Axiomatizing Flat Iteration

Flat iteration is a variation on the original binary version of the Kleene star operation P*Q, obtained by restricting the first argument to be a sum of atomic actions. It generalizes prefix iteration, in which the first argument is a single action. Complete finite equational axiomatizations are given for five notions of bisimulation congruence over basic CCS with flat iteration, viz. strong congruence, branching congruence, eta-congruence, delay congruence and weak congruence. Such axiomatizations were already known for prefix iteration and are known not to exist for general iteration. The use of flat iteration has two main advantages over prefix iteration: 1.The current axiomatizations generalize to full CCS, whereas the prefix iteration approach does not allow an elimination theorem for an asynchronous parallel composition operator. 2.The greater expressiveness of flat iteration allows for much shorter completeness proofs. In the setting of prefix iteration, the most convenient way to obtain the completeness theorems for eta-, delay, and weak congruence was by reduction to the completeness theorem for branching congruence. In the case of weak congruence this turned out to be much simpler than the only direct proof found. In the setting of flat iteration on the other hand, the completeness theorems for delay and weak (but not eta-) congruence can equally well be obtained by reduction to the one for strong congruence, without using branching congruence as an intermediate step. Moreover, the completeness results for prefix iteration can be retrieved from those for flat iteration, thus obtaining a second indirect approach for proving completeness for delay and weak congruence in the setting of prefix iteration.Comment: 15 pages. LaTeX 2.09. Filename: flat.tex.gz. On A4 paper print with: dvips -t a4 -O -2.15cm,-2.22cm -x 1225 flat. For US letter with: dvips -t letter -O -0.73in,-1.27in -x 1225 flat. More info at http://theory.stanford.edu/~rvg/abstracts.html#3

Cyclic Datatypes modulo Bisimulation based on Second-Order Algebraic Theories

Cyclic data structures, such as cyclic lists, in functional programming are tricky to handle because of their cyclicity. This paper presents an investigation of categorical, algebraic, and computational foundations of cyclic datatypes. Our framework of cyclic datatypes is based on second-order algebraic theories of Fiore et al., which give a uniform setting for syntax, types, and computation rules for describing and reasoning about cyclic datatypes. We extract the "fold" computation rules from the categorical semantics based on iteration categories of Bloom and Esik. Thereby, the rules are correct by construction. We prove strong normalisation using the General Schema criterion for second-order computation rules. Rather than the fixed point law, we particularly choose Bekic law for computation, which is a key to obtaining strong normalisation. We also prove the property of "Church-Rosser modulo bisimulation" for the computation rules. Combining these results, we have a remarkable decidability result of the equational theory of cyclic data and fold.Comment: 38 page

Completeness for Flat Modal Fixpoint Logics

This paper exhibits a general and uniform method to prove completeness for certain modal fixpoint logics. Given a set \Gamma of modal formulas of the form \gamma(x, p1, . . ., pn), where x occurs only positively in \gamma, the language L\sharp (\Gamma) is obtained by adding to the language of polymodal logic a connective \sharp\_\gamma for each \gamma \epsilon. The term \sharp\_\gamma (\varphi1, . . ., \varphin) is meant to be interpreted as the least fixed point of the functional interpretation of the term \gamma(x, \varphi 1, . . ., \varphi n). We consider the following problem: given \Gamma, construct an axiom system which is sound and complete with respect to the concrete interpretation of the language L\sharp (\Gamma) on Kripke frames. We prove two results that solve this problem. First, let K\sharp (\Gamma) be the logic obtained from the basic polymodal K by adding a Kozen-Park style fixpoint axiom and a least fixpoint rule, for each fixpoint connective \sharp\_\gamma. Provided that each indexing formula \gamma satisfies the syntactic criterion of being untied in x, we prove this axiom system to be complete. Second, addressing the general case, we prove the soundness and completeness of an extension K+ (\Gamma) of K\_\sharp (\Gamma). This extension is obtained via an effective procedure that, given an indexing formula \gamma as input, returns a finite set of axioms and derivation rules for \sharp\_\gamma, of size bounded by the length of \gamma. Thus the axiom system K+ (\Gamma) is finite whenever \Gamma is finite

On Kleene Algebra vs. Process Algebra

We try to clarify the relationship between Kleene algebra and process algebra, based on the very recent work on Kleene algebra and process algebra. Both for concurrent Kleene algebra (CKA) with communications and truly concurrent process algebra APTC with Kleene star and parallel star, the extended Milner's expansion law $a\parallel b=a\cdot b+b\cdot a+a\parallel b +a\mid b$ holds, with $a,b$ being primitives (atomic actions), $\parallel$ being the parallel composition, $+$ being the alternative composition, $\cdot$ being the sequential composition and the communication merge $\mid$ with the background of computation. CKA and APTC are all the truly concurrent computation models, can have the same syntax (primitives and operators), maybe have the same or different semantics

Extended modular operad

This paper is a sequel to [LoMa] where moduli spaces of painted stable curves were introduced and studied. We define the extended modular operad of genus zero, algebras over this operad, and study the formal differential geometric structures related to these algebras: pencils of flat connections and Frobenius manifolds without metric. We focus here on the combinatorial aspects of the picture. Algebraic geometric aspects are treated in [Ma2].Comment: 38 pp., amstex file, no figures. This version contains additional references and minor change