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====== Principles for Software Composition ======

{{:magistraleinformatica:psc:psc.png?250 | }}

**PSC 2021/22 (375AA, 9 CFU)**

Lecturer: **Roberto Bruni**\\
[[http://www.di.unipi.it/~bruni|web]] - [[mailto:bruni@di.unipi.it|email]] - [[https://teams.microsoft.com/l/team/19%3azTPYdF4TogSCVHaDQMahbTuiXN8HIrzE73OP01DoHh01%40thread.tacv2/conversations?groupId=78a9a5a2-4b1e-4f22-8d16-328ef5b60b2a&tenantId=c7456b31-a220-47f5-be52-473828670aa1|Microsoft Teams channel]]

Office hours: **Tuesday 16:00-18:00 or by appointment**

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==== Objectives ====

The objective of the course is to present:
  * different models of computation,
  * their programming paradigms,
  * their mathematical descriptions, both //concrete// and //abstract//,
  * some intellectual tools/techniques for reasoning on models.

The course will cover the basic techniques for assigning meaning to programs with higher-order, concurrent and probabilistic features (e.g., domain theory, logical systems, well-founded induction, structural recursion, labelled transition systems, Markov chains, probabilistic reactive systems, stochastic process algebras) and for proving their fundamental properties, such as termination, normalisation, determinacy, behavioural equivalence and logical equivalence. Temporal and modal logics will also be studied for the specification and analysis of programs. In particular, some emphasis will be posed on modularity and compositionality, in the sense of guaranteeing some property of the whole by proving simpler properties of its parts.


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==== Prerequisites ====

There are no prerequisites, but the students are expected to have some familiarity with discrete mathematics, first-order logic, context-free grammars, and code fragments in imperative and functional style.

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==== Textbook(s) ====

Main text:

  * Roberto Bruni, Ugo Montanari, "[[http://www.springer.com/978-3-319-42898-7|Models of Computation]]", Springer Texts in Computer Science, 2017.

Other readings:

  * Graham Hutton, "[[http://www.cs.nott.ac.uk/~pszgmh/pih.html|Programming in Haskell]]", 2nd edition, Cambridge University Press (2016). Chapters: 1-8, 14, 15.
  * Joe Armstrong, [[https://pragprog.com/titles/jaerlang2/programming-erlang-2nd-edition/|Programming Erlang]], 2nd edition. The Pragmatic Bookshelf (2013). Chapters: 1-5, 8, 10-12.
  * Caleb Doxsey, [[http://shop.oreilly.com/product/0636920046516.do|Introducing Go]], O'Reilly Media (2016). Chapters: 1-4, 6-7, 10.
  * Robin Milner, "[[https://books.google.it/books/about/Communication_and_Concurrency.html?id=QPyyQgAACAAJ&redir_esc=y|Communication and Concurrency]]", Prentice Hall (1989). Chapters: 1-7, 10.
  * Luca Aceto, et al, "[[https://www.cambridge.org/core/books/reactive-systems/9B91FFD3996FC77D7AF0C309F8EE4CF7#|Reactive Systems]]", Cambridge University Press (2011). Chapters: 1-7.
  * Jane Hillston, [[http://www.dcs.ed.ac.uk/pepa/book.pdf|A Compositional Approach to Performance Modelling]], Cambridge University Press (1996). Chapters 1-3.

External resources:

  * [[https://www.haskell.org/|Haskell]]
  * [[http://www.erlang.org/|Erlang]]
  * [[http://golang.org/|Google Go]]
  * [[http://caal.cs.aau.dk/|CAAL]]
  * [[http://www.dcs.ed.ac.uk/pepa/|PEPA]]



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==== Exam ====

Normally, the evaluation would have been based on written and oral exams.

Due to the covid-19 countermeasures, for the current period, the evaluation will be solely based on oral exams. As an alternative to the assignment of exercises during the oral exam, students can develop a small project before the exam (contact the teacher if interested).

Registration to exams (mandatory): [[https://esami.unipi.it/esami/|Exams registration system]]

During the oral exam the student must demonstrate 
  * knowledge: his/her knowledge of the course material, and 
  * problem solving: the ability to solve some simple exercises, and
  * understanding: the ability to discuss the reading matter thoughtfully and with propriety of expression.

