SlideShare une entreprise Scribd logo
1  sur  32
Télécharger pour lire hors ligne
Intuitionistic First-Order Logic
Categorical semantics via the Curry-Howard isomorphism




                                       Marco Benini

                                    M.Benini@leeds.ac.uk

                              Department of Pure Mathematics
                                    University of Leeds


                                   14th November 2012
Introduction




An observation: in part D of P. Johnstone’s Sketches of an Elephant, there
is a categorical semantics for the simply typed λ-calculus. In the very same
class of models, one can give a semantics to the corresponding fragment of
propositional logic.

The problem:
is it possible to do the same for full first-order intuitionistic logic?




 2 of 32
Introduction



Johstone’s account comes from Lambek and Scott’s Introduction to
Higher-Order Categorical Logic. The considered propositional logic is
minimal logic limited to conjunction and implication.

In Taylor, Practical Foundation of Mathematics, one finds that the treatment
of disjunction requires distributive categories in order to follow the same
pattern as the previous works.

To my knowledge, no categorical semantics appears in literature which
models the full first-order intuitionistic logic AND the corresponding
λ-calculus.




 3 of 32
The λ-calculus


Definition 1 (Lambda signature)
A λ-signature Σ = 〈S , F , R , Ax〉 is a structure where
1. 〈S , F , R 〉 is a logical signature, i.e.,
  1.1 a set S of sort symbols;
  1.2 a set F of function symbols, each one decorated as f : s1 × · · · × sn → s0 ,
      with s0 , . . . , sn ∈ S;
  1.3 a set R of relation symbols, each one decorated as r : s1 × · · · × sn , with
      s1 , . . . , sn ∈ S;
2. Ax is the set of axiom symbols, each one decorated as a : A → B where
   A, B ∈ λTypes(Σ) and FV(A → B ) = .
We call LTerms(Σ) the collection of logical terms constructed from the
signature Σ, assuming to have a denumerable set of variables Vs for each
s ∈ S.


 4 of 32
The λ-calculus


Definition 2 (Lambda type)
Fixed a λ-signature Σ, the λ-types on Σ are inductively defined along with
their free variables as follows:
1. 0, 1 ∈ λTypes(Σ) and FV(0) = FV(1) = ;
2. if p : s1 × · · · × sn ∈ R and t1 : s1 , . . . , tn : sn ∈ LTerms(Σ), then
   p(t1 , . . . , tn ) ∈ λTypes(Σ) and FV(p(t1 , . . . , tn )) = n=1 FV(ti : si );
                                                                     i
3. if A, B ∈ λTypes(Σ) then A × B , A + B , A → B ∈ λTypes(Σ) and
   FV(A × B ) = FV(A + B ) = FV(A → B ) = FV(A) ∪ FV(B );
4. if x ∈ Vs and A ∈ λTypes(Σ) then ∀x : s. A, ∃x : s. A ∈ λTypes(Σ) and
   FV(∀x : s. A) = FV(∃x : s. A) = FV(A)  {x : s}.




  5 of 32
The λ-calculus


Definition 3 (Lambda term)
Fixed a λ-signature Σ = 〈S , F , R , Ax〉, for each type t ∈ λTypes(Σ), we
assume there is a denumerable set Wt of (typed) variables.
A λ-term is inductively defined together with its free variables as:
1. if x ∈ Wt then x : t ∈ λTerms(Σ) and FV(x : t ) = {x : t };
2. if f : A → B ∈ Ax and t : A ∈ λTerms(Σ) then f (t ): B ∈ λTerms(Σ) and
   FV(f (t ): B ) = FV(t : A);
3. if s : A, t : B ∈ λTerms(Σ) then 〈s, t 〉 : A × B ∈ λTerms(Σ) and
   FV(〈s, t 〉 : A × B ) = FV(s : A) ∪ FV(t : B );
4. if t : A × B ∈ λTypes(Σ) then fst(t ): A ∈ λTerms(Σ),
   snd(t ): B ∈ λTerms(Σ) and
   FV(fst(t ): A) = FV(snd(t ): B ) = FV(t : A × B );                       →

