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Process Algebra (2IF45) Dr. Suzana Andova Practical issues • Lecturer - Suzana Andova - Group: Software Engineering and Technology group - Section: Model Driven Software Engineering • My coordinates: • office HG 5.36 • email [email protected] • phone: 5089 1 Process Algebra (2IF45) Organization • Course information - http://www.win.tue.nl/~andova/education/2IF45/201112/201112.html • Course material • book: − Jos C.M. Baeten, T. Basten, M.A. Reniers “Process Algebra: Equational Theories of Communicating Processes” • lecture notes “Probabilistic Process Algebra” available at http://www.win.tue.nl/~andova/education/2IF45/lnpa.pdf • distributed papers • slides 2 Process Algebra (2IF45) Organization Lectures • Quarter 3 (06-02-2012 – 02-04-2012): laplace-gebouw -1.19 • Quarter 4 (23-04-2012 – 18-06-2012): HG 6.09 Course activities • group assignments − 3 assignments = 40% of the final grade • exam = 60% of the final grade • home works − not compulsory but useful • active participation, discussion • few questionnaires – evaluation 3 Process Algebra (2IF45) Content of the course • Equational theories and Operational semantics • Equational theory of communication processes • Equational theories for probabilistic processes Learning objectives: • be able to develop extensions of a process algebraic language, axiomatically and semantically • be able to establish and analyze relations and discover "inconsistencies" between a given equational theory and its operational semantics • be able to decide the most suitable construct(s) to specify particular system behavior • be able to specify and analyze probabilistic process specifications 4 Process Algebra (2IF45) Questions for you • Where did you do your bachelor: • TU/e (CS, ES, WIN) • TU/e (Mechanical eng., Electrical eng. ) • HBO • Manipal • other university? • Who knows what formal methods are about? • Who is familiar with labeled transition systems? • Who is familiar with (any level) mCRL2, Chi? • Who has knowledge (any level) on model checking? • Who has knowledge on bisimulation relations? • Who is acquainted with probability theory? 5 Process Algebra (2IF45) Questions for me? 6 Process Algebra (2IF45) Process Algebra (2IF45) Introduction Dr. Suzana Andova Foundations (Example) • Natural numbers N = {0, 1, 2, …} • Operations: + and • The Peano axioms define the arithmetical properties of natural numbers • “ingredients” to build the set of natural numbers N − 0 constant and − s unary operator (successor function) Axiom If n is a natural number then s(n) is natural number, n N s(n) N 8 Process Algebra (2IF45) Foundations (Example - cont.) Addition of natural numbers • “addition” a: N x N → N is axiomatized as − a(x,0) = x − a(x,s(y)) = s(a(x,y)) Multiplication of natural numbers • “multiplication” m: N x N → N is axiomatized as − m(x,0) = 0 − m(x,s(y)) = a(m(x,y),x) 9 Process Algebra (2IF45) Foundations (Example - cont.) Derivation of other equalities • use the axioms • derive more equalities using the following rules: − reflexivity x = x − symmetry x= y y = x − transitivity x = y y = z x = z Example of a theorem: s(s(0)) = m(s(s(0)), s(0)) 10 Process Algebra (2IF45) Foundations (Example – recap ) The Peano axioms define the arithmetical properties of natural numbers − − − − 0 constant and S unary operator “addition” a: N x N → N binary function “multiplication” m: N x N → N binary function Signature − Terms: s(s(0)), a(s(0),m(s(0),s(s(s(0))))), 0, − − − − − n N s(n) N a(x,0) = x a(x,s(y)) = s(a(x,y)) m(x,0) = 0 m(x,s(y)) = a(m(x,y),x) Axioms − reflexivity x = x − symmetry x= y y = x − transitivity x = y y = z x = z 11 Process Algebra (2IF45) Relation (derivation rules) Foundation • Axiom is any mathematical statement that serves as a starting point from which other statements are logically derived “absolute truth” • Derivation rules are also part of the theory used to form new “truths” from the old once. • Theorems are mathematical statements that can be derived from the axioms by derivation. • Interpretation and models of an equational theory 12 Process Algebra (2IF45) Alternative Representation of numbers: unary semantics • If we would like to represent numbers as s(s(0)) 1 s(0) 1 0 13 Process Algebra (2IF45) Alternative Representation of numbers: unary semantics 1 s(x) x 0 1 y y’ 1 a(x,y) a(x, y’) 1 x x’ , y 1 a(x,y) x’ x, y a(x,y) a( s(s(0)), s(s(s(0))) ) 