Justification Logic Research Articles

Decidability for some justification logics with negative introspection

AbstractJustification logics are modal logics that include justifications for the agent's knowledge. So far, there are no decidability results available for justification logics with negative introspection. In this paper, we develop a novel model construction for such logics and show that justification logics with negative introspection are decidable for finite constant specifications.

Realization for justification logics via nested sequents: Modularity through embedding

Justification logics are refinements of modal logics, where justification terms replace modalities. Modal and justification logics are connected via the so-called realization theorems. We develop a general constructive method of proving the realization of a modal logic in an appropriate justification logic by means of cut-free modal nested sequent systems. We prove a constructive realization theorem that uniformly connects every normal modal logic formed from the axioms d, t, b, 4, and 5 with one of its justification counterparts. We then generalize the notion of embedding introduced by Fitting for justification logics, which enables us to extend our realization theorem to all natural justification counterparts. As a result, we obtain a modular realization theorem that provides several justification counterparts based on various axiomatizations of a modal logic. We also prove that these justification counterparts realize the same modal logic if and only if they belong to the same equivalence class induced by our embedding relation, thereby demonstrating that the embedding provides the right level of granularity among justification logics.

The Ontology of Justifications in the Logical Setting

Justification Logic provides an axiomatic description of justifications and delegates the question of their nature to semantics. In this note, we address the conceptual issue of the logical type of justifications: we argue that justifications in the logical setting are naturally interpreted as sets of formulas which leads to a class of epistemic models that we call modular models. We show that Fitting models for Justification Logic naturally encode modular models and can be regarded as convenient pre-models of the former.

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 Π2p-hard, reproving and generalizing an earlier result by Milnikel.

Tableaux and hypersequents for justification logics

Justification logic is a new generation of epistemic logics which along with the traditional modal knowledge/belief operators also consider justification assertions ‘t is a justification for F.’ In this paper, we introduce a prefixed tableau system for one of the major logics of this kind S4LPN, which combines the Logic of Proofs (LP) and epistemic logic S4 with an explicit negative introspection principle ¬t:F→□¬t:F. We show that the prefixed tableau system for S4LPN is sound and complete with respect to Fitting-style semantics. We also introduce a hypersequent calculus HS4LPN and show that HS4LPN is complete. We establish this fact by using a translation from the prefixed tableau system to the hypersequent calculus. This completeness result gives us a semantic proof of cut-admissibility for S4LPN. We conclude the paper by discussing a subsystems of S4LPN, namely S4LP.

Prefixed tableaus and nested sequents

Nested sequent systems for modal logics are a relatively recent development, within the general area known as deep reasoning. The idea of deep reasoning is to create systems within which one operates at lower levels in formulas than just those involving the main connective or operator. Prefixed tableaus go back to 1972, and are modal tableau systems with extra machinery to represent accessibility in a purely syntactic way. We show that modal nested sequents and prefixed modal tableaus are notational variants of each other, roughly in the same way that Gentzen sequent calculi and tableaus are notational variants. This immediately gives rise to new modal nested sequent systems which may be of independent interest. We discuss some of these, including those for some justification logics that include standard modal operators.

Justification Logic as a foundation for certifying mobile computation

We explore an intuitionistic fragment of Artëmov’s Justification Logic as a type system for a programming language for mobile units. Such units consist of both a code and a certificate component. Our language, the Certifying Mobile Calculus, caters for code and certificate development in a unified theory. In the same way that mobile code is constructed out of code components and extant type systems track local resource usage to ensure the mobile nature of these components, our system additionally ensures correct certificate construction out of certificate components. We present proofs of type safety and strong normalization for a run-time system based on an abstract machine.

Multi-agent Justification Logic: communication and evidence elimination

This paper presents a logic combining Dynamic Epistemic Logic, a framework for reasoning about multi-agent communication, with a new multi-agent version of Justification Logic, a framework for reasoning about evidence and justification. This novel combination incorporates a new kind of multi-agent evidence elimination that cleanly meshes with the multi-agent communications from Dynamic Epistemic Logic, resulting in a system for reasoning about multi-agent communication and evidence elimination for groups of interacting rational agents.

Justifications for common knowledge

Justification logics are epistemic logics that explicitly include justifications for the agents' knowledge. We develop a multi-agent justification logic with evidence terms for individual agents as well as for common knowledge. We define a Kripke-style semantics that is similar to Fitting's semantics for the Logic of Proofs LP. We show the soundness, completeness, and finite model property of our multi-agent justification logic with respect to this Kripke-style semantics. We demonstrate that our logic is a conservative extension of Yavorskaya's minimal bimodal explicit evidence logic, which is a two-agent version of LP. We discuss the relationship of our logic to the multi-agent modal logic S4 with common knowledge. Finally, we give a brief analysis of the coordinated attack problem in the newly developed language of our logic.

Knowledge, Time, and the Problem of Logical Omniscience

It is well known that Modal Epistemic logic (MEL) suffers from the problem of logical omniscience. In this paper, we will argue that in order to solve the problem, the temporal dimension of knowledge has to be revealed and following this analysis, we present a general epistemic framework, timed Modal Epistemic Logic (tMEL), modified from MEL, such that the time at which a formula is known by an agent based on his reasoning procedure is explicitly stated. With the help of the additional temporal devices, we are able to determine what is actually known by the agent within a reasonable time of reasoning. The discussions will focus on tS4, the tMEL counterpart of S4, but the method can be uniformly generalized to the study of other tMEL logics. Both the semantics and axiomatic proof systems will be provided, accompanied by soundness and completeness results, and other technical features of tMEL are also examined. This work originates from the study of Justification Logic, which shapes many aspects of this paper, and is also a direct response to the request to utilize the use of awareness functions such that time can be added to the picture. A generalized awareness function is employed in the semantics to trace when a formula is deduced.

Dynamic logics of knowledge and access

A recurring issue in any formal model representing agents’ (changing) informational attitudes is how to account for the fact that the agents are limited in their access to the available inference steps, possible observations and available messages. This may be because the agents are not logically omniscient and so do not have unlimited reasoning ability. But it can also be because the agents are following a predefined protocol that explicitly limits statements available for observation and/or communication. Within the broad literature on epistemic logic, there are a variety of accounts that make precise a notion of an agent’s “limited access” (for example, Awareness Logics, Justification Logics, and Inference Logics). This paper interprets the agents’ access set of formulas as a constraint on the agents’ information gathering process limiting which formulas can be observed.

Justification logics and hybrid logics

Hybrid logics internalize their own semantics. Members of the newer family of justification logics internalize their own proof methodology. It is an appealing goal to combine these two ideas into a single system, and in this paper we make a start. We present a hybrid/justification version of the modal logic T. We give a semantics, a proof theory, and prove a completeness theorem. In addition, we prove a Realization Theorem, something that plays a central role for justification logics generally. Since justification logics are newer and less well known than hybrid logics, we sketch their background, and give pointers to their range of applicability. We conclude with suggestions for future research. Indeed, the main goal of this paper is to encourage others to continue the investigation begun here.

Public communication in justification logic

Justification Logic is a framework for reasoning about evidence and justification in multi-agent systems. Most accounts of Justification Logic are essentially static, in that the (justified) beliefs of agents are immutable. In this article, we add public communication, a dynamic operation of belief change studied in the area of Dynamic Epistemic Logic, to the language of Justification Logic. Introducing notions of bisimulation for the languages of Justification Logic with and without public communication, we catalogue the expressive relationships that exist between almost all of the well-known static fragments of Justification Logic and then determine whether the addition of public communication affects the various expressive relationships existing between these fragments.

Realizations and LP

LP can be seen as a logic of knowledge with justifications. See [S. Artemov, The logic of justification, The Review of Symbolic Logic 1 (4) (2008) 477–513] for a recent comprehensive survey of justification logics generally. Artemov’s Realization Theorem says justifications can be extracted from validities in the more conventional Hintikka-style logic of knowledge S4, in which they are not explicitly present. Justifications, however, are far from unique. There are many ways of realizing each theorem of S4 in the logic LP. If the machinery of justifications is to be applied to artificial intelligence, or better yet, to everyday reasoning, we will need to work with whatever justifications we may have at hand—one version may not be interchangeable with another, even though they realize the same S4 formula. In this paper we begin the process of providing tools for reasoning about justifications directly. The tools are somewhat complex, but in retrospect this should not be surprising. Among other things, we provide machinery for combining two realizations of the same formula, and for replacing subformulas by equivalent subformulas. (The second of these is actually weaker than just stated, but this is not the place for a detailed formulation.) The results are algorithmic in nature—semantics for LP plays no role. We apply our results to provide a new algorithmic proof of Artemov’s Realization Theorem itself. This paper is a much extended version of [M.C. Fitting, Realizations and LP, in: S. Artemov, A. Nerode (Eds.), Logical Foundations of Computer Science—New York ’07, in: Lecture Notes in Computer Science, vol. 4514, Springer-Verlag, 2007, pp. 212–223].

Self-Referential Justifications in Epistemic Logic

This paper is devoted to the study of self-referential proofs and/or justifications, i.e., valid proofs that prove statements about these same proofs. The goal is to investigate whether such self-referential justifications are present in the reasoning described by standard modal epistemic logics such as $\mathsf{S4}$. We argue that the modal language by itself is too coarse to capture this concept of self-referentiality and that the language of justification logic can serve as an adequate refinement. We consider well-known modal logics of knowledge/belief and show, using explicit justifications, that $\mathsf{S4}$, $\mathsf{D4}$, $\mathsf{K4}$, and $\mathsf{T}$with their respective justification counterparts $\mathsf{LP}$, $\mathsf{JD4}$, $\mathsf{J4}$, and $\mathsf{JT}$describe knowledge that is self-referential in some strong sense. We also demonstrate that self-referentiality can be avoided for $\mathsf{K}$and $\mathsf{D}$. In order to prove the former result, we develop a machinery of minimal evidence functions used to effectively build models for justification logics. We observe that the calculus used to construct the minimal functions axiomatizes the reflected fragments of justification logics. We also discuss difficulties that result from an introduction of negative introspection.

THE LOGIC OF JUSTIFICATION

We describe a general logical framework, Justification Logic, for reasoning about epistemic justification. Justification Logic is based on classical propositional logic augmented by justification assertions t: F that read t is a justification for F. Justification Logic absorbs basic principles originating from both mainstream epistemology and the mathematical theory of proofs. It contributes to the studies of the well-known Justified True Belief vs. Knowledge problem. We state a general Correspondence Theorem showing that behind each epistemic modal logic, there is a robust system of justifications. This renders a new, evidence-based foundation for epistemic logic. As a case study, we offer a resolution of the Goldman–Kripke ‘Red Barn’ paradox and analyze Russell’s ‘prime minister example’ in Justification Logic. Furthermore, we formalize the well-known Gettier example and reveal hidden assumptions and redundancies in Gettier’s reasoning.

Justification logics, logics of knowledge, and conservativity

Several justification logics have been created, starting with the logic LP, (Artemov, Bull Symbolic Logic 7(1):1---36, 2001). These can be thought of as explicit versions of modal logics, or of logics of knowledge or belief, in which the unanalyzed necessity (knowledge, belief) operator has been replaced with a family of explicit justification terms. We begin by sketching the basics of justification logics and their relations with modal logics. Then we move to new material. Modal logics come in various strengths. For their corresponding justification logics, differing strength is reflected in different vocabularies. What we show here is that for justification logics corresponding to modal logics extending T, various familiar extensions are actually conservative with respect to each other. Our method of proof is very simple, and general enough to handle several justification logics not directly corresponding to distinct modal logics. Our methods do not, however, allow us to prove comparable results for justification logics corresponding to modal logics that do not extend T. That is, we are able to handle explicit logics of knowledge, but not explicit logics of belief. This remains open.

The topology of justification

Justification Logic is a family of epistemic logical systems obtained from modal logics of knowledge by adding a new type of formula t:F, which is read t is a justification for F. The principal epistemic modal logic S4 includes Tarski’s well-known topological interpretation, according to which the modality 2X is read the Interior of X in a topological space (the topological equivalent of the ‘knowable part of X’). In this paper, we extend Tarski’s topological interpretation from S4 to Justification Logic systems with both modality and justification assertions. The topological semantics interprets t:X as a reachable subset of X (the topological equivalent of ‘test t confirms X’). We establish a number of soundness and completeness results with respect to Kripke topology and the real topology for S4-based systems of Justification Logic.

Interacting Explicit Evidence Systems

Logic of proofs $\mathsf{LP}$, introduced by S. Artemov, originally designed for describing properties of formal proofs, now became a basis for the theory of knowledge with justification (cf. S. Artemov, Evidence-based common knowledge, Technical report TR–2004018, CUNY Ph.D. Program in Computer Science, 2005). So far, in epistemic systems with justification the corresponding “evidence part”, even for multi-agent systems, consisted of a single explicit evidence logic. In this paper we introduce logics describing two interacting explicit evidence systems. We find an appropriate formalization of the intended semantics and prove the completeness of these logics with respect to both symbolic and arithmetical models. Also, we find the forgetful projections for the two-agent justification logics which are extensions of the bimodal logic $\mathsf{S4}^{2}$.

Feasible Operations on Proofs: The Logic of Proofs for Bounded Arithmetic

This paper presents a semantics for the logic of proofs $\mathsf{LP}$ in which all the operations on proofs are realized by feasibly computable functions. More precisely, we will show that the completeness of $\mathsf{LP}$ for the semantics of proofs of Peano Arithmetic extends to the semantics of proofs in Buss’ bounded arithmetic $\mathsf{S}^{1}_{2}$. In view of applications in epistemology of $\mathsf{LP}$ in particular and justification logics in general this result shows that explicit knowledge in the propositional framework can be made computationally feasible.