Logical Frameworks

1
Logical Frameworks
FORMALWARE Engineering CISM Udine (Italy),
September 24-28, 2001

• Furio Honsell
• Professor of Foundations of Informatics
• Università di Udine, Italyhonsell_at_dimi.uniud.it

2
Outline of the Course

• Some introductory reflections on Logical
  Frameworks
• Theoretical and Practical motivations for
  computerized proof assistants based on
  Constructive Type Theory
• Encoding formal systems and Program Logics and
  Calculi in Type Theory
• Higher Order Abstract Syntax and Higher Order
  Natural Deduction
• Applications using the proof assistant for CIC
  COQ

3
The epistemological and technological milieu

• Dramatic need for dependable IT - e.g.
  life-critical software, mass-produced embedded
  software, digital woes (Wiener)
• Running tests is not enough how can we achieve
  greater reliability? Absolute validation?
• The only viable answer appears to be through
  rigorous analysis based on formal proofs using
  formal logic validated by mathematical models.
• Hence, the plethoric (?) multitude of Formal
  Methods in Computer Science specification
  languages, program logics and calculi,
  denotational and operational models, automata,
  etc.

4
A brief (western) history of proofs

• Plato (427-347 BC) Menos slave is made to
  recall a proof of an instance of Pythagoras
  Theorem(500 BC)
• Thales (546 BC), Aristotle (384-322 BC), Euclid
  (325-265 BC)
• Crucial in Mathematics and Logic, but not an
  object of formal mathematical study itself until
  the early XXth century Frege, Hilbert, Gentzen
• It becomes a first-class mathematical citizen
  only with Intuitionism (ca 1920) and Constructive
  Mathematics Brouwer, Heyting, Bishop
• De Bruijn (1968) Computer Assisted Formal Proof
  Checking
• Martin-Löf (1982) Computational contents of
  Proofs.

5
Absolute vs relative reliability

• Complete certainty is utopian Formal Methods can
  only increase our confidence
• At best we can prove formally that a given piece
  of software meets its formal specifications
• The exact correspondence of the formal
  specification to the real world is beyond any
  formal justification

6
Automated Deduction vs CAFR

• CAFR i.e. Computer Aided Formal Reasoning
• Building proofs, let alone formal proofs, is a
  creative activity, hence very difficult to
  automate.
• On the other hand, formal proof checking is
  routine, although it is a highly error-prone
  activity for the human mind, because of the often
  tedious and unperspicuous logical granularity.
• Semi-automated interactive proof assistants,
  such as Coq, are a good compromise
• Coq official site http//coq.inria.fr
  

7
Universal vs special-purpose formal
systems

• There are a number of universal computational
  models and Turing complete languages
• But experience in FM has indicated that there is
  no chance to come up with the ultimate, unique
  logical system.
• We have to give up any hope of finding a
  Reductionist Paradise in Computer Science
• We have to learn to live, with a multitude of
  different models/calculi/logics and
  special-purpose formalisms, and we have to be
  ready to develop new conceptually irreducibile
  frameworks.
• E.g. formalisms for concurrency, security,
  mobility, in the era of global computing

8
A plethora of formal systems

• l-calculus (Church 1936), p-calculus (Milner,
  Parrow, Walker 1992), n-calculus (Pitts,Stark
  1993), spi-calculus (Abadi,Gordon 1997), Ambient
  calculus (Cardelli,Gordon 1998), blu calculus
  etc.
• Process Algebras, CCS, CSP, Petri Nets
• Operational, Denotational, Logical Semantics
• Hoares, Modal, and Temporal Logics
• Constructive Type Theories
• We do not want to start over from first
  principles the meta-theory of each and everyone
  of these, nor to re-implement from scratch
  interactive tools for each of these.

9
How many abstraction levels?

• It appears to be convenient to distinguish at
  least two metalevels above the object one where
  hardware and software systems live.
• The semantical and syntactical formal systems, in
  the previous slide, which are normally used to
  specify and analyze the object level systems
  appear on this first metalevel.
• If we want to avoid duplication of efforts in
  carrying out proofs of metatheoretical properties
  of such systems or in developing tools for
  manipulating them, we need to conceive formal
  systems at yet a higher metalevel

10
Computational Metamodels

• This is the level where commonalities across
  different systems can be factored out and
  focused. This is the level where Computational
  Metamodels or Logical Frameworks live.
• But, even at this level uniqueness cannot be
  achieved. If we want to keep the mathematical
  overhead in representations to a minimum and have
  simple and transparent encodings we have to
  entertain more than one computational metamodel.

11
Metamodels, examples

• Logical Frameworks based on Constructive Type
  Theory (Martin-Lof, Plotkin, Harper, Honsell
  1880s)
• ASM (Gurevich 1990s)
• Action Structures and Calculi (Milner 1990s),
• PVS (Shankar)
• Rewriting Logics (Meseguer),
• Graph Grammars, as general term rewriting systems
  (Montanari, Ehrig),
• Many different kinds of double and enriched
  categories tiles (Montanari)

12
Framework theories

• Why not use FOL, HOL, Category Theory, Set
  Theory, as computational frameworks?
• Of course, these could be utilized, and indeed
  have been FOL, HOL, SETL,
• But such a broad generalty brings about yet more
  opaque encodings. It appears to be more
  convenient to focus on computation-specific
  pervasive ideas, especially if we want to keep an
  eye at building interactive tools. Some
  mechanisms are best factored out, i.e. abstracted
  away, or highlighted.

13
Pervasive concepts

• The inferential machinery of (higher order) rule
  application and hypothetico-general derivation
  combined with the generalization of algebraic
  context-free grammars, known as Higher Order
  Abstract Syntax (HOAS), in the case of Logical
  Frameworks
• Labeled Transition Systems and observation
  semantics based on bisimulation in the case of
  Action Calculi, Co-algebraic and Final
  Semantics, Tile Logics
• Term rewriting in the case of Graph Grammars
• ASM ? PVS?