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==== Oral Exams: schedule ====

See the channel **Exams** in the Microsoft Teams platform

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==== Announcements ====
 
  * The lecture of **tuesday march 8 is canceled** and will be rescheduled in the upcoming weeks

  * Due to the covid-19 emergency situation, there will be no **mid-term exams**

  * as the course starts:\\ Each student must subscribe the [[https://teams.microsoft.com/l/team/19%3azTPYdF4TogSCVHaDQMahbTuiXN8HIrzE73OP01DoHh01%40thread.tacv2/conversations?groupId=78a9a5a2-4b1e-4f22-8d16-328ef5b60b2a&tenantId=c7456b31-a220-47f5-be52-473828670aa1|Microsoft Teams channel]] of the course and then send an email to the professor from his/her favourite email account with subject **PSC21-22** and the following data\\ (by doing so, the account will be included in the class mailing-list, where important announcements can be sent):
    - **first name** and **last name** (please clarify which is which, to avoid ambiguities)
    - **enrolment number** (numero di matricola)
    - bachelor degree (**course of study** and **university**)
    - your favourite topics in computer science (optional)

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==== Lectures (1st part) ====

[[https://teams.microsoft.com/l/team/19%3azTPYdF4TogSCVHaDQMahbTuiXN8HIrzE73OP01DoHh01%40thread.tacv2/conversations?groupId=78a9a5a2-4b1e-4f22-8d16-328ef5b60b2a&tenantId=c7456b31-a220-47f5-be52-473828670aa1|Virtual classroom]]: To join a lecture enter the course team on Microsoft Teams, go to the Lectures channel and click on the scheduled lecture.