 6 of 32
The λ-calculus
 → (Lambda term)
 5. if t : A ∈ λTerms(Σ) then inlB (t ): A + B ∈ λTerms(Σ),
    inrB (t ): B + A ∈ λTerms(Σ) and
    FV(inlB (t ): A + B ) = FV(inrB (t ): B + A) = FV(t : A);
 6. if s : A + B , t : A → C , r : B → C ∈ λTerms(Σ) then
    when(s, t , r ): C ∈ λTerms(Σ) and
    FV(when(s, t , r ): C ) = FV(s : A + B ) ∪ FV(t : A → C ) ∪ FV(r : B → C );
 7. if x ∈ WA and t : B ∈ λTerms(Σ) then (λx : A. t ): A → B ∈ λTerms(Σ) and
    FV((λx : A. t ): A → B ) = FV(t : B )  {x : A};
 8. if s : A → B , t : A ∈ λTerms(Σ) then s · t : B ∈ λTerms(Σ) and
    FV(s · t : B ) = FV(s : A → B ) ∪ FV(t : A);
 9. ∗ : 1 ∈ λTerms(Σ) and FV(∗ : 1) = ;
10. FA : 0 → A ∈ λTerms(Σ) and FV(FA : 0 → A) = ;                                 →

  7 of 32
The λ-calculus
 → (Lambda term)
11. if x ∈ Vs and t : A ∈ λTerms(Σ) where x : s ∈ FV∗ (t : A), then
    allI(λx : s. t ):(∀x : s. A) ∈ λTerms(Σ) and
    FV(allI(λx : s. t ):(∀x : s. A)) = FV(t : A);
12. if t :(∀x : s. A) ∈ λTerms(Σ) and r : s ∈ LTerms(Σ) then
    allE(t , r ):(A[r /x ]) ∈ λTerms(Σ) and
    FV(allE(t , r ):(A[r /x ])) = FV(t :(∀x : s. A));
13. if x ∈ Vs , r : s ∈ LTerms(Σ) and t :(A[r /x ]) ∈ λTerms(Σ) then
    exIx (t ):(∃x : s. A) ∈ λTerms(Σ) and
    FV(exIx (t ):(∃x : s. A)) = FV(t :(A[r /x ]));
14. if t :(∃x : s. A), ∈ λTerms(Σ) and r : A → B ∈ λTerms(Σ) where
    x : s ∈ FV∗ (r : A → B ), then exE(t , (λx : s. r )): B ∈ λTerms(Σ) and
    FV(exE(t , (λx : s. r )): B ) = FV(t :(∃x : s. A)) ∪ FV(r : A → B ).
 In the previous definition, x : s ∈ FV∗ (t : A) if and only if there is
 r ∈ λTypes(Σ) and y ∈ Wr such that x : s ∈ FV(r ) and y : r ∈ FV(t : A).
   8 of 32
The λ-calculus

    Definition 4 (Lambda calculus)
    A derivation is inductively defined by the following inference rules, whose
    antecedents and consequents are equalities-in-context within a fixed
    λ-signature Σ:
(eq0 ) x : A. s =C t    y : B . s[r1 /x1 , . . . , rn /xn ] =C t [r1 /x1 , . . . , rn /xn ] where, for
         any 1 ≤ i ≤ n, y : B . ri : Ai is a term-in-context;
                                   
            (x : A. s1 =B1 t1 ) 
                                
                      .
                      .
(eq1 )                .                  x : A. r [s/y ] =C r [t /y ];
                                   
           (x : A. sm =Bm tm )
                                   

(eq2 )      x : A. x =A x;
(eq3 ) x : A, y : A. x =A y       x : A, y : A. y =A x;
           (x : A, y : A, z : A. x =A y )
(eq4 )                                           x : A, y : A, z : A. x =A z;
           (x : A, y : A, z : A. y =A z )
                                                                                                         →
      9 of 32
The λ-calculus

    → (Lambda calculus)
(eq5 ) x : A. s =C t       x : A. (λy : B . s) =B →C (λy : B . t );
(eq6 ) x : A. r =C t      x : A. allI(λy : s. r ) =(∀y : s. C ) allI(λy : s. t );
(eq7 ) x : A. u =C v         x : A. exE(t , (λy : s. u )) =C exE(t , (λy : s. v ));
(×0 )       x : 1. x =1 ∗;
(×1 )       x : A, y : B . fst(〈x , y 〉) =A x;
(×2 )       x : A, y : B . snd(〈x , y 〉) =B y ;
(×3 )       z : A × B . 〈fst(z ), snd(z )〉 =A×B z;
(+0 )       x : A. when(inlB (a), t , s) =C t · a;
(+1 )       x : A. when(inrD (b), t , s) =C s · b;
                                                                                      →


