scholarly journals A Complete Equational Axiomatization for Prefix Iteration with Silent Steps

1995 ◽  
Vol 2 (5) ◽  
Author(s):  
Luca Aceto ◽  
Anna Ingólfsdóttir
Keyword(s):  
Process Algebra ◽  
Axiom System ◽  
Subject Classification ◽  
Ams Subject Classification ◽  
Star Operation ◽  
Atomic Action ◽  
Keywords And Phrases ◽  
Bisimulation Equivalence

Fokkink ((1994) Inf. Process. Lett. 52: 333{337) has recently proposed a complete<br />equational axiomatization of strong bisimulation equivalence for MPA_delta^*(A_tau),<br />i.e., the language obtained by extending Milner's basic CCS with prefix iteration.<br />Prefix iteration is a variation on the original binary version of the Kleene star operation<br />p*q obtained by restricting the first argument to be an atomic action. In this<br />paper, we extend Fokkink's results to a setting with the unobservable action by<br />giving a complete equational axiomatization of Milner's observation congruence over<br />MPA_delta^*(A_tau ). <br />The axiomatization is obtained by extending Fokkink's axiom system<br />with two of Milner's standard tau-laws and the following three equations that describe<br />the interplay between the silent nature of tau and prefix iteration:<br />tau . x = tau*x<br />a*(x+tau.y) = a*(x+tau.y + a.y)<br />tau.(a*x) = a*(tau.a*x) .<br />Using a technique due to Groote, we also show that the resulting axiomatization is<br />omega-complete, i.e., complete for equality of open terms.<br />AMS Subject Classification (1991): 68Q40, 68Q42.<br />CR Subject Classification (1991): D.3.1, F.3.2, F.4.2.<br />Keywords and Phrases: Minimal Process Algebra, prefix iteration, equational<br />logic, omega-completeness, observation congruence.

scholarly journals Axiomatizing Prefix Iteration with Silent Steps

1995 ◽  
Vol 2 (56) ◽  
Author(s):  
Luca Aceto ◽  
Willem Jan Fokkink ◽  
Rob J. van Glabbeek ◽  
Anna Ingólfsdóttir
Keyword(s):  
Process Algebra ◽  
Direct Proof ◽  
Completeness Theorem ◽  
Subject Classification ◽  
Star Operation ◽  
Weak Bisimulation ◽  
Keywords And Phrases ◽  
Completeness Result ◽  
Considerable Simplification ◽  
Branching Bisimulation

<p>Prefix iteration is a variation on the original binary version of the Kleene star operation P*Q, obtained by restricting the first argument to be an atomic action. The interaction of prefix iteration with silent steps is studied in the setting of Milner's basic CCS. Complete equational axiomatizations are given for four notions of behavioural congruence over basic CCS with prefix iteration, viz. branching congruence, eta-congruence, delay congruence and weak congruence. The completeness proofs for eta-, delay, and weak congruence are obtained by reduction to the completeness theorem for branching congruence. It is also argued that the use of the completeness result for branching congruence in obtaining the completeness result for weak congruence leads to a considerable simplification with respect to the only direct proof presented in the literature. The preliminaries and the completeness proofs focus on open terms, i.e. terms that may contain process variables. As a by-product, the omega-completeness of the axiomatizations is obtained as well as their completeness for closed terms.</p><p> </p><p>AMS Subject Classification (1991): 68Q10, 68Q40, 68Q55.<br />CR Subject Classification (1991): D.3.1, F.1.2, F.3.2.<br />Keywords and Phrases: Concurrency, process algebra, basic CCS, prefix iteration, branching bisimulation, eta-bisimulation, delay bisimulation, weak bisimulation, equational logic, complete axiomatizations.</p>

scholarly journals A Note on an Expressiveness Hierarchy for Multi-exit Iteration

2002 ◽  
Vol 9 (40) ◽  
Author(s):  
Luca Aceto ◽  
Willem Jan Fokkink ◽  
Anna Ingólfsdóttir
Keyword(s):  
Positive Integer ◽  
Process Algebra ◽  
Expressive Language ◽  
Basic Process ◽  
Star Operation ◽  
The Family ◽  
Bisimulation Equivalence

Multi-exit iteration is a generalization of the standard binary Kleene star operation that allows for the specification of agents that, up to bisimulation equivalence, are solutions of systems of recursion equations of the form<br />X_1 = P_1 X_2 + Q_1 <br /><br /> X_n = P_n X_1 + Q_n <br /><br /> where n is a positive integer, and the P_i and the Q_i are process terms. The addition of multi-exit iteration to Basic Process Algebra (BPA) yields a more expressive language than that obtained by augmenting BPA with the standard binary Kleene star. This note offers an expressiveness hierarchy, modulo bisimulation equivalence, for the family of multi-exit iteration operators proposed by Bergstra, Bethke and Ponse.