^ N ^ Date ^^ Time ^ Room ^ Lecture notes ^ Links ^
|  1 | Mon | 14/02 | 11:00-13:00 | M1 | 01 - Introduction to the course\\ \\ 02 - Preliminaries:\\ //from syntax to semantics, the role of formal semantics// | {{ :magistraleinformatica:psc:2022-02-14_-_01_-_intro.pdf |Lecture 01}}\\ {{ :magistraleinformatica:psc:2022-02-15_-_02_-_semantics.pdf |Lecture 02}} |
|  2 | Tue | 15/02 | 14:00-16:00 | M1 | 02 - Preliminaries (ctd):\\ //SOS approach, small-step operational semantics, big-step operational semantics, denotational semantics, compositionality principle, normalisation, determinacy, consistency, equivalence, congruence// | {{ :magistraleinformatica:psc:2022-02-15_-_02_-_semantics.pdf |Lecture 02}} |
|  3 | Thu | 17/02 | 14:00-16:00 | M1 | 03 - Unification:\\ //inference process, signatures, substitutions, unification problem, most general unifiers, unification algorithm//\\ \\ 04 - Logical systems:\\ //logical systems, derivations, theorems// | {{ :magistraleinformatica:psc:2022-02-17_-_03_-_unification.pdf |Lecture 03}}\\ {{ :magistraleinformatica:psc:2022-02-21_-_04_-_logic.pdf |Lecture 04}} |
|  4 | Mon | 21/02 | 11:00-13:00 | M1 | 04 - Logical systems (ctd.):\\ //logic programs, goal-oriented derivations//\\ \\ 05 - Induction:\\ //precedence relation, infinite descending chains, well-founded relations, well-founded induction, mathematical induction, proof of induction principle// | {{ :magistraleinformatica:psc:2022-02-21_-_04_-_logic.pdf |Lecture 04}}\\ {{ :magistraleinformatica:psc:2022-02-22_-_05a_-_induction.pdf |Lecture 05a}} |
|  5 | Tue | 22/02 | 14:00-16:00 | M1 | 05 - Induction (ctd.):\\ //structural induction, termination of arithmetic expressions, determinacy of arithmetic expressions// | {{ :magistraleinformatica:psc:2022-02-22_-_05a_-_induction.pdf |Lecture 05a}}\\ {{ :magistraleinformatica:psc:2022-02-24_-_05b_-_more_induction.pdf |Lecture 05b}} |
|  6 | Thu | 24/02 | 14:00-16:00 | M1 | Exercises:\\ //unification, goal-oriented derivations//\\ \\ 05 - Induction (ctd.):\\ //many-sorted signatures, arithmetic and  boolean expressions// | {{ :magistraleinformatica:psc:01_-_inference_-_2022.pdf |Exercises 01}}\\ {{ :magistraleinformatica:psc:2022-02-24_-_05b_-_more_induction.pdf |Lecture 05b}} |
|  7 | Mon | 28/02 | 11:00-13:00 | M1 | 05 - Induction (ctd.):\\ //structural induction over many-sorted signatures, termination of boolean expressions, memories, update operation, operational semantics of commands, divergence, rule for divergence, limits of structural induction, induction on derivations, rule induction, determinacy of commands// | \\ {{ :magistraleinformatica:psc:2022-02-24_-_05b_-_more_induction.pdf |Lecture 05b}}\\ {{ :magistraleinformatica:psc:2022-02-28_-_05c_-_rule_induction.pdf |Lecture 05c}} |
|  8 | Tue | 01/03 | 14:00-16:00 | M1 | 06 - Equivalence:\\ //operational equivalence, concrete equivalences, parametric equivalences, equivalence and divergence//\\ \\ 07 - Induction and recursion:\\ //well-founded recursion, lexicographic precedence relation, Ackermann function, denotational semantics of arithmetic expressions// | {{ :magistraleinformatica:psc:2022-03-01_-_06_-_equivalence.pdf |Lecture 06}}\\ {{ :magistraleinformatica:psc:2022-03-01_-_07_-_recursion.pdf |Lecture 07}} |
|  9 | Thu | 03/03 | 14:00-16:00 | M1 | Exercises:\\ //induction, termination, determinacy, divergence//\\ \\ 07 - Induction and recursion (ctd.):\\ //consistency of operational and denotational semantics for arithmetic expressions, fixpoint equations//\\ \\ 08 - Partial orders and fixpoints:\\ //partial orders, Hasse diagrams// | {{ :magistraleinformatica:psc:02_-_properties_-_2022.pdf |Exercises 02}}\\ {{ :magistraleinformatica:psc:2022-03-01_-_07_-_recursion.pdf |Lecture 07}}\\ {{ :magistraleinformatica:psc:2022-03-07_-_08a_-_cpo.pdf |Lecture 08a}} |
