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A Complete Equational Axiomatization for MPA with String Iteration
 DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE, AALBORG UNIVERSITY
, 1995
"... We study equational axiomatizations of bisimulation equivalence for the language obtained by extending Milner's basic CCS with string iteration. String 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 none ..."
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Cited by 13 (5 self)
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We study equational axiomatizations of bisimulation equivalence for the language obtained by extending Milner's basic CCS with string iteration. String 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 nonempty sequence of atomic actions. We show that, for every positive integer k, bisimulation equivalence over the set of processes in this language with loops of length at most k is finitely axiomatizable. We also offer a countably infinite equational theory that completely axiomatizes bisimulation equivalence over the whole language. We prove that this result cannot be improved upon by showing that no finite equational axiomatization of bisimulation equivalence over basic CCS with string iteration can exist, unless the set of actions is empty.
On Equivalents of Wellfoundedness  An experiment in Mizar
, 1998
"... Four statements equivalent to wellfoundedness (wellfounded induction, existence of recursively defined functions, uniqueness of recursively defined functions, and absence of descending omegachains) have been proved in Mizar and the proofs mechanically checked for correctness. It seems not to be w ..."
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Cited by 13 (3 self)
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Four statements equivalent to wellfoundedness (wellfounded induction, existence of recursively defined functions, uniqueness of recursively defined functions, and absence of descending omegachains) have been proved in Mizar and the proofs mechanically checked for correctness. It seems not to be widely known that the existence (without the uniqueness assumption) of recursively defined functions implies wellfoundedness. In the proof we used regular cardinals, a fairly advanced notion of set theory. The theory of cardinals in Mizar was developed earlier by G. Bancerek. With the current state of the Mizar system, the proofs turned out to be an exercise with only minor additions at the fundamental level. We would like to stress the importance of a systematic development of a mechanized data base for mathematics in the spirit of the QED Project.
A Complete Equational Axiomatization for Prefix Iteration with Silent Steps
 DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE, AALBORG UNIVERSITY
, 1995
"... Fokkink ((1994) Inf. Process. Lett. 52: 333337) has recently proposed a complete equational axiomatization of strong bisimulation equivalence for MPA i.e., the language obtained by extending Milner's basic CCS with prefix iteration. p q obtained by restricting the first argument to be an at ..."
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Cited by 8 (2 self)
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Fokkink ((1994) Inf. Process. Lett. 52: 333337) has recently proposed a complete equational axiomatization of strong bisimulation equivalence for MPA i.e., the language obtained by extending Milner's basic CCS with prefix iteration. p q obtained by restricting the first argument to be an atomic action. In this paper, we extend Fokkink's results to a setting with the unobservable action by giving a complete equational axiomatization of Milner's observation congruence over with two of Milner's standard laws and the following three equations that describe the interplay between the silent nature of and prefix iteration: Using a technique due to Groote, we also show that the resulting axiomatization is !complete, i.e., complete for equality of open terms.
A Note on "How to Write a Proof"
, 1996
"... We believe that mechanical checking of reallife proofs can become practical and therefore we use Mizar  a proof checking system for proofs written in a style of traditional mathematics. In the beginning of 1994 we came across a copy of L. Lamport's [8] paper in which "a method for writing proofs ..."
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Cited by 3 (1 self)
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We believe that mechanical checking of reallife proofs can become practical and therefore we use Mizar  a proof checking system for proofs written in a style of traditional mathematics. In the beginning of 1994 we came across a copy of L. Lamport's [8] paper in which "a method for writing proofs is proposed that makes it much harder to prove things that are not true." For Mizar users the issue of How to Write a Proof? is an important one, as Mizar is a proof checker and not an automated prover. We have tested Mizar fitness for writing structured proofs in Lamport's style by rewriting his proof of the irrationality of p 2 into Mizar. It was not surprising to notice that formatting conventions help in presenting and reading proofs. However, such conventions do not, as they cannot, guarantee the correctness of the written proof, our little test being a case in point. We advocate development and employment of mechanical checkers for proofs.
The Binary Decision Machine: A mathematical verification redone in PVS
 in PVS
, 1996
"... This paper presents a mathematical verification mechanically checked with the Prototype Verification System (PVS)[Owre96]. Prior to the mechanical proof a complete mathematical verification was done by us with use of the Funmath notation [Bout93]. One of the key ideas behind Funmath is the descri ..."
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Cited by 1 (1 self)
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This paper presents a mathematical verification mechanically checked with the Prototype Verification System (PVS)[Owre96]. Prior to the mechanical proof a complete mathematical verification was done by us with use of the Funmath notation [Bout93]. One of the key ideas behind Funmath is the description of mathematical notions as functions. The object of this verification is the binary decision machine (BDM) by R.T. Boute [Bout76], a hardware component which is used to realize systems described by binary decision diagrams [Lee59] [Aker78]. The proof in PVS was given with a conservative extension of the logic, i.e. no inconsistencies were introduced into the PVS logic. It is shown that this could easily be done (except in one case) given the original definitional style of the proof (i.e. using functions to describe objects), and the underlying (extensional) higherorder logic of PVS. 1 1 Introduction The object of this study is the binary decision machine (BDM) [Bout76], a bi...