Regular terms with the Kleene operations $ \cup $, ; and $ * $ can be thought of as operators on languages, generating other languages. An equation $\tau _1 = \tau _2 $ between two such terms is said to be satisfiable just in case languages exist which make this equation true. We show that the satisfiability problem even for $ * $-free regular terms is undecidable. Similar techniques are used to show that a very natural extension of the Process Logic of Harel, Kozen and Parikh is undecidable.

We define a propositional logic of games which lies in expressive power between the Propositional Dynamic Logic of Fischer and Ladner [FL] and the µ-calculus of Kozen [K]. We show that the logic is decidable and give a very simple, complete set of axioms, one of the rules being Brouwer's bar induction. Even though decidable, this logic is powerful enough to define well orderings. We state some other results, open questions and indicate directions for further research.

Regular terms with the Kleene operations ∪,;, and * can be thought of as operators on languages, generating other languages. An equation r1 = r2 between two such terms is said to be satisfiable just in case languages exist which make this equation true. We show that the satisfiability problem even for *-free regular terms is undecidable. Similar techniques are used to show that a very natural extension of the Process Logic of Harel, Kozen and Parikh is undecidable.

It is well known that regularity of a language implies certain properties known as pumping lemmas or iteration theorems. However, the question of a converse result has been open. We show that the usual form of pumping is very far from implying regularity but that a strengthened pumping property, the block pumping property, is equivalent to regularity. The proof involves use of the finite version of Ramsey’s theorem. We compare our results with recent results of Jaffe and Beauquier and state some open questions.

We define a process logic PL that subsumes Pratt's process logic, Parikh's SOAPL, Nishimura's process logic, and Pnueli's 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.

We indicate below the various results in this paper and the sections where these results are fully described. δ1 is introductory).(δ2). The partial correctness assertion (PCA) A{α}B is expressed in PDL in the form A → [α]B. We shall consider the question of when a given finite set of PCAs implies another and give optimal complexity bounds. We also give other results of an algebraic nature.(δ3). DPDL is less well understood than PDL since the filtration technique of F-L that applies to the latter falls for the former. We point out a translation from PDL to DPDL which leads to a lower bound for the latter.(δ4). What do models of PDL look like? We show the existence of canonical and universal models and show how an infinite model can be related to its F-L factors.

We study the expressive power of various versions of Dynamic Logic and compare them with each other as well as with standard languages in the logical literature. One version of Dynamic Logic is equivalent to the infinitary logic LCKω1ω, but regular Dynamic Logic is strictly less expressive. In particular, the ordinals ωω and ω ω.2 are indistinguishable by formulas of regular Dynamic Logic.

We prove the decidability of the validity problem for a rather general language for talking about computations. As corollaries of our result, we obtain some decidability results of Pratt, Constable, Fischer-Ladner, and Pnueli and also a new decidability result for deterministic propositional dynamic logic.

“From two integers k, l one passes immediately to kl; 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?

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: The language generated by a context-free grammmar is linear in a sense that is defined precisely. 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. The result that not every CF language is a finite automaton (FA) language is improved in the following way. There exists a CF language L such that for any L′ ⊆ L , if L′ is FA, an L″ ⊆ L can be found such that L″ is also FA, L′ ⊆ L″ and L″ contains infinitely many sentences not in L′ . 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.