| 10 | Mon | 07/03 | 11:00-13:00 | M1 | Exercises:\\ //x++, arithmetic expressions with side-effects//\\ \\ 08 - Partial orders and fixpoints (ctd):\\ //chains, least element, minimal element, bottom element, upper bounds, least upper bound, chains, limits, complete partial orders, powerset completeness// | {{ :magistraleinformatica:psc:2022-03-07_-_08a_-_cpo.pdf |Lecture 08a}}\\ {{ :magistraleinformatica:psc:2022-03-10_-_08b_-_kleene.pdf |Lecture 08b}} |
| - | Tue | 08/03 | 14:00-16:00 |  | **Canceled** |  |
| 11 | Thu | 10/03 | 14:00-16:00 | M1 | 08 - Partial orders and fixpoints (ctd.):\\ //prefix independence, CPO of partial functions, monotonicity, continuity, Kleene's fixpoint theorem, McCarthy's 91 function, recursive definitions of partial functions as logical systems, immediate consequences operator// | {{ :magistraleinformatica:psc:2022-03-10_-_08b_-_kleene.pdf |Lecture 08b}}\\ {{ :magistraleinformatica:psc:2022-03-14_-_08c_-_ico.pdf |Lecture 08c}} |
| 12 | Mon | 14/03 | 11:00-13:00 | M1 | 08 - Partial orders and fixpoints (ctd.):\\ //set of theorems as fixpoint//\\ \\ 09 - Denotational semantics:\\ //lambda-notation, free variables, capture-avoiding substitutions, alpha-conversion, beta rule, conditionals, denotational semantics of commands, fixpoint computation//\\ \\ Exercises:\\ //posets, semantics// | {{ :magistraleinformatica:psc:2022-03-14_-_08c_-_ico.pdf |Lecture 08c}}\\ {{ :magistraleinformatica:psc:2022-03-14_-_09_-_denotational_imp.pdf |Lecture 09}}\\ {{ :magistraleinformatica:psc:03_-_poset_-_2022.pdf |Exercises 03}} |
| 13 | Tue | 15/03 | 14:00-16:00 | M1 | 10 - Consistency:\\ //denotational equivalence, congruence, compositionality principle, consistency of commands, correctness, completeness//\\ \\ Exercises:\\ //well-founded recursion, posets, semantics// | {{ :magistraleinformatica:psc:2022-03-15_-_10_-_consistency_imp.pdf |Lecture 10}}\\ {{ :magistraleinformatica:psc:03_-_poset_-_2022.pdf |Exercises 03}} |
| 14 | Thu | 17/03 | 14:00-16:00 | M1 | 11 - Haskell:\\ //an overview//\\ \\ Haskell ghci:\\ //basics, tuples, lists, list comprehension, guards, pattern matching, lambda, partial application, zip, exercises// | [[https://www.haskell.org/|Haskell]]\\ {{ :magistraleinformatica:psc:2022-03-17_-_11_-_haskell.pdf |Lecture 11}}\\ {{ :magistraleinformatica:psc:ghci-session1-2022.txt.zip |ghci session 01}} |
| 15 | Mon | 21/03 | 11:00-13:00 | M1 | Haskell ghci (ctd):\\ //recursive definitions, tail recursion, let-in, where, map, filter, exercises// | [[https://www.haskell.org/|Haskell]]\\ {{ :magistraleinformatica:psc:ghci-session2-2022.txt.zip |ghci session 02}} |
| 16 | Tue | 22/03 | 14:00-16:00 | M1 | Haskell ghci (ctd.):\\ //fixpoint operator, folds, application, function composition, data types, type classes, recursive data structures, derived instances, exercises// | [[https://www.haskell.org/|Haskell]]\\ {{ :magistraleinformatica:psc:ghci-session3-2022.txt.zip |ghci session 03}} |
| 17 | Thu | 24/03 | 14:00-16:00 | M1 | Exercises:\\ //Haskell//\\ \\ 12 - HOFL:\\ //syntax, pre-terms, types, types judgements// | {{ :magistraleinformatica:psc:04_-_haskell-2022.pdf |Exercises 04}}\\ {{ :magistraleinformatica:psc:2022-03-24_-_12a_-_hofl_types.pdf |Lecture 12a}} |
| 18 | Mon | 28/03 | 11:00-13:00 | M1 | 12 - HOFL (ctd.):\\ //type system, type checking, type inference, principal type, canonical forms, operational semantics, lazy vs eager evaluation// | {{ :magistraleinformatica:psc:2022-03-24_-_12a_-_hofl_types.pdf |Lecture 12a}}\\ {{ :magistraleinformatica:psc:2022-03-28_-_12b_-_hofl_operational.pdf |Lecture 12b}} |
| 19 | Tue | 29/03 | 14:00-16:00 | M1 | 13 - Domain theory:\\ //Integers with bottom, cartesian product, projections, switching lemma, functional domains// | {{ :magistraleinformatica:psc:2022-03-29_-_13a_-_cartesian_domains.pdf |Lecture 13a}}\\ {{ :magistraleinformatica:psc:2022-03-29_-_13b_-_functional_domains.pdf |Lecture 13b}} |