     10 of 32
The λ-calculus
    → (Lambda calculus)
(+2 ) when y : A1 ∈ FV(x1 : A1 + A2 ) ∪ FV(x3 : B1 → C ) ∪ FV(x4 : B2 → C ) and
      y : A2 ∈ FV(x2 : A1 + A2 ) ∪ FV(x3 : B1 → C ) ∪ FV(x4 : B2 → C )
                        x0 : A1 + A2 , x1 : A1 → (B1 + B2 ), x2 : A2 → (B1 + B2 ),
                        x3 : B1 → C , x4 : B2 → C .
                         when(when(x0 , x1 , x2 ), x3 , x4 ) =C
                         =C when(x0 , (λy : A1 . when(x1 · y , x3 , x4 )),
                                         (λy : A2 . when(x2 · y , x3 , x4 ))) ;
 (+3 )      x : A, y : 0. FA ·y =A x;
(→0 )       x : A. (λy : C . s) · t =B s[t /y ];
(→1 )       x : A. (λy : C . t · y ) =C →B t where y : C ∈ FV(t : C → B );
 (∀0 )      x : A. allE(allI(λz : s. t ), r ) =B [r /z ] t [r /z ];
(∀1 )     x : A. allE(u , r ) =B allE(v , r ) r : s∈LTerms(Σ)   x : A. u =(∀z : s. B ) v ;
                                                                                             →
     11 of 32
The λ-calculus


   → (Lambda calculus)
(∃0 )      x : A. exE(exIz (t ), (λz : s. v )) =B (v [r /z ]) · t;
(∃1 ) x : A. exE(u , (λz : s. r )) =B exE(u , (λz : s. t ))          x : A. r =C →B t where
      FV(r : C → B ) = FV(t : C → B );
(∃2 )      v :(∃y : s. A). w =B exE(v , (λy : s. (λz : A. w [exIy (z )/v ]))) with
        z : A ∈ FV(w : B );
(∃3 )          x : A. exE(exE(a, (λy : s. (λz : D . b))), (λy : s. c )) =C
                      =C exE(a, (λy : s. (λz : D . exE(b, (λy : s. c )))));
(∃4 )      x : A. exE(a, (λy : s. (λz : C . b[exIy (z )/w ]))) =B b[a/w ] with
        z : C ∈ FV(b : B ).




    12 of 32
Semantics

Definition 5 (Logically distributive category)
Fixed a λ-signature Σ = 〈S , F , R , Ax〉, a category C together with a map
M : λTypes(Σ) → Obj C is said to be logically distributive if it satisfies the
following seven conditions:
1. C has finite products;
2. C has finite co-products;
3. C has exponentiation;
4. C is distributive, i.e., for every A, B , C ∈ Obj C, the arrow
   ∆ = [1A × ι1 , 1A × ι2 ]: (A × B ) + (A × C ) → A × (B + C ) has an inverse,
   where [_, _] is the co-universal arrow of the (A × B ) + (A × C ) co-product,
   _ × _ is the product arrow, 1A is the identity arrow on A, and
   ι1 : B → B + C, ι2 : C → B + C are the canonical injections of the B + C
   co-product.
                                                                                →
 13 of 32
Semantics


→ (Logically distributive category)
For every s ∈ S, A ∈ λTypes(Σ), and x ∈ Vs , let
ΣA (x : s): LTerms(Σ)(s) → C be the functor from the discrete category
LTerms(Σ)(s) = t : s | t : s ∈ LTerms(Σ) to C defined by t : s → M (A[t /x ]).

Also, for every s ∈ S, A ∈ λTypes(Σ), and x ∈ Vs , let C(∀x : s. A) be the
subcategory of C whose objects are the vertices of the cones on ΣA (x : s)
such that they are of the form MB for some B ∈ λTypes(Σ) and
x : s ∈ FV(B ). Moreover, the arrows of C(∀x : s. A) , apart identities, are the
arrows in the category of cones over ΣA (x : s) having the objects of
C(∀x : s. A) as domain and M (∀x : s. A) as co-domain.                           →




 14 of 32
Semantics


→ (Logically distributive category)
Finally, for every s ∈ S, A ∈ λTypes(Σ), and x ∈ Vs , let C(∃x : s. A) be the
subcategory of C whose objects are the vertices of the co-cones on
ΣA (x : s) such that they are of the form MB for some B ∈ λTypes(Σ) and
x : s ∈ FV(B ). Moreover, the arrows of C(∃x : s. A) , apart identities, are the
arrows in the category of co-cones over ΣA (x : s) having the objects of
C(∃x : s. A) as co-domain and M (∃x : s. A) as domain.

