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Principles for Software Composition

PSC 2023/24 (375AA, 9 CFU)

Lecturer: Roberto Bruni
web - email - Microsoft Teams channel

Office hours: By appointment (preferably on Tuesday 14:00-16:00)


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:


The evaluation will be solely based on oral exams, which can involve the assignment of written exercises.

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

See the channel Exams in the Microsoft Teams platform


  • as the course starts:
    Each student must subscribe the Microsoft Teams channel of the course and then fill the form Students information to provide the following contact data and info about her/his background:
    1. first name
    2. last name
    3. enrolment number (numero di matricola), optional
    4. email
    5. bachelor degree (course of study and university)
    6. MSc course (if Computer Science, specify which curriculum)
  • then, fill the (optional) form about your familiarity with some of the subjects of the course: Familiar subjects

Lectures (1st part)

N Date Time Room Lecture notes Links
1 Tue 20/02 16:00-18:00 C1 01 - Introduction to the course

02 - Preliminaries:
from syntax to semantics, the role of formal semantics, SOS approach, small-step operational semantics, big-step operational semantics
Lecture 01
Lecture 02
2 Thu 22/02 16:00-18:00 M1 02 - Preliminaries (ctd):
denotational semantics, compositionality principle, normalisation, determinacy, consistency, equivalence, congruence

03 - Unification:
inference process, signatures, substitutions, most general than relation, unification problem
Lecture 02
Lecture 03
3 Fri 23/02 14:00-16:00 C1 03 - Unification (ctd):
unification problem, most general unifiers, unification algorithm

04 - Logical systems:
logical systems, derivations, theorems, logic programs, goal-oriented derivations
Lecture 03
Lecture 04
4 Tue 27/02 16:00-18:00 C1 Exercises:
unification, goal-oriented derivations

05 - Induction:
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
Exercises 01
Lecture 05a
Lecture 05b
5 Thu 29/02 16:00-18:00 X3 Exercises:

05 - Induction:
many-sorted signatures, arithmetic and boolean expressions, structural induction over many-sorted signatures, termination of boolean expressions, memories, update operation, operational semantics of commands
Lecture 05b
6 Fri 01/03 14:00-16:00 C1 05 - Induction (ctd.):
divergence, rule for divergence, limits of structural induction, induction on derivations, rule induction, determinacy of commands
Lecture 05c
7 Tue 05/03 16:00-18:00 C1 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, fixpoint equations, consistency of operational and denotational semantics for arithmetic expressions
Lecture 06
Lecture 07
Lecture 08a
8 Thu 07/03 16:00-18:00 X3 Exercises:
induction, termination, determinacy, divergence

08 - Partial orders and fixpoints (ctd.):
partial orders, Hasse diagrams, chains, least element, minimal element, bottom element, upper bounds, least upper bound, limits, complete partial orders, powerset completeness, prefix independence, CPO of partial functions, monotonicity, continuity, Kleene's fixpoint theorem
Exercises 02
Lecture 08b
9 Fri 08/03 14:00-16:00 C1 08 - Partial orders and fixpoints (ctd.):
McCarthy's 91 function, recursive definitions of partial functions as logical systems, immediate consequences operator, 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
Lecture 08c
Lecture 09
10 10 - Consistency:
denotational equivalence, congruence, compositionality principle, consistency of commands, correctness, completeness

posets, semantics, well-founded recursion, posets, semantics
Lecture 10
Exercises 03
11 11 - Haskell:
an overview

Haskell ghci:
basics, tuples, lists, list comprehension, guards, pattern matching, lambda, partial application, zip, exercises
Lecture 11
ghci session 01
12 Haskell ghci (ctd):
recursive definitions, tail recursion, let-in, where, map, filter, fixpoint operator, folds, exercises
ghci session 02
13 Haskell ghci (ctd.):
folds, application, function composition, data types, type classes, recursive data structures, derived instances, exercises
ghci session 03
14 12 - HOFL:
syntax, pre-terms, types, types judgements, type system, type checking, type inference, principal type
Lecture 12a
15 Exercises:

12 - HOFL (ctd.):
canonical forms, operational semantics, lazy vs eager evaluation
Exercises 04
Lecture 12b
16 13 - Domain theory:
Integers with bottom, cartesian product, projections, switching lemma, functional domains
Lecture 13a
Lecture 13b
17 13 - Domain theory (ctd.):
lifting, let notation, continuity theorems, apply, fix

14 - Denotational semantics of HOFL:
definition and examples, type consistency
Lecture 13c
Lecture 14
18 13 - Domain theory (ctd.):
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, lifted vs unlifted semantics

HOFL, domains
Lecture 13c
Lecture 14
Lecture 15
Exercises 05
19 Exercises:
HOFL, domains

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
Exercises 05
Lecture 16
erl session
20 17 - CCS:
Syntax, operational semantics, value passing
Lecture 17a

Lectures (2nd part)

N Date Time Room Lecture notes Links
21 17 - CCS (ctd.):
finitely branching processes, guarded processes

18 - Bisimulation:
abstract semantics, graph isomorphism, trace equivalence, bisimulation game, strong bisimulation
Lecture 17b
Lecture 18a
Lecture 18b
22 18 - Bisimulation (ctd.):
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, strong bisimilarity as a fixpoint, Phi operator, Phi is monotone, Phi is continuous (on finitely branching processes), Knaster-Tarski's fixpoint theorem
Lecture 18b
Lecture 18c
23 19 - Hennessy-Milner logic:
modalities, HML syntax, formula satisfaction, converse of a formula, HML equivalence

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)
Lecture 19
Lecture 20
24 21 - CCS at work (ctd.):
modelling imperative programs with CCS, playing with CCS (using CAAL), modelling and verification of mutual exclusion algorithms with CCS and CAAL

CAAL session (copy the text and paste it in the Edit panel)
Lecture 21
CAAL session
25 Exercises:
Erlang, CCS

22 - Temporal and modal logics:
linear temporal logic (LTL), linear structures models, shifting, LTL satisfaction, equivalence of formulas, automata-like models
Exercises 06
Lecture 22a
26 22 - Temporal and modal logics (ctd.):
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
Lecture 22a
Lecture 22b
27 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
Lecture 23
Google Go
go session
28 24 - Pi-calculus:
name mobility, free names, bound names, syntax and operational semantics, scope extrusion, early and late bisimilarities, weak semantics
Lecture 24
29 Exercises:
logics, GoogleGo, pi-calculus

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, ergodic DTMC, steady state distribution
Exercises 07
Lecture 25a
30 25 - Measure theory and Markov chains (ctd):
finite path probability, negative exponential distribution, CTMC, embedded DTMC, infinitesimal generator matrix, CTMC stationary distribution

26 - Probabilistic bisimilarities:
bisimilarity revisited, reachability predicate, CTMC bisimilarity, DTMC bisimilarity, Markov chains with actions, probabilistic reactive systems, bisimilarity for reactive systems, Larsen-Skou logic
Lecture 25a
Lecture 25b
Lecture 26
31 27 - PEPA:
motivation, basic ideas, PEPA workflow, PEPA syntax, cooperation combinator, bounded capacity, apparent rate, PEPA operational semantics, performance analysis, reward structures
Lecture 27
32 Exercises:
Markov chains, probabilistic systems, PEPA
Exercises 08
33 Mini-projects discussion

Past courses

magistraleinformatica/psc/start.txt · Ultima modifica: 05/03/2024 alle 01:13 (4 ore fa) da Roberto Bruni