Dexter Kozen

Results on the propositional μ-calculus

Abstract In this paper we define and study a propositional μ-calculus L μ, which consists essentially of propositional modal logic with a least fixpoint operator. L μ is syntactically simpler yet strictly more expressive than Propositional Dynamic Logic (PDL). For a restricted version we give an exponential-time decision procedure, small model property, and complete deductive system, theory subsuming the corresponding results for PDL.

A completeness theorem for Kleene algebras and the algebra of regular events

A finitary axiomatization of the algebra of regular events involving only equations and equational implications that is sound for all interpretations over Kleene algebras is given. Axioms for Kleene algebra are presented, and some basic consequences are derived. Matrices over a Kleene algebra are considered. The notion of an automaton over an arbitrary Kleen algebra is defined and used to derive the classical results of the theory of finite automata as a result of the axioms. The completeness of the axioms for the algebra of regular events is treated. Open problems are indicated. >

Results on the Propositional µ-Calculus

We define a propositional version of the Μ-calculus, and give an exponential-time decision procedure, small model property, and complete deductive system. We also show that it is strictly more expressive than PDL. Finally, we give an algebraic semantics and prove a representation theorem.

Kleene algebra with tests

We introduce Kleene algebra with tests, an equational system for manipulating programs. We give a purely equational proof, using Kleene algebra with tests and commutativity conditions, of the following classical result: every while program can be simulated by a while program can be simulated by a while program with at most one while loop. The proof illustrates the use of Kleene algebra with tests and commutativity conditions in program equivalence proofs.

Semantics of probabilistic programs

Two complementary but equivalent semantic interpretations of a high level probabilistic programming language are given. One of these interprets programs as partial measurable functions on a measurable space. The other interprets programs as continuous linear operators on a Banach space of measures. It is shown how the ordered domains of Scott and others are embedded naturally into these spaces. Two general results about probabilistic programs are proved.

Lower bounds for natural proof systems

Two decidable logical theories are presented, one complete for deterministic polynomial time, one complete for polynomial space. Both have natural proof systems. A lower space bound of n/log(n) is shown for the proof system for the PTIME complete theory and a lower length bound of 2cn/log(n) is shown for the proof system for the PSPACE complete theory.

NetKAT: semantic foundations for networks

Recent years have seen growing interest in high-level languages for programming networks. But the design of these languages has been largely ad hoc, driven more by the needs of applications and the capabilities of network hardware than by foundational principles. The lack of a semantic foundation has left language designers with little guidance in determining how to incorporate new features, and programmers without a means to reason precisely about their code. This paper presents NetKAT, a new network programming language that is based on a solid mathematical foundation and comes equipped with a sound and complete equational theory. We describe the design of NetKAT, including primitives for filtering, modifying, and transmitting packets; union and sequential composition operators; and a Kleene star operator that iterates programs. We show that NetKAT is an instance of a canonical and well-studied mathematical structure called a Kleene algebra with tests (KAT) and prove that its equational theory is sound and complete with respect to its denotational semantics. Finally, we present practical applications of the equational theory including syntactic techniques for checking reachability, proving non-interference properties that ensure isolation between programs, and establishing the correctness of compilation algorithms.

A probabilistic PDL

Abstract In this paper we give a probabilistic analog PPDL of Propositional Dynamic Logic. We prove a small model property and give a polynomial space decision procedure for formulas involving well-structured programs. We also give a deductive calculus and illustrate its use by calculating the expected running time of a simple random walk.

On the power of the compass (or, why mazes are easier to search than graphs)

About the same time, A.N. Shah gave a finite autom~ton with 5 pebbles which could search an arbitrary maze (the autonlaton may drop a pebble on a cell it is visiting, then upon returning to that cell later on can sense the pebbles presence, and if desired pick it up and move it to a new cell). Shah also conjectured that fewer than 5 pebbles would not suffice. The first of our two main results is that, contrary to Shahs conjecture, the search can be implemented with just two pebbles. The question is still open whether a finite automaton with just one pebble can search any maze.

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.