==== Lectures (2nd part) ====

[[https://teams.microsoft.com/l/team/19%3azTPYdF4TogSCVHaDQMahbTuiXN8HIrzE73OP01DoHh01%40thread.tacv2/conversations?groupId=78a9a5a2-4b1e-4f22-8d16-328ef5b60b2a&tenantId=c7456b31-a220-47f5-be52-473828670aa1|Virtual classroom]]: To join a lecture enter the course team on Microsoft Teams, go to the Lectures channel and click on the scheduled lecture.


^ N ^ Date ^^ Time ^ Room ^ Lecture notes ^ Links ^
| 20 | Thu | 31/03 | 14:00-16:00 | M1 | 13 - Domain theory (ctd.):\\ //functional domains, lifting, let notation, continuity theorems//\\ \\ 14 - Denotational semantics of HOFL:\\ //definition and examples, type consistency// | {{ :magistraleinformatica:psc:2022-03-31_-_13c_-_continuity_theorems.pdf |Lecture 13c}}\\ {{ :magistraleinformatica:psc:2022-03-31_-_14_-_hofl_denotational.pdf |Lecture 14}} |
| 21 | Mon | 04/04 | 11:00-13:00 | M1 | 13 - Domain theory (ctd.):\\ //apply, fix, curry, uncurry//\\ \\ 14 - Denotational semantics of HOFL (ctd.):\\ //substitution lemma, compositionality and other properties//\\ \\ 15 - Consistency of HOFL:\\ //Counterexample to completeness, correctness of the operational semantics, operational convergence, denotational convergence, operational convergence implies denotational convergence (and vice versa), operational and denotational equivalence, correspondence for type int, unlifted semantics// | {{ :magistraleinformatica:psc:2022-03-31_-_13c_-_continuity_theorems.pdf |Lecture 13c}}\\ {{ :magistraleinformatica:psc:2022-03-31_-_14_-_hofl_denotational.pdf |Lecture 14}}\\ {{ :magistraleinformatica:psc:2022-04-04_-_15_-_consistency_hofl.pdf |Lecture 15}} |
| 22 | Tue | 05/04 | 14:00-16:00 | M1 | 15 - Consistency of HOFL (ctd.):\\ //lifted vs unlifted semantics//\\ \\ 16 - Erlang:\\ //an overview//\\ \\ Erlang erl:\\ //numbers, atoms, tuples, lists, terms, variables, term comparison, pattern matching, list comprehension, modules, functions, guards, higher order, recursion, pids, spawn, self, send, receive, examples// | {{ :magistraleinformatica:psc:2022-04-04_-_15_-_consistency_hofl.pdf |Lecture 15}}\\ {{ :magistraleinformatica:psc:2022-04-05_-_16_-_erlang.pdf |Lecture 16}}\\ [[http://www.erlang.org/|Erlang]]\\ {{ :magistraleinformatica:psc:erl_2022.zip |erl session}} |
| 23 | Thu | 07/04 | 14:00-16:00 | M1 | Exercises:\\ //HOFL, domains//\\ \\ 17 - CCS:\\ //Syntax, operational semantics//  | {{ :magistraleinformatica:psc:05_-_hofl-2022.pdf |Exercises 05}}\\ {{ :magistraleinformatica:psc:2022-04-07_-_17a_-_ccs.pdf |Lecture 17a}} |
| 24 | Mon | 11/04 | 11:00-13:00 | M1 | 17 - CCS (ctd.):\\ //value passing, finitely branching processes, guarded processes//\\ \\ 18 - Bisimulation:\\ //abstract semantics, graph isomorphism, trace equivalence// | {{ :magistraleinformatica:psc:2022-04-07_-_17a_-_ccs.pdf |Lecture 17a}}\\ {{ :magistraleinformatica:psc:2022-04-11_-_17b_-_ccs_guarded.pdf |Lecture 17b}}\\ {{ :magistraleinformatica:psc:2022-04-11_-_18a_-_ccs_abstract.pdf |Lecture 18a}} |
| 25 | Tue | 12/04 | 14:00-16:00 | M1 | 18 - Bisimulation (ctd.):\\ //bisimulation game, strong bisimulation, strong bisimilarity, strong bisimilarity is an equivalence, strong bisimilarity is a bisimulation, strong bisimilarity is the coarsest strong bisimulation, strong bisimilarity is a congruence, some laws for strong bisimilarity// | {{ :magistraleinformatica:psc:2022-04-12_-_18b_-_ccs_bisimulation.pdf |Lecture 18b}} |
| 26 | Thu | 14/04 | 14:00-16:00 | M1 | 18 - Bisimulation (ctd.):\\ //strong bisimilarity as a fixpoint, Phi operator, Phi is monotone, Phi is continuous (on finitely branching processes), Knaster-Tarski's fixpoint theorem//\\ \\ 19 - Hennessy-Milner logic:\\ //modalities, HML syntax, formula satisfaction, converse of a formula, HML equivalence// | {{ :magistraleinformatica:psc:2022-04-14_-_18c_-_ccs_bis_as_fix.pdf |Lecture 18c}}\\ {{ :magistraleinformatica:psc:2022-04-14_-_19_-_hml.pdf |Lecture 19}} |
| 27 | Tue | 26/04 | 14:00-16:00 | M1 | 20 - Weak Semantics:\\ //weak transitions, weak bisimulation, weak bisimilarity, weak bisimilarity is not  a congruence, weak observational congruence, Milner's tau-laws//\\ \\ 21 - CCS at work:\\ //modelling imperative programs with CCS, playing with CCS (using CAAL)// | {{ :magistraleinformatica:psc:2022-04-26_-_20_-_weak.pdf |Lecture 20}}\\ {{ :magistraleinformatica:psc:2022-04-26_-_21_-_ccs_at_work.pdf |Lecture 21}} |
