- •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?
60 |
CHAPTER 8. COHERENCE SPACES |
There remains another solution:
F3(ftg; ftg) = F3(ffg; ftg) = F3(ftg; ffg) = ftg
F3(ffg; ffg) = ffg
? otherwise.
The stability condition was used to eliminate the case of:
F0(ftg; ?) = F0(?; ftg) = ftg
What have we got against this example? It violates a principle of least data: we have F0(ftg; ftg) = ftg; we seek to nd a least approximant to the pair of arguments ftg; ftg which already gives ftg; now we have at our disposal ?; ftg and ftg; ? which are minimal (?; ? does not work) and distinct.
Of course, knowing that we always have a distinguished (least) solution (rather than many minimal solutions) for a problem of this kind radically simpli es a lot of calculations.
8.4Direct product of two coherence spaces
A function F of two arguments, mapping A; B to C is stable when:
i) a0 a 2 A ^ b0 b 2 B ) F (a0; b0) F (a; b)
" |
bj) = |
" |
F (ai; bj) (directed union) |
ii) F (Si"2I ai; Sj2J |
S(i;j)2I J |
iii) a1 [ a2 2 A ^ b1 [ b2 2 B ) F (a1 \ a2; b \ b2) = F (a1; b1) \ F (a2; b2)
Likewise we de ne stability in any number of arguments. Observe that, whereas separate continuity su ces for joint continuity, stability in two arguments is equivalent to stability in each separately, together with the additional condition that the pullback
I@
@
@
(a; b0) |
(a0; b) |
@I
(a0; b0) (where a0 a 2 A and b0 b 2 B) be preserved.
We would like to avoid studying stable functions of two (or more) variables and so reduce them to the unary case. For this we shall introduce the (direct) product A N B of two coherence spaces. The notation comes from linear logic.