Chair of Software Engineering 1 Concurrent Object-Oriented Programming Arnaud Bailly, Bertrand Meyer and Volkan Arslan

Chair of Software Engineering 2 Lecture 6: Tutorial on Formal Logics for Computation and Concurrency.

Chair of Software Engineering 3 Formal Logics  Have a syntax defining (legal) terms.  Allow reasoning by providing:  Axioms defining equality on terms, and/or  A semantics defining the meaning of terms accompanied by a semantic equivalence.  Soundness/Completeness properties may relate the two approaches.

Chair of Software Engineering 4 Example: Integer Calculus  Syntax in (Enhanced) Backus-Naur Form.  Digit ::= 0 | 1 | … |9  Integer ::= |  Term ::=  Equality is symmetric, transitive, reflexive.  Operators can be associative, commutative, distributive.  Allows axiomatic reasoning, by tautology.  How to define “reasonable” equality? + | - | *

Chair of Software Engineering 5 Formalisms are Useful for…  Understanding (precise definitions).  Documentation.  Static analysis:  protocol verification,  code optimization (e.g. partial evaluation),  code validation.  Code generation.  Test case generation.  Execution monitoring.

Chair of Software Engineering 6 Functions and Functional Programming  Mathematical function:  taking (positional) parameters,  potentially calling other functions,  returning a result.  The result depends only on.  Calls with same parameters yield same result.  A function defines an input-output relation.  Functional programming:  Functions are 1 st -class entities,  Extensional equivalence with termination.  Programs are more concrete than functions!

Chair of Software Engineering 7 Lambda-Calculus [Church 1934]  Calculus of (sequential) functional programs.  Two constructions (operators):  lambda-abstraction,  function application.  Infinite but enumerable set of variables.  Lambda acts as a binding operator.  The scope of a lambda abstraction extends as far to the right as possible.

Chair of Software Engineering 8 Syntactical Aspects (1)  Occurrences of a variable under a lambda are bound.  Other occurrences are free.

Chair of Software Engineering 9 Syntactical Aspects (2)  The substitution of by in is noted :  Alpha-conversion (renaming) is defined as:  We shall assume from now that all terms are considered modulo alpha-conversion.

Chair of Software Engineering 10 Operational Semantics  Given as a set of rewriting rules.  Only 1 rule, Beta-reduction:  Defines a transition relation.  Repeatedly convert and reduce a term to have its semantics given by transitive closure.  A reducible part of a term is called a redex.  Functional application associates to the left.  Values are non-reducible terms (terminated computations).

Chair of Software Engineering 11 Encoding Booleans (1)

Chair of Software Engineering 12 Encoding Booleans (2)

Chair of Software Engineering 13 Encoding Pairs (or lists, records…)  A encloses two items and.  There exists two projection functions and.  and.  where ?

Chair of Software Engineering 14 Encoding Naturals (1) A natural just applies times a given function to an argument

Chair of Software Engineering 15 Encoding Naturals (2)

Chair of Software Engineering 16 What Can You Do?  Reason exactly about naturals.  Encode objects! (almost with what we have seen)  Write any sequential program (Church-Turing)  Haskell, ML, Lisp, Scheme, etc.  Prove properties?

Chair of Software Engineering 17 Imperative Programming  Problem: encoding a state.  Similar to semantic function for lambda terms.  Operational semantics:  give a function from states to states.  Repeat it as long as possible.  Computation finished when:  semantic function not applicable, or  Function yields always the same state (fixpoint).  The result is contained in variable Result.  In practice: becomes where associates variables to values.

Chair of Software Engineering 18 Example of State Encoding function fact (int n) is int result := 1; while n > 1 do result := result * n; n := n – 1; od; return result; end; function fact (int n) is return fact_rec (n, 1) function fact_rec (int n, int result) is if (n > 1) then return fact_rec (n - 1, n * result); else return result; end

Chair of Software Engineering 19 Modeling Concurrency  The choice operator can be used for expansion: x := 1 y := 1 Cobegin x := y + 1 || y := x + 1 Coend

Chair of Software Engineering 20 Non-Determinism in Lambda

Chair of Software Engineering 21 Encoding Non-Determinism?  It does not work!  Because of one property named either diamond, Church-Rosser, or confluence.  In short: the order of reductions does not matter.

Chair of Software Engineering 22 Example  Although a lambda-term can have more than one redex:  One can hardly say this models concurrency!

Chair of Software Engineering 23 Calculi for Concurrency  Have (internal) non-deterministic operators:  CCS,  CSP,  Pi-Calculus,  Ambients…  Next Time!