| 28 | Thu | 28/04 | 14:00-16:00 | M1 | 21 - CCS at work:\\ //modelling and verification of mutual exclusion algorithms with CCS and CAAL//\\ \\ CAAL session (copy the text and paste it in the Edit panel) | [[http://caal.cs.aau.dk/|CAAL]]\\ {{ :magistraleinformatica:psc:caal-session-2022.zip |CAAL session}} |
| 29 | Mon | 02/05 | 11:00-13:00 | M1 | Exercises:\\ //Erlang, CCS//\\ \\ 22 - Temporal and modal logics:\\ //linear temporal logic (LTL), linear structures models, shifting, LTL satisfaction, equivalence of formulas, automata-like models// | {{ :magistraleinformatica:psc:06_-_erlang_-_ccs-2022.pdf |Exercises 06}}\\ {{ :magistraleinformatica:psc:2022-05-02_-_22a_-_ltl_ctl.pdf |Lecture 22a}} |
| 30 | Tue | 03/05 | 14:00-16:00 | M1 | 22 - Temporal and modal logics:\\ //computational tree logic (CTL* and CTL), infinite trees, infinite paths, branching structure, CTL* satisfaction, equivalence of formulas, CTL formulas, expressiveness comparison, mu-calculus, positive normal form, least and greatest fixpoints, invariant properties, possibly properties, mu-calculus with labels// | {{ :magistraleinformatica:psc:2022-05-03_-_22b_-_mu_calculus.pdf |Lecture 22b}} |
| 31 | Thu | 05/05 | 14:00-16:00 | M1 | 23 - GoogleGo:\\ //an overview//\\ \\ GoogleGo playground:\\ Go principles, variable declaration, type conversion, multiple assignments, type inference, imports, packages and public names, named return values, naked return, multiple results, conditionals and loops, pointers, struct, receiver arguments and methods, interfaces, goroutines, bidirectional channels, channel types, send, receive, asynchronous communication with buffering, close, select, communicating communication means, range, handling multiple senders, concurrent prime sieve | {{ :magistraleinformatica:psc:2022-05-05_-_23_-_google_go.pdf |Lecture 23}}\\ [[http://golang.org/|Google Go]]\\ {{ :magistraleinformatica:psc:go-session-2022.zip |go session}} |
| 32 | Mon | 09/05 | 14:00-16:00 | M1 | 24 - Pi-calculus:\\ //name mobility, free names, bound names, syntax and operational semantics, scope extrusion, early and late bisimilarities, weak semantics// | {{ :magistraleinformatica:psc:2022-05-09_-_24_-_pi_calculus.pdf |Lecture 24}} |
| 33 | Tue | 10/05 | 14:00-16:00 | M1 | 25 - Measure theory and Markov chains:\\ //probability space, random variables, stochastic processes, homogeneous Markov chains, DTMC, DTMC as matrices, DTMC as PTS, next state probability, finite path probability, ergodic DTMC, steady state distribution//   | {{ :magistraleinformatica:psc:2022-05-10_-_25a_-_dtmc.pdf |Lecture 25a}} |
| 34 | Thu | 12/05 | 14:00-16:00 | M1 | Exercises:\\ //logics, GoogleGo, pi-calculus//\\ \\ 25 - Measure theory and Markov chains:\\ //negative exponential distribution, CTMC, embedded DTMC, infinitesimal generator matrix, CTMC stationary distribution//\\ \\ 26 - Probabilistic bisimilarities:\\ //bisimilarity revisited, reachability predicate, CTMC bisimilarity.// | {{ :magistraleinformatica:psc:07_-_logics_-_googlego_-_pi_-_2022.pdf |Exercises 07}}\\ {{ :magistraleinformatica:psc:2022-05-12_-_25b_-_ctmc.pdf |Lecture 25b}}\\ {{ :magistraleinformatica:psc:2022-05-12_-_26_-_probabilistic_bisimilarities.pdf |Lecture 26}} |
| 35 | Mon | 16/05 | 11:00-13:00 | M1 | 26 - Probabilistic bisimilarities (ctd.):\\ //DTMC bisimilarity, Markov chains with actions, probabilistic reactive systems, bisimilarity for reactive systems, Larsen-Skou logic//\\ \\ 27 - PEPA:\\ //motivation, basic ideas, PEPA workflow, PEPA syntax, cooperation combinator, bounded capacity, apparent rate, PEPA operational semantics, performance analysis, reward structures// | {{ :magistraleinformatica:psc:2022-05-12_-_26_-_probabilistic_bisimilarities.pdf |Lecture 26}}\\ {{ :magistraleinformatica:psc:2022-05-16_-_27_-_pepa.pdf |Lecture 27}}\\ [[http://www.dcs.ed.ac.uk/pepa/|PEPA]] |
| 36 | Tue | 17/05 | 14:00-16:00 | M1 | Exercises:\\ //Markov chains, probabilistic systems, PEPA// | {{ :magistraleinformatica:psc:08_-_probability_-_2022.pdf |Exercises 08}}\\  |
| End |  |  |  |  |  |  |