1 14 1 a( s(s(0)), 0 ) 1 s(0) a( s(s(0)), s(s(0)) ) 1 a( s(s(0)), s(0) ) 1 0 Alternative Representation of numbers: unary semantics a( s(s(0)), s(s(s(0))) ) 1 a( s(0), s(s(s(0))) ) 1 a( s(s(0)), s(s(0)) ) a( s(0), s(s(0)) ) 1 a( s(0), s(0) ) 1 1 1 1 a( s(0), 0 ) 15 1 1 1 a( s(s(0)), 0 ) 1 1 a( s(s(0)), s(0) ) 1 a( 0, s(s(s(0))) ) 1 a(0, s(s(0)) ) 1 a(0, s(0) ) 1 1 a(0,0) Process Algebra (2IF45) Representation of Reactive systems Dr. Suzana Andova Reactive systems • Reactive systems execute by reacting to stimuli from its environment • Many of them are control crucial and/or safety critical • These systems are large and usually consist of a number of components which interact with each other • Modeling reactive systems • abstract model of the system • unambiguous description • methods and tools for model analysis (verification of qualitative properties, performance analysis) 17 Process Algebra (2IF45) Representation as Labeled transition systems x:= 1; y:= x+1; out(y). in(x); y:= x+1; while (true) { out(y); }. ?x ?x !x y:=x+1 ?y y:=x+1 !y !y 18 out(x); in(y). Process Algebra (2IF45) Representation as Labeled transition systems VM1 VM2 VM3 ?return ?coin !coffee ?coin ?return !tea ?coin !coffee !tea !tea !coffee User !coin 19 error ?coffee Process Algebra (2IF45) Representation as Labeled transition systems VM1 !coffee ?coin !tea composition VM1 and User User coin !coin ?coffee coffee !coffee 20 ?coin !tea Process Algebra (2IF45) Representation as Labeled transition systems VM2 VM2’ ?return ?return ?coin !coffee ?coin ?coffee ?tea !coffee !tea !tea User !coin 21 !coffee ?coffee Process Algebra (2IF45) Representation as Labeled transition systems VM1’’ VM1’ ?coin !coffee Using VM1’ !tea ?coin ?coin !coffee !tea Using VM1’’ coin coin coffee coffee 22 Process Algebra (2IF45) coin Questions • When modeling a system, is an LTS a model to start with or is it something to be obtained as a final or side product? • What entities do we need to have predefined, to be able to produce an LTS? • What is a state? • What is a transition? • How do we know drawing a transition from a state s to a state s’ is right? How do we know which label to assign to it? • How do we combine LTSs? 23 Process Algebra (2IF45) Use of LTS representations In (model checking) tools manipulating the state space (LTSs): UPPAAL, Prism, MRMC manipulating the specification (language): mCRL2, Chi, CADP, FDR, PEPA, MRMC +IMC reduction on specification components’ specifications … reduction on specification the whole system specification No! reduction on LTSs verification model checking the state space SSpace generation property specification Yes! 24 Process Algebra (2IF45) composition by axiom Equational theory in place In (model checking) tools manipulating the state space (LTSs): UPPAAL, Prism, MRMC manipulating the specification (language): mCRL2, Chi, CADP, FDR, PEPA, MRMC updated IMC reduction on specification components’ specifications … reduction on specification the whole system specification No! reduction on LTSs verification model checking 25 Yes! the state space composition by axiom SS generation by the SOS rules property specification Process Algebra (2IF45) Equational theory in place In (model checking) tools manipulating the state space (LTSs): UPPAAL, Prism, MRMC manipulating the specification (language): mCRL2, Chi, CADP, FDR, PEPA, MRMC updated IMC reduction on specification components’ specifications consistent … reduction on specification the whole system specification No! reduction on LTSs verification model checking 26 Yes! the state space composition by axiom SS generation by the SOS rules property specification Process Algebra (2IF45) Equational theory in place In this course we will learn HOW to build a consistent Process Algebra = checking) specification In (model tools language + axioms manipulating manipulating + SOS rules the state space (LTSs): the specification (language): + reduction equivalence relations UPPAAL, Prism, MRMC mCRL2, Chi, CADP, FDR, PEPA, MRMC updated IMC so that the initial specification and the model checked LTS, they both reduction describe the same system! on specification components’ specifications consistent … reduction on specification the whole system specification No! reduction on LTSs verification model checking 27 Yes! the state space composition by axiom SS generation by the SOS rules property specification Process Algebra (2IF45)