics and then we deduce complexity properties for the justification logic systems. On the other hand, it is similarly notable how certain lower complexity bounds -- the NEXP-hardness bound and the general Σp2-hardness bound we present -- are more dependent on the behavior of the justifications. The complexity results are interesting for Modal Logic as well, as we give hardness results that hold even for the diamond-free, 1-variable fragments of these multi-modal logics and then we determine the complexity of these logics in a general case

Lower complexity bounds in justification logic

Justification Logic studies epistemic and provability phenomena by introducing justifications/proofs into the language in the form of justification terms. Pure justification logics serve as counterparts of traditional modal epistemic logics, and hybrid logics combine epistemic modalities with justification terms. The computational complexity of pure justification logics is typically lower than that of the corresponding modal logics. Moreover, the so-called reflected fragments, which still contain complete information about the respective justification logics, are known to be in~NP for a wide range of justification logics, pure and hybrid alike. This paper shows that, under reasonable additional restrictions, these reflected fragments are NP-complete, thereby proving a matching lower bound. The proof method is then extended to provide a uniform proof that the corresponding full pure justification logics are $\Pi^p_2$-hard, reproving and generalizing an earlier result by Milnikel