Joshua Sack

Temporal Languages for Epistemic Programs

This paper adds temporal logic to public announcement logic (PAL) and dynamic epistemic logic (DEL). By adding a previous-time operator to PAL, we express in the language statements concerning the muddy children puzzle and sum and product. We also express a true statement that an agent’s beliefs about another agent’s knowledge flipped twice, and use a sound proof system to prove this statement. Adding a next-time operator to PAL, we provide formulas that express that belief revision does not take place in PAL. We also discuss relationships between announcements and the new knowledge agents thus acquire; such relationships are related to learning and to Fitch’s paradox. We also show how inverse programs and hybrid logic each can be used to help determine whether or not an arbitrary structure represents the play of a game. We then add a past-time operator to DEL, and discuss the importance of adding yet another component to the language in order to prove completeness.

A general framework for probabilistic characterizing formulae

Recently, a general framework on characteristic formulae was proposed by Aceto et al. It offers a simple theory that allows one to easily obtain characteristic formulae of many non-probabilistic behavioral relations. Our paper studies their techniques in a probabilistic setting. We provide a general method for determining characteristic formulae of behavioral relations for probabilistic automata using fixed-point probability logics. We consider such behavioral relations as simulations and bisimulations, probabilistic bisimulations, probabilistic weak simulations, and probabilistic forward simulations. This paper shows how their constructions and proofs can follow from a single common technique.

Logic for update products and steps into the past

Abstract This paper provides a sound and complete proof system for a language L e + Y that adds to Dynamic Epistemic Logic (DEL) a discrete previous-time operator as well as single symbol formulas that partially reveal the most recent event that occurred. The completeness theorem is by filtration followed by model unraveling and other model transformations. Decidability follows from the completeness proof. The degree to which it is important to include the additional single symbol formulas is addressed in a discussion about the difficulties of the completeness for a language L Y that only adds the previous-time operator to DEL. Discussion is also given regarding the completeness for a language obtained by removing common knowledge operators from L e + Y .

Characteristic Formulae for Fixed-Point Semantics: A General Framework

The literature on concurrency theory offers a wealth of examples of characteristic-formula constructions for various behavioural relations over finite labelled transition systems and Kripke structures that are defined in terms of fixed points of suitable functions. Such constructions and their proofs of correctness have been developed independently, but have a common underlying structure. This study provides a general view of characteristic formulae that are expressed in terms of logics with a facility for the recursive definition of formulae. It is shown how several examples of characteristic-formula constructions from the literature can be recovered as instances of the proposed general framework, and how the framework can be used to yield novel constructions.

Deriving the correctness of quantum protocols in the probabilistic logic for quantum programs

This paper presents a sound axiomatization for a probabilistic modal dynamic logic of quantum programs. The logic can express whether a state is separable or entangled, information that is local to a subsystem of the whole quantum system, and the probability of positive answers to quantum tests of certain properties. The power of this axiomatization is demonstrated with proofs of properties concerning bases of a finite-dimensional Hilbert space, composite systems, entangled and separable states, and with proofs of the correctness of two probabilistic quantum protocols (the quantum leader election protocol and the BB84 quantum key distribution protocol).

Logics of temporal-epistemic actions

We present Dynamic Epistemic Temporal Logic, a framework for reasoning about operations on multi-agent Kripke models that contain a designated temporal relation. These operations are natural extensions of the well-known “action models” from Dynamic Epistemic Logic (DEL). Our “temporal action models” may be used to define a number of informational actions that can modify the “objective” temporal structure of a model along with the agents’ basic and higher-order knowledge and beliefs about this structure, including their beliefs about the time. In essence, this approach provides one way to extend the domain of action model-style operations from atemporal Kripke models to temporal Kripke models in a manner that allows actions to control the flow of time. We present a number of examples to illustrate the subtleties involved in interpreting the effects of our extended action models on temporal Kripke models. We also study preservation of important epistemic-temporal properties of temporal Kripke models under temporal action model-induced operations, provide complete axiomatizations for two theories of temporal action models, and connect our approach with previous work on time in DEL.

A Modal Logic for Mixed Strategies

Modal logics have proven to be a very successful tool for reasoning about games. However, until now, although logics have been put forward for games in both normal form and games in extensive form, and for games with complete and incomplete information, the focus in the logic community has hitherto been on games with pure strategies. This paper is a first to widen the scope to logics for games that allow mixed strategies. We present a modal logic for games in normal form with mixed strategies, and demonstrate its soundness and strong completeness. Characteristic for our logic is a number of infinite rules.

Resource bisimilarity and graded bisimilarity coincide

Resource bisimilarity has been proposed in the literature on concurrency theory as a notion of bisimilarity over labeled transition systems that takes into account the number of choices that a system has. Independently, g-bisimilarity has been defined over Kripke models as a suitable notion of bisimilarity for graded modal logic. This note shows that these two notions of bisimilarity coincide over image-finite Kripke frames.

Compositional Reasoning for Multi-modal Logics

We provide decomposition and quotienting results for multi-modal logic with respect to a composition operator, traditionally used for epistemic models, due to van Eijck et al. (Journal of Applied Non-Classical Logics 21(3–4):397–425, 2011), that involves sets of atomic propositions and valuation functions from Kripke models. While the composition operator was originally defined only for epistemic S5 n models, our results apply to the composition of any pair of Kripke models. In particular, our quotienting result extends a specific result in the above mentioned paper by van Eijck et al. for the composition of epistemic models with disjoint sets of atomic propositions to compositions of any two Kripke models regardless of their sets of atomic propositions. We also explore the complexity of the formulas we construct in our decomposition result.

Sigma algebras in probabilistic epistemic dynamics

This paper extends probabilistic dynamic epistemic logic from a finite setting to an infinite setting, by introducing σ-algebras to the probability spaces in the models. This may extend the applicability of the logic to a real world setting with infinitely many possible measurements. It is shown that the dynamics preserves desirable properties of measurability and that completeness of the proof system holds with the extended semantics.