- •Sense, Denotation and Semantics
- •Sense and denotation in logic
- •The algebraic tradition
- •The syntactic tradition
- •The two semantic traditions
- •Tarski
- •Heyting
- •Natural Deduction
- •The calculus
- •The rules
- •Interpretation of the rules
- •The Curry-Howard Isomorphism
- •Lambda Calculus
- •Types
- •Terms
- •Operational significance
- •Conversion
- •Description of the isomorphism
- •Relevance of the isomorphism
- •The Normalisation Theorem
- •The weak normalisation theorem
- •Proof of the weak normalisation theorem
- •Degree and substitution
- •Degree and conversion
- •Conversion of maximal degree
- •Proof of the theorem
- •The strong normalisation theorem
- •Sequent Calculus
- •The calculus
- •Sequents
- •Structural rules
- •The intuitionistic case
- •Logical rules
- •Some properties of the system without cut
- •The last rule
- •Subformula property
- •Asymmetrical interpretation
- •Sequent Calculus and Natural Deduction
- •Properties of the translation
- •Strong Normalisation Theorem
- •Reducibility
- •Properties of reducibility
- •Atomic types
- •Product type
- •Arrow type
- •Reducibility theorem
- •Pairing
- •Abstraction
- •The theorem
- •The calculus
- •Types
- •Terms
- •Intended meaning
- •Conversions
- •Normalisation theorem
- •Expressive power: examples
- •Booleans
- •Integers
- •Expressive power: results
- •Canonical forms
- •Representable functions
- •Coherence Spaces
- •General ideas
- •Coherence Spaces
- •The web of a coherence space
- •Interpretation
- •Stable functions
- •Parallel Or
- •Direct product of two coherence spaces
- •The Function-Space
- •The trace of a stable function
- •Representation of the function space
- •The Berry order
- •Partial functions
- •Denotational Semantics of T
- •Simple typed calculus
- •Types
- •Terms
- •Properties of the interpretation
- •Booleans
- •Integers
- •Sums in Natural Deduction
- •Defects of the system
- •Standard conversions
- •The need for extra conversions
- •Subformula Property
- •Extension to the full fragment
- •Commuting conversions
- •Properties of conversion
- •The associated functional calculus
- •Empty type
- •Sum type
- •Additional conversions
- •System F
- •The calculus
- •Comments
- •Representation of simple types
- •Booleans
- •Product of types
- •Empty type
- •Sum type
- •Existential type
- •Representation of a free structure
- •Free structure
- •Representation of the constructors
- •Induction
- •Representation of inductive types
- •Integers
- •Lists
- •Binary trees
- •Trees of branching type U
- •The Curry-Howard Isomorphism
- •Coherence Semantics of the Sum
- •Direct sum
- •Lifted sum
- •dI-domains
- •Linearity
- •Characterisation in terms of preservation
- •Linear implication
- •Linearisation
- •Linearised sum
- •Tensor product and units
- •Cut Elimination (Hauptsatz)
- •The key cases
- •The principal lemma
- •The Hauptsatz
- •Resolution
- •Strong Normalisation for F
- •Idea of the proof
- •Reducibility candidates
- •Remarks
- •Reducibility with parameters
- •Substitution
- •Universal abstraction
- •Universal application
- •Reducibility theorem
- •Representation Theorem
- •Representable functions
- •Numerals
- •Total recursive functions
- •Provably total functions
- •Proofs into programs
- •Formulation of HA2
- •Translation of HA2 into F
- •Representation of provably total functions
- •Semantics of System F
- •What is Linear Logic?
Chapter 3
The Curry-Howard Isomorphism
We have seen that Heyting's ideas perform very well in the framework of natural deduction. We shall exploit this remark by establishing a formal system of typed terms for discussing the functional objects which lie behind the proofs. The signi cance of the system will be given by means of the functional equations we have written down. In fact, these equations may be read in two di erent ways, which re-iterate the dichotomy between sense and denotation:
as the equations which de ne the equality of terms, in other words the equality of denotations (the static viewpoint).
as rewrite rules which allows us to calculate terms by reduction to a normal form. That is an operational, dynamic viewpoint, the only truly fruitful view for this aspect of logic.
Of course the second viewpoint is under-developed by comparison with therst one, as was the case in Logic! For example denotational semantics of programs (Scott's semantics, for example) abound: for this kind of semantics, nothing changes throughout the execution of a program. On the other hand, there is hardly any civilised operational semantics of programs (we exclude ad hoc semantics which crudely paraphrase the steps toward normalisation). The establishment of a truly operational semantics of algorithms is perhaps the most important problem in computer science.
The correspondence between types and propositions was set out in [Howard].
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