5. All the subcategories C(∀x : s. A) have terminal objects, and all the
   subcategories C(∃x : s. A) have initial objects;
                                                                                   →




 15 of 32
Semantics

→ (Logically distributive category)
6. The M map is such that
  6.1 M (0) = 0, the initial object of C;
  6.2 M (1) = 1, the terminal object of C;
  6.3 M (A × B ) = MA × MB, the binary product in C;
  6.4 M (A + B ) = MA + MB, the binary co-prooduct in C;
  6.5 M (A → B ) = MB MA , the exponential object in C;
  6.6 M (∀x : s. A) is the terminal object in the subcategory C(∀x : s. A) ;
  6.7 M (∃x : s. A) is the initial object in the subcategory C(∃x : s. A) ;
7. For every x ∈ Vs , A, B ∈ λTypes(Σ) with x : s ∈ FV(A), MA × M (∃x : s. B )
   is an object of C(∃x : s. A×B ) since, if M (∃x : s. B ), {δt }t : s∈LTerms(Σ) is a
   co-cone over ΣB (x : s), and there is one by condition (5), then
    MA × M (∃x : s. B ), {1MA ×δt }t : s∈LTerms(Σ) is a co-cone over ΣA×B (x : s).
   Thus, there is a unique arrow !: M (∃x : s. A × B ) → MA × M (∃x : s. B ) in
   C(∃x : s. A×B ) . Our last condition requires that the arrow ! has an inverse.

 16 of 32
Semantics




Definition 6 (Σ-structure)
Given a λ-signature Σ = 〈S , F , R , Ax〉, a Σ-structure is a triple 〈C, M , MAx 〉
such that C together with M forms a logically distributive category and MAx
is a map from Ax such that MAx (a : A → B ) ∈ HomC (MA, MB ).




 17 of 32
Semantics

Definition 7 (λ-terms semantics)
Fixed a Σ-structure 〈C, M , MAx 〉, let A ≡ A1 × · · · An , and let
x ≡ x1 : A1 , . . . , xn : An be a context. The semantics of a term-in-context
x . t : B, notation x . t : B , is an arrow in HomC (MA, MB ) inductively defined
as follows:
1.    x . xi : Ai = πi , the i-th projector of the product MA = MA1 × · · · × MAn ;
2. if a : C → B ∈ Ax then x . a(t ): B = MAx a ◦ x . t : C ;
3.    x . 〈s, t 〉 : B × C = ( x . s : B , x . t : C ) where (_, _) is the universal arrow
     of the product MB × MC;
4.    x . fst(t ): B = π1 ◦ x . t : B × C where π1 is the first canonical projector
     of the product MA × MB;
5.    x . snd(t ): C = π2 ◦ x . t : B × C where π2 is the second canonical
     projector of the product MA × MB;
                                                                                      →
 18 of 32
Semantics


→ (λ-terms semantics)
6.    x . (λz : C . t ): C → B is the exponential transpose of
      x , z : C . t : B : MA × MC → MB;
7.    x . s · t : B = ev ◦ ( x . s : C → B , x . t : C ) where ev is the exponential
     evaluation arrow;
8.    x . inlB (t ): C + B = ι1 ◦ x . t : C with ι1 the first canonical injection of the
     co-product MC + MB;
9.    x . inrC (t ): C + B = ι2 ◦ x . t : B with ι2 the second canonical injection of
     the co-product MC + MB;
                                                                                       →




 19 of 32
Semantics
 → (λ-terms semantics)
10. calling [_, _] the co-universal arrow of (MA × MC1 ) + (MA × MC2 ), (_, _)
    the universal arrow of MA × (MC1 + MC2 ), and noticing that the arrow
    ∆ : (MA × MC1 ) + (MA × MC2 ) → MA × (MC1 + MC2 ) has an inverse
    because C with M is logically distributive

               x . when(t , u , v ): B = [ev ◦ ( x . u : C1 → B × 1MC1 ) ,
                                         ev ◦ ( x . v : C2 → B × 1MC2 )] ◦
                                        ◦ ∆−1 ◦ (1MA , x . t : C1 + C2 ) ;