scholarly journals An Equational Axiomatization for Multi-Exit Iteration

1996 ◽  
Vol 3 (22) ◽  
Author(s):  
Luca Aceto ◽  
Willem Jan Fokkink
Keyword(s):  
Positive Integer ◽  
Process Algebra ◽  
Expressive Language ◽  
General Structure ◽  
Basic Process ◽  
Star Operation ◽  
The Family ◽  
Bisimulation Equivalence

<p>This paper presents an equational axiomatization of bisimulation equivalence over the language of Basic Process Algebra (BPA) with multi-exit iteration. Multi-exit iteration is a generalization of the standard binary Kleene star operation that allows for the specification of agents that, up to bisimulation equivalence, are solutions<br />of systems of recursion equations of the form</p><p>X1 = P1 X2 + Q1<br />...<br />Xn = Pn X1 + Qn</p><p>where n is a positive integer, and the Pi and the Qi are process terms. The addition<br />of multi-exit iteration to BPA yields a more expressive language than that obtained by augmenting BPA with the standard binary Kleene star (BPA). As a<br />consequence, the proof of completeness of the proposed equational axiomatization<br />for this language, although standard in its general structure, is much more involved<br />than that for BPA. An expressiveness hierarchy for the family of k-exit iteration operators proposed by Bergstra, Bethke and Ponse is also offered.</p><p> </p>

scholarly journals On the Finitary Bisimulation

1995 ◽  
Vol 2 (59) ◽  
Author(s):  
Luca Aceto ◽  
Anna Ingólfsdóttir
Keyword(s):  
Denotational Semantics ◽  
Subject Classification ◽  
Process Algebras ◽  
Transition Systems ◽  
Behavioural Observation ◽  
Labelled Transition Systems ◽  
Full Abstraction ◽  
Ams Subject Classification ◽  
Keywords And Phrases

<p>The finitely observable, or finitary, part of bisimulation is a key tool in establishing full abstraction results for denotational semantics for process algebras with respect to bisimulation-based preorders. A bisimulation-like characterization of this relation for arbitrary transition systems is given, relying on Abramsky's characterization in terms of the finitary domain logic. More informative behavioural, observation-independent characterizations of the finitary bisimulation are also offered for several interesting classes of transition systems. These include transition systems with countable action sets, those that have bounded convergent sort and the sort-finite ones. The result for sort-finite transition systems sharpens a previous behavioural characterization of the finitary bisimulation for this class of structures given by Abramsky.</p><p><br />AMS Subject Classification (1991): 68Q10 (Modes of computation), 68Q55<br />(Semantics), 03B70 (Logic of Programming), 68Q90 (Transition nets).<br />Keywords and Phrases: Concurrency, labelled transition systems with divergence,<br />bisimulation preorder, finitary relations, domain logic for transition systems.</p>

scholarly journals A Menagerie of Non-Finitely Based Process Semantics over BPA*: From Ready Simulation Semantics to Completed Traces

1996 ◽  
Vol 3 (23) ◽  
Author(s):  
Luca Aceto ◽  
Willem Jan Fokkink ◽  
Anna Ingólfsdóttir
Keyword(s):  
Process Algebra ◽  
Linear Time ◽  
Subject Classification ◽  
Regular Expressions ◽  
Basic Process ◽  
Branching Time ◽  
Finitely Based ◽  
Trace Semantics ◽  
Bisimulation Equivalence ◽  
Trace Equivalence

