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Principles for Software Composition

PSC 2020/21 (375AA, 9 CFU)

Lecturer: Roberto Bruni
web - email - Microsoft Teams channel

Office hours: Tuesday 15:00-17:00 or by appointment


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.


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.


Main text:

Other readings:

External resources:


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.

Registration to exams (mandatory): 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.

Oral Exams: schedule

Date Time Name Place
weekday DD/MM HH:MM Student Name Microsoft Teams


  • Due to the covid-19 emergency situation, there will be no mid-term exams
  • as the course starts:
    Each student must subscribe the Microsoft Teams channel of the course and then send an email to the professor from his/her favourite email account with subject PSC20-21 and the following data
    (by doing so, the account will be included in the class mailing-list, where important announcements can be sent):
    1. first name and last name (please clarify which is which, to avoid ambiguities)
    2. enrolment number (numero di matricola)
    3. bachelor degree (course of study and university)
    4. your favourite topics in computer science (optional)

Lectures (1st part)

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 15/02 16:00-18:00 Teams 01 - Introduction to the course Lecture 01
2 Wed 17/02 14:00-16:00 Teams 02 - Preliminaries:
from syntax to semantics, the role of formal semantics, SOS approach, small-step operational semantics, big-step operational semantics
Lecture 02
3 Fri 21/02 14:00-16:00 Teams 02 - Preliminaries (ctd.):
denotational semantics, compositionality principle, normalisation, determinacy, consistency, equivalence, congruence, signatures

03 - Unification:
inference process, signatures, substitutions,unification problem, unification algorithm
Lecture 03
4 Mon 22/02 16:00-18:00 Teams 04 - Logical systems:
logical systems, derivations, logic programs, goal-oriented derivations
Lecture 04
5 Wed 24/02 14:00-16:00 Teams 05 - Induction and recursion:
precedence relation, infinite descending chains, well-founded relations, well-founded induction, mathematical induction, proof of induction principle, structural induction, termination of arithmetic expressions, determinacy of arithmetic expressions
Lecture 05a
6 Fri 26/02 14:00-16:00 Teams 05 - Induction and recursion:
termination of arithmetic expressions

unification, goal-oriented derivations
Exercises 01
7 Mon 01/03 16:00-18:00 Teams 05 - Induction and recursion:
determinacy of arithmetic expressions, structural induction over many-sorted signatures, termination of boolean expressions, operational semantics of commands, divergence, induction on derivations
8 Wed 03/03 14:00-16:00 Teams 05 - Induction and recursion:
rule induction, determinacy of commands, well-founded recursion,
denotational semantics of arithmetic expressions,
consistency of operational and denotational semantics for arithmetic expressions
lexicographic precedence relation, Ackermann function, fixpoint equations
9 Fri 05/03 14:00-16:00 Teams Exercises:
induction and recursion, termination, determinacy, divergence

Partial orders and fixpoints:
partial orders, Hasse diagrams, chains, least element, minimal element,
bottom element, upper bounds, least upper bound, chains, limits
10 Teams Partial orders and fixpoints:
complete partial orders, powerset completeness,
CPO of partial functions, monotonicity, continuity, Kleene's fixpoint theorem
11 Teams Partial orders and fixpoints:
immediate consequences operator, denotational semantics of commands,
fixpoint computation, semantic equivalence, consistency of commands
12 Teams Exercises:
well-founded recursion, posets, semantics

Partial orders and fixpoints:
consistency of commands, completeness
13 Teams Lambda-notation:
free variables, capture-avoiding substitutions

an overview
ghci session 1
14 Teams Haskell:
lambda, guards, pattern matching, recursion, exercises
ghci session 2
15 Teams Haskell:
recursive definitions, folds, application, function composition
ghci session 3
16 Teams Haskell:
data types, type classes, recursive data structures, derived instances
ghci session 4

17 Teams Exercises:
ghci session 5

syntax, type system, type checking, type inference, principal type
18 Teams HOFL:
canonical forms, operational semantics, lazy vs eager evaluation

Domain theory:
Integers with bottom, cartesian product, projections
19 Teams Self-assessment test

Lectures (2nd part)

N Date Time Room Lecture notes Links
20 Teams Correction of self-assessment test
21 Teams Domain theory:
switching lemma, functional domains, lifting
22 Teams Domain theory:
continuity theorems, apply, fix, let notation

Denotational semantics of HOFL:
definition and examples, type consistency
23 Teams Denotational semantics of HOFL:
substitution lemma, compositionality and other properties

Consistency of HOFL:
Counterexample to completeness, correctness of the operational semantics,
operational convergence, denotational convergence,
operational convergence implies denotational convergence,
denotational convergence implies operational convergence (optional),
operational and denotational equivalence, correspondence for type int,
unlifted semantics
24 Teams Exercises:
HOFL, domains
25 Teams Erlang:
an overview
erl session
26 Teams CCS:
Syntax, operational semantics, finitely branching processes,
guarded processes, value passing, abstract semantics, graph isomorphism
27 Teams CCS:
trace equivalence, bisimulation game, strong bisimulation,
strong bisimilarity, strong bisimilarity is an equivalence,
strong bisimilarity as a fixpoint
28 Teams CCS:
strong bisimilarity as a fixpoint, Knaster-Tarski's fixpoint theorem,
strong bisimilarity is a congruence, some laws for strong bisimilarity,
Hennessy-Milner logic
29 Teams CCS:
weak bisimulation, weak bisimilarity, weak observational congruence,
Milner's tau-laws, modelling with CCS
30 Teams CCS:
modelling and playing with CCS (using CAAL)
CAAL session (copy the text and paste it in the Edit panel)
31 Teams Exercises:
Erlang, CCS

Temporal and modal logics:
linear temporal logic (LTL)
32 Teams Temporal and modal logics:
computational tree logic (CTL* and CTL), mu-calculus
33 Teams GoogleGo:
an overview
Google Go
34 Teams Pi-calculus:
syntax and operational semantics, examples
35 Teams Exercises:
logics, GoogleGo, pi-calculus

early and late bisimilarities, weak bisimilarities
36 Teams 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, negative exponential distribution
37 Teams Measure theory and Markov chains:
CTMC, embedded DTMC, infinitesimal generator matrix,
CTMC stationary distribution, bisimilarity revisited, reachability predicate,
CTMC bisimilarity, DTMC bisimilarity

Markov chains with actions:
reactive systems, bisimilarity for reactive systems, Larsen-Skou logic

motivation, basic ideas, PEPA workflow
38 Teams PEPA:
PEPA syntax, apparent rate, PEPA operational semantics,
performance analysis

probabilistic systems
- Teams Oral Exam
Exams registration system
- Teams Oral Exam
Exams registration system
- Teams Oral Exam
Exams registration system
- Teams Oral Exam
Exams registration system
- Teams Oral Exam
Exams registration system
- Teams Oral Exam
Exams registration system

Past courses

magistraleinformatica/psc/start.txt · Ultima modifica: 26/02/2021 alle 15:13 (2 giorni fa) da Roberto Bruni