11.    x . ∗ : 1 =!: MA → 1, the universal arrow of the terminal object;
12.    x . FB : 0 → B is the exponential transpose of
      (!: 0 → MB ) ◦ (πn+1 : MA × 0 → 0);
                                                                             →

  20 of 32
Semantics
 → (λ-terms semantics)
13.    x . allI(λz : s. t ):(∀z : s. B ) = β ◦ α where
      α ≡ 1MAi1 × · · · × 1MAik : MA → MA with A ≡ Ai1 × · · · × Aik , where
      x ≡ {xi1 : Ai1 , . . . , xik : Aik } = FV(t : B ), and β : MA → M (∀z : s. B ) is the
      universal arrow from MA to the terminal object in C∀z : s. B ;
14.     x . allE(t , r ): B [r /z ] = pr ◦ x . t :(∀z : s. B ) where
      pr : M (∀z : s. B ) → M (B [r /z ]) is the r -th projector of the unique cone on
      ΣB (z : s) whose vertex is M (∀z : s. B ).
      It is worth noticing that pr = w :(∀z : s. B ). allE(w , r ): B [r /z ] ;
15.     x . exIz (t ):(∃z : s. B ) = jr ◦ x . t : B [r /z ] where
      jr : M (B [r /z ]) → M (∃z : s. B ) is the r -th injection of the unique co-cone
      on ΣB (z : s) whose vertex is M (∃z : s. B ).
      It is worth noticing that jr = w : B [r /z ]. exIz (w ):(∃z : s. B ) ;
                                                                                              →

  21 of 32
Semantics




 → (λ-terms semantics)
16.  x . exE(t , (λz : s. r )): B = γ ◦ β−1 ◦ (α, x . t :(∃z : s. C ) ) where
  16.1 α ≡ 1MAi1 × · · · × 1MAik : MA → MA with A ≡ Ai1 × · · · × Aik , where
        x ≡ {xi1 : Ai1 , . . . , xik : Aik } = FV(t :(∃z : s. C )) ∪ FV(r : C → B );
  16.2 β : M (∃z : s. A × C ) → MA × M (∃z : s. C ) is the co-universal arrow in the
        subcategory C∃z : s. A ×C ;
  16.3 γ : M (∃z : s. A × C ) → MB is the co-universal arrow in C∃z : s. A ×C .




  22 of 32
Soundness



Definition 8 (Validity)
An equality-in-context x . s =A t is valid in the λ-theory T , a set of
equalities-in-context, when, in every logically distributive category C, each
model M of T is also a model of x . s =A t.
A Σ-structure M in C is a model of a theory T when it is a model of each φ
in T .
Finally, M is a model of an equality-in-context x . t =A s if x . t : A = x . s : A .

Theorem 9 (Soundness)
If an equation-in-context x . s =A t is derivable from a λ-theory T , then
x . s =A t is valid in each model of T in every logically distributive category.




 23 of 32
Completeness



Definition 10 (Syntactical equivalence)
Given a λ-theory T , the syntactical equivalence of two terms-in-context is
defined by fixing the generated equivalence classes. Precisely, the
equivalence class [x : A. t : B ] is defined as the minimal set, composed by
terms-in-context, such that
1. x : A. t : B ∈ [x : A. t : B ]—reflexivity;
2. if T y : D . s =C r , where y : D . s =C r is an equality-in-context, and
   y : D . s : C ∈ [x : A. t : B ], then y : D . r : C ∈ [x : A. t : B ]—closure under
   provable equivalence;
                                                                                         →



 24 of 32
Completeness
→ (Syntactical equivalence)
3. if y : D . s : C is a term-in-context and, for some 1 ≤ i < m and
   z : Di × Di +1 ∈ FV(s : C ) ∪ y1 : D1 , . . . , ym : Dm , it happens that

            y1 : D1 , . . . , yi −1 : Di −1 , z : Di × Di +1 , yi +1 : Di +2 ,
                      . . . , ym : Dm . s[fst(z )/yi ][snd(z )/yi +1 ]: C ∈ [x : A. t : B ] ,

    then y : D . s : C ∈ [x : A. t : B ]—closure under associativity in contexts;
4. if y : D . s : C is a term-in-context and, for some 1 ≤ i < m and
   z : Di +1 × Di ∈ FV(s : C ) ∪ y1 : D1 , . . . , ym : Dm , it happens that

            y1 : D1 , . . . , yi −1 : Di −1 , z : Di +1 × Di , yi +1 : Di +2 ,
                      . . . , ym : Dm . s[snd(z )/yi ][fst(z )/yi +1 ]: C ∈ [x : A. t : B ] ,

    then y : D . s : C ∈ [x : A. t : B ]—closure under commutativity in contexts;
                                                                                                →
 25 of 32
Completeness