<p>Fokkink and Zantema ((1994) Computer Journal 37:259-267) have shown that<br />bisimulation equivalence has a finite equational axiomatization over the language<br />of Basic Process Algebra with the binary Kleene star operation (BPA*). In light<br />of this positive result on the mathematical tractability of bisimulation equivalence<br />over BPA*, a natural question to ask is whether any other (pre)congruence relation<br />in van Glabbeek's linear time/branching time spectrum is finitely (in)equationally<br />axiomatizable over it. In this paper, we prove that, unlike bisimulation equivalence,<br />none of the preorders and equivalences in van Glabbeek's linear time/branching time<br />spectrum, whose discriminating power lies in between that of ready simulation and<br />that of completed traces, has a finite equational axiomatization. This we achieve by<br />exhibiting a family of (in)equivalences that holds in ready simulation semantics, the<br />finest semantics that we consider, whose instances cannot all be proven by means of<br />any finite set of (in)equations that is sound in completed trace semantics, which is<br />the coarsest semantics that is appropriate for the language BPA*. To this end, for<br />every finite collection of (in)equations that are sound in completed trace semantics, we<br />build a model in which some of the (in)equivalences of the family under consideration<br />fail. The construction of the model mimics the one used by Conway ((1971) Regular<br />Algebra and Finite Machines, page 105) in his proof of a result, originally due to<br />Redko, to the effect that infinitely many equations are needed to axiomatize equality<br />of regular expressions.</p><p>Our non-finite axiomatizability results apply to the language BPA* over an arbitrary<br />non-empty set of actions. In particular, we show that completed trace equivalence<br />is not finitely based over BPA* even when the set of actions is a singleton.<br />Our proof of this result may be easily adapted to the standard language of regular expressions to yield a solution to an open problem posed by Salomaa ((1969) Theory<br />of Automata, page 143).<br />Another semantics that is usually considered in process theory is trace semantics.<br />Trace semantics is, in general, not preserved by sequential composition, and is<br />therefore inappropriate for the language BPA*. We show that, if the set of actions<br />is a singleton, trace equivalence and preorder are preserved by all the operators in<br />the signature of BPA, and coincide with simulation equivalence and preorder, respectively.<br />In that case, unlike all the other semantics considered in this paper, trace<br />semantics have nite, complete equational axiomatizations over closed terms.</p><p> </p><p>AMS Subject Classification (1991): 08A70, 03C05, 68Q10, 68Q40, 68Q45,<br />68Q55, 68Q68, 68Q70.<br />CR Subject Classification (1991): D.3.1, F.1.1, F.1.2, F.3.2, F.3.4, F.4.1.<br />Keywords and Phrases: Concurrency, process algebra, Basic Process Algebra<br />(BPA*), Kleene star, bisimulation, ready simulation, simulation, completed trace semantics,<br />ready trace semantics, failure trace semantics, readiness semantics, failures<br />semantics, trace semantics, equational logic, complete axiomatizations.</p><p> </p>

scholarly journals Bisimilarity is not Finitely Based over BPA with Interrupt

2005 ◽  
Vol 12 (33) ◽  
Author(s):  
Luca Aceto ◽  
Willem Jan Fokkink ◽  
Anna Ingólfsdóttir ◽  
Sumit Nain
Keyword(s):  
Process Algebra ◽  
Sharp Contrast ◽  
Basic Process ◽  
Finitely Based ◽  
Bisimulation Equivalence

This paper shows that bisimulation equivalence does not afford a finite equational axiomatization over the language obtained by enriching Bergstra and Klop's Basic Process Algebra with the interrupt operator. Moreover, it is shown that the collection of closed equations over this language is also not finitely based. In sharp contrast to these results, the collection of closed equations over the language BPA enriched with the disrupt operator is proven to be finitely based.

Designs Robust Against Aberrations

1993 ◽  
Vol 43 (1-2) ◽  
pp. 95-108 ◽  
Author(s):  
N. K. Mandal ◽  
K. R. Shah
Keyword(s):  
Sufficient Conditions ◽  
Subject Classification ◽  
Alternative Formulation ◽  
Block Designs ◽  
Ams Subject Classification ◽  
Secondary 62K05

In this paper, we obtain sufficient conditions for a design to be robust against aberrations in the sense of Box and Draper. Block designs, row-column designs and fractional designs are considered here. An alternative formulation of robustness is also presented. AMS Subject Classification: Primary 62K99; Secondary 62K05.

scholarly journals Two Higher Order Iterative Methods for Solving Nonlinear Equations

2018 ◽  
Vol 14 (1) ◽  
pp. 179-187
Author(s):  
Jivandhar Jnawali ◽  
Chet Raj Bhatta
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Order Of Convergence ◽  
Higher Order ◽  
Secant Method ◽  
Subject Classification ◽  
Numerical Examples ◽  
Ams Subject Classification

 The main purpose of this paper is to derive two higher order iterative methods for solving nonlinear equations as variants of Mir, Ayub and Rafiq method. These methods are free from higher order derivatives. We obtain these methods by amalgamating Mir, Ayub and Rafiq method with standard secant method and modified secant method given by Amat and Busquier. The order of convergence of new variants are four and six. Also, numerical examples are given to compare the performance of newly introduced methods with the similar existing methods. 2010 AMS Subject Classification: 65H05 Journal of the Institute of Engineering, 2018, 14(1): 179-187

scholarly journals Reinforcement Number of a Graph with respect to Half-Domination

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
G. Muhiuddin ◽  
N. Sridharan ◽  
D. Al-Kadi ◽  
S. Amutha ◽  
M. E. Elnair
Keyword(s):  
Upper Bounds ◽  
Subject Classification ◽  
Ams Subject Classification

In this paper, we introduce the concept of reinforcement number with respect to half-domination and initiate a study on this parameter. Furthermore, we obtain various upper bounds for this parameter. AMS subject classification: 05C38, 05C40, 05C69.

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