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

Complexity Jumps In Multiagent Justification Logic Under Interacting Justifications

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 two-agent 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

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

Two Variable vs. Linear Temporal Logic in Model Checking and Games

Model checking linear-time properties expressed in first-order logic has non-elementary complexity, and thus various restricted logical languages are employed. In this paper we consider two such restricted specification logics, linear temporal logic (LTL) and two-variable first-order logic (FO2). LTL is more expressive but FO2 can be more succinct, and hence it is not clear which should be easier to verify. We take a comprehensive look at the issue, giving a comparison of verification problems for FO2, LTL, and various sublogics thereof across a wide range of models. In particular, we look at unary temporal logic (UTL), a subset of LTL that is expressively equivalent to FO2; we also consider the stutter-free fragment of FO2, obtained by omitting the successor relation, and the expressively equivalent fragment of UTL, obtained by omitting the next and previous connectives. We give three logic-to-automata translations which can be used to give upper bounds for FO2 and UTL and various sublogics. We apply these to get new bounds for both non-deterministic systems (hierarchical and recursive state machines, games) and for probabilistic systems (Markov chains, recursive Markov chains, and Markov decision processes). We couple these with matching lower-bound arguments. Next, we look at combining FO2 verification techniques with those for LTL. We present here a language that subsumes both FO2 and LTL, and inherits the model checking properties of both languages. Our results give both a unified approach to understanding the behaviour of FO2 and LTL, along with a nearly comprehensive picture of the complexity of verification for these logics and their sublogics.Comment: 37 pages, to be published in Logical Methods in Computer Science journal, includes material presented in Concur 2011 and QEST 2012 extended abstract

On lower bounds for circuit complexity and algorithms for satisfiability

This work is devoted to explore the novel method of proving circuit lower bounds for the class NEXP by Ryan Williams. Williams is able to show two circuit lower bounds: A conditional lower bound which says that NEXP does not have polynomial size circuits if there exists better-than-trivial algorithms for CIRCUIT SAT and an inconditional lower bound which says that NEXP does not have polynomial size circuits of the class ACC^0. We put special emphasis on the first result by exposing, in as much as of a self-contained manner as possible, all the results from complexity theory that Williams use in his proof. In particular, the focus is put in an efficient reduction from non-deterministic computations to satisfiability of Boolean formulas. The second result is also studied, although not as thoroughly, and some pointers with regards to the relationship of Williams' method and the known complexity theory barriers are given