Rohit Parikh

On Context-Free Languages

In this report, certain properties of context-free (CF or type 2) grammars are investigated, like that of Chomsky. In particular, questions regarding structure, possible ambiguity and relationship to finite automata are considered. The following results are presented:<list><item>The language generated by a context-free grammmar is linear in a sense that is defined precisely. </item><item>The requirement of unambiguity—that every sentence has a unique phrase structure—weakens the grammar in the sense that there exists a CF language that cannot be generated unambiguously by a CF grammar. </item><item>The result that not every CF language is a finite automaton (FA) language is improved in the following way. There exists a CF language <italic>L</italic> such that for any <italic>L′</italic> ⊆ <italic>L</italic>, if <italic>L′</italic> is FA, an <italic>L″</italic> ⊆ <italic>L</italic> can be found such that <italic>L″</italic> is also FA, L′ ⊆ <italic>L″</italic> and <italic>L″</italic> contains infinitely many sentences not in <italic>L′</italic>. </item><item>A type of grammar is defined that is intermediate between type 1 and type 2 grammars. It is shown that this type of grammar is essentially stronger than type 2 grammars and has the advantage over type 1 grammars that the phrase structure of a grammatical sentence is unique, once the derivation is given. </item></list>

Existence and feasibility in arithmetic

“From two integers k, l one passes immediately to k l ; this process leads in a few steps to numbers which are far larger than any occurring in experience, e.g., 67 (257729) . Intuitionism, like ordinary mathematics, claims that this number can be represented by an arabic numeral. Could not one press further the criticism which intuitionism makes of existential assertions and raise the question: What does it mean to claim the existence of an arabic numeral for the foregoing number, since in practice we are not in a position to obtain it?

States of Knowledge

Abstract When a group of people are thinking about some true formula A, one of them may know it, all of them may know it, or it might be common knowledge. But there are also many other possible states of knowledge and they can change when communication takes place. What states of knowledge are possible? What is the dynamics of such changes? And how do they affect possible co-ordinated action? We describe some developments in this domain.

A Knowledge Based Semantics of Messages

We investigate the semantics of messages, and argue that the meaning ofa message is naturally and usefully given in terms of how it affects theknowledge of the agents involved in the communication. We note thatthis semantics depends on the protocol used by the agents, and thus not only the message itself, but also the protocol appears as a parameter in the meaning. Understanding this dependence allows us to give formal explanations of a wide variety of notions including language dependence, implicature, and the amount of information in a message.

The logic of games and its applications

Abstract We develop a logic in which the basic objects of concern are games, or equivalently, monotone predicate transforms. we give completeness and decision results and extend to certain kinds of many-person games. Applications to a cake cutting algorithm and to a protocol for exchanging secrets, are given.

Distributed Processes and the Logic of Knowledge

We establish a very natural connection between distributed processes and the logic of knowledge which promises to shed light on both areas.

An elementary proof of the completeness of PDL

Abstract We give an elementary proof of the completeness of the Segerberg axions for Propositional Dynamic Logic.

Process logic: Expressiveness, decidability, completeness

We define a process logic PL that subsumes Pratts process logic, Parikhs SOAPL, Nishimuras process logic, and Pnuelis Temporal Logic in expressiveness. The language of PL is an extension of the language of Propositional Dynamic Logic (PDL). We give a deductive system for PL which includes the Segerberg axioms for PDL and prove that it is complete. We also show that PL is decidable.

Communication, consensus, and knowledge☆

Abstract We continue the work of Aumann (Ann. Statist. 4 (1976), 1236–1239), Geanakoplos and Polemarchakis, (J. Econ. Theory 28 (1982), 192–200), Cave (Econ. Lett. 12 (1983), 147–152), and Bacharach (J. Econ. Theory 37 (1985), 167–190) on common knowledge and consesus, extending the results of Geanakoplos and Polemarchakis, Cave, and Bacharach to the case where the number n of communicants is greater than two, but communication is in pairs. When n > 2, then communication is pairs does not lead to common knowledge for the group. We show that, nevertheless, conditional probabilities will converge in any fair protocol for communication, i.e., when none of the participants is blocked from communication. We show that the Cave-Bacharach generalisation of the results of Geanakoplos and Polemarchakis and of Aumann does not hold for pairwise communication when n > 2, and hence any proof for this case must use methods different from theirs. In particular, our proof uses a convexity condition that is obeyed by conditional probabilities, but is not implied by Caves union consistency principle.

Game Logic - An Overview

Game Logic is a modal logic which extends Propositional Dynamic Logic by generalising its semantics and adding a new operator to the language. The logic can be used to reason about determined 2-player games. We present an overview of meta-theoretic results regarding this logic, also covering the algebraic version of the logic known as Game Algebra.