==== Exam sessions ====

^ Date ^^ Time ^ Room ^ Info ^ 
| ... | ... | ... | Teams | Exam\\ [[https://esami.unipi.it/esami/|Exams registration system]]\\ The actual date of the oral exam will be agreed with the teacher |
| ... | ... | ... | Teams | Exam\\ [[https://esami.unipi.it/esami/|Exams registration system]]\\ The actual date of the oral exam will be agreed with the teacher |
| ... | ... | ... | Teams | Exam\\ [[https://esami.unipi.it/esami/|Exams registration system]]\\ The actual date of the oral exam will be agreed with the teacher |
| ... | ... | ... | Teams | Exam\\ [[https://esami.unipi.it/esami/|Exams registration system]]\\ The actual date of the oral exam will be agreed with the teacher |
| ... | ... | ... | Teams | Exam\\ [[https://esami.unipi.it/esami/|Exams registration system]]\\ The actual date of the oral exam will be agreed with the teacher |
| ... | ... | ... | Teams | Exam\\ [[https://esami.unipi.it/esami/|Exams registration system]]\\ The actual date of the oral exam will be agreed with the teacher |


----

==== Past courses ====

  * [[magistraleinformatica:psc:2020-21:|A.A. 2020/21]]
  * [[magistraleinformatica:psc:2019-20:|A.A. 2019/20]]
  * [[magistraleinformatica:psc:2018-19:|A.A. 2018/19]]
  * [[magistraleinformatica:psc:2017-18:|A.A. 2017/18]]
  * [[magistraleinformatica:mod:|MOD (A.A. 2016/17)]]
  * [[magistraleinformatica:mod:2015-16:|MOD (A.A. 2015/16)]]
  * [[magistraleinformatica:mod:2014-15:|MOD (A.A. 2014/15)]]
  * [[magistraleinformatica:mod:2013-14:|MOD (A.A. 2013/14)]]

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