→ (Syntactical equivalence)
5. if y : D . s : C ∈ [x : A. y : B ] and z : Di ∈ FV(s : C ) ∪ y1 : D1 , . . . , ym : Dm for
   some 1 ≤ i ≤ m, then

               y1 : D1 , . . . , yi −1 : Di −1 , z : Di , yi +1 : Di +1 , . . . , ym : Dm . s[z /yi ]: C

    is in [x : A. t : B ]—closure under α-renaming in contexts.




 26 of 32
Completeness



Definition 11 (Syntactical category)
Given a λ-theory T , the syntactical category CT has λTypes(Σ) as objects,
where Σ is the λ-signature of T , and the equivalence classes
[x : A. t : B ]: A → B as arrows.
Identities are given by the classes [x : A. x : A]: A → A for each λ-type A,
and composition is given by substitution:

                [y : B . s : C ] ◦ [x : A. t : B ] = [x : A. s[t /y ]: C ] .

Moreover, the map MT : λTypes(Σ) → Obj CT is defined as MT A = A.




 27 of 32
Completeness




Proposition 12
The CT category is logically distributive.

Proposition 13
Given a λ-theory T on the Σ signature, the Σ-structure 〈CT , MT , MAx 〉 on
the corresponding syntactical category is defined by MAx which maps
f : A → B ∈ Ax to [x : A. f (x ): B ].
This Σ-structure is a model for T and, moreover, it satisfies exactly those
equalities-in-context which are provable in T .




 28 of 32
Completeness



Proposition 14
For every logically distributive category C, there is a biijection between
equivalence classes, modulo natural equivalences, of structure-preserving
functors CT → C and equivalence classes, modulo isomorphisms, of
models of T in C, induced by the map F → F (MT ).

Theorem 15 (Completeness)
If x . s =A t is an equality-in-context valid in every model for T in each
logically distributive category, then T x . s =A t.




 29 of 32
Soundness and Completeness in Logic



Definition 16 (Valid type)
A λ-type A is valid in the model N = 〈N, N , NAx 〉 when there exists an
arrow 1 → NA in N.

A λ-type A is a logical consequence in the model N of the λ-types
B1 , . . . , Bn when there exists N (B1 × · · · × Bn ) → NA in N.

A λ-type A is a logical consequence of B1 , . . . , Bn when it is a logical
consequence of B1 , . . . , Bn in every model in every logically distributive
category.




 30 of 32
Soundness and Completeness in Logic




Proposition 17
A λ-type A is a logical consequence of B1 , . . . , Bn if and only if there exists a
term-in-context x : B1 × · · · × Bn . t : A.

Corollary 18
A λ-type A is a logical consequence of B1 , . . . , Bn if and only if there is a
proof of A from the hypotheses B1 , . . . , Bn , when λ-types are interpreted as
logical formulae and λ-terms as logical proofs, according to the
Curry-Howard isomorphism.




 31 of 32
The end




           Questions?




32 of 32

Contenu connexe

Tendances

Notes parabolas
Notes   parabolasNotes   parabolas
Notes parabolasLori Rapp
 
16 partial derivatives
16 partial derivatives16 partial derivatives
16 partial derivativesmath267
 
12 graphs of second degree functions x
12 graphs of second degree functions x12 graphs of second degree functions x
12 graphs of second degree functions xmath260
 
27 triple integrals in spherical and cylindrical coordinates
27 triple integrals in spherical and cylindrical coordinates27 triple integrals in spherical and cylindrical coordinates
27 triple integrals in spherical and cylindrical coordinatesmath267
 
4 areas in polar coordinates
4 areas in polar coordinates4 areas in polar coordinates
4 areas in polar coordinatesmath267
 
2 polar graphs
2 polar graphs2 polar graphs
2 polar graphsmath267
 
1 polar coordinates
1 polar coordinates1 polar coordinates
1 polar coordinatesmath267
 
13 graphs of factorable polynomials x
13 graphs of factorable polynomials x13 graphs of factorable polynomials x
13 graphs of factorable polynomials xmath260
 
11 graphs of first degree functions x
11 graphs of first degree functions x11 graphs of first degree functions x
11 graphs of first degree functions xmath260
 
t7 