We define an epistemic logic or logic of knowledge, PL, and a formalism to undertake privacy-centric reasoning in security protocols, over a Dolev-Yao model. We are able to automatically verify all the privacy requirements that are commonplace in security-protocol verification (i.e., strong secrecy, anonymity, various types of unlinkablity including weak unlinkability), as well as privacy notions that are less studied (i.e., privacy regarding lists' membership). Our methodology does not vary with the property: it is uniform no matter the kind of privacy requirement specified and/or verified. We operate in the setting of a bounded number of protocol-sessions. We also implement Phoebe – a proof-of-concept model checker for this methodology. We use Phoebe to check all the aforementioned properties, and we also show-case it on the “benchmark” anonymity and unlinkability requirements of several well-known protocols.

In the symbolic verification of cryptographic protocols, a central problem is deciding whether a protocol admits an execution which leaks a designated secret to the malicious intruder. In [1], it is shown that, when considering finitely many sessions, this “insecurity problem” is NP-complete. Central to their proof strategy is the observation that any execution of a protocol can be simulated by one where the intruder only communicates terms of bounded size. However, when we consider models where, in addition to terms, one can also communicate logical statements about terms, the analysis of the insecurity problem becomes tricky when both these inference systems are considered together. In this paper we consider the insecurity problem for protocols with logical statements that include equality on terms and existential quantification. Witnesses for existential quantifiers may be unbounded, and obtaining small witness terms while maintaining equality proofs complicates the analysis considerably. We extend techniques from [1] to show that this problem is also in NP.

First order modal logic (𝖥𝖮𝖬𝖫) is built by extending First Order Logic (𝖥𝖮) with modal operators. A typical formula is of the form \(\forall x \exists y \Box P(x,y)\) . Not only is 𝖥𝖮𝖬𝖫 undecidable, even simple fragments like that of restriction to unary predicate symbols, guarded fragment and two variable fragment, which are all decidable for 𝖥𝖮 become undecidable for 𝖥𝖮𝖬𝖫. In this paper we study Term Modal logic (𝖳𝖬𝖫) which allows modal operators to be indexed by terms. A typical formula is of the form \(\forall x \exists y~\Box _x P(x,y)\) . There is a close correspondence between 𝖳𝖬𝖫 and 𝖥𝖮𝖬𝖫 and we explore this relationship in detail in the paper. In contrast to 𝖥𝖮𝖬𝖫, we show that the two variable fragment (without constants, equality) of 𝖳𝖬𝖫 is decidable. Further, we prove that adding a single constant makes the two variable fragment of 𝖳𝖬𝖫 undecidable. On the other hand, when equality is added to the logic, it loses the finite model property.

Bundled products are often offered as good deals to customers. When we bundle quantifiers and modalities together (as in $\exists x \Box$, $\Diamond \forall x$ etc.) in first-order modal logic (FOML), we get new logical operators whose combinations produce interesting fragments of FOML without any restriction on the arity of predicates, the number of variables, or the modal scope. It is well-known that finding decidable fragments of FOML is hard, so we may ask: do bundled fragments that exploit the distinct expressivity of FOML constitute good deals in balancing the expressivity and complexity? There are a few positive earlier results on some particular fragments. In this paper, we try to fully map the terrain of bundled fragments of FOML in (un)decidability, and in the cases without a definite answer yet, we show that they lack the finite model property. Moreover, whether the logics are interpreted over constant domains (across states/worlds) or increasing domains presents another layer of complexity. We also present the \textit{loosely bundled fragment}, which generalizes the bundles and yet retain decidability (over increasing domain models).

Abstract We study procedures for the derivability problem of fragments of intuitionistic logic. Intuitionistic logic is known to be PSPACE-complete, with implication being one of the main contributors to this complexity. In fact, with just implication alone, we still have a PSPACE-complete logic. We study fragments of intuitionistic logic with restricted implication and develop algorithms for these fragments which are based on the proof rules. We identify a core fragment whose derivability is solvable in linear time. Adding disjunction elimination to this core gives a logic which is solvable in co-NP. These sub-procedures are applicable to a wide variety of logics with rules of a similar flavour. We also show that we cannot do better than co-NP whenever disjunction elimination interacts with other rules.

Abstract We study definability in the first-order theory of graph order: i.e. the set of all isomorphism types of simple finite graphs ordered by either the minor, subgraph or induced subgraph relation. Natural graph families like cycles and trees are definable in these orders, as also notions like connectivity, maximum degree, etc. This machinery allows us to show mutual interpretability with arithmetic for all orders. We discuss implications for formalizing statements of graph theory in such theories of order. 1

Term modal logics (TML) are modal logics with unboundedly many modalities, with quantification over modal indices, so that we can have formulas of the form $\exists y. \forall x. (\Box_x P(x,y) \supset\Diamond_y P(y,x))$. Like First order modal logic, TML is also "notoriously" undecidable, in the sense that even very simple fragments are undecidable. In this paper, we show the decidability of one interesting fragment, that of two variable TML. This is in contrast to two-variable First order modal logic, which is undecidable.

Quantified modal logic is notorious for being undecidable, with very few known decidable fragments such as the monodic ones. For instance, even the two-variable fragment over unary predicates is undecidable. In this paper, we study a particular fragment, namely the bundled fragment, where a first-order quantifier is always followed by a modality when occurring in the formula, inspired by the proposal of [Yanjing Wang, 2017] in the context of non-standard epistemic logics of know-what, know-how, know-why, and so on. As always with quantified modal logics, it makes a significant difference whether the domain stays the same across possible worlds. In particular, we show that the predicate logic with the bundle "forall Box" alone is undecidable over constant domain interpretations, even with only monadic predicates, whereas having the "exists Box" bundle instead gives us a decidable logic. On the other hand, over increasing domain interpretations, we get decidability with both "forall Box" and "exists Box" bundles with unrestricted predicates, where we obtain tableau based procedures that run in PSPACE. We further show that the "exists Box" bundle cannot distinguish between constant domain and variable domain interpretations.

We study repeated normal form games where the number of players is large and suggest that it is useful to consider a neighbourhood structure on the players. The structure is given by a graph G whose nodes are players and edges denote visibility. The neighbourhoods are maximal cliques in G. The game proceeds in rounds where in each round the players of every clique X of G play a strategic form game among each other. A player at a node v strategises based on what she can observe, i.e., the strategies and the outcomes in the previous round of the players at vertices adjacent to v. Based on this, the player may switch strategies in the same neighbourhood, or migrate to another neighbourhood. Player types, giving the rationale for such switching, are specified in a simple modal logic.

We study games in which the choices available to players are not fixed, and may change during the course of play. Specifically, we consider a model in which players may switch strategies, and a global (social) decision may remove some choices, based on the strategies being adopted by players. We propose a logical formalism in which such choices are specified, and a model of bounded memory strategies in which the eventual implications of such choices can be computed, and present preliminary results.

We contend that reasoning about knowledge is both natural and pragmatic for verification of electronic voting protocols. We present a model in which desirable properties of elections are naturally expressed using standard knowledge operators, and show that the associated logic is decidable (under reasonable assumptions of bounded agents and nonces).

An important problem in the analysis of security protocols is that of checking whether a protocol preserves secrecy, i.e., no secret owned by the honest agents is unintentionally revealed to the intruder. This problem has been proved to be undecidable in several settings. In particular, Durgin et a l. prove the undecidability of the secrecy problem in the presence of an unbounded set of nonces, even when the message length is bounded. In this paper we prove that even in the presence of an unbounded set of nonces the secrecy problem is decidable for a reasonable subclass of protocols, which we call context-explicit protocols.

We study linear time temporal logics of multiple agents, where the temporal modalities are local. These modalities not only refer to local next-instants and local eventuality, but also global views of agents at any local instant, which are updated due to communication from other agents. Thus agents also reason about the future, present and past of other agents in the system. The models for these logics are simple: runs of networks of synchronizing automata. Problems like gossiping in interconnection networks am naturally described in the logics proposed here. We present solutions to the satisfiability and model checking problems for these logics. Further since formulas are insensitive to different interleavings of runs, partial order based verification methods become applicable for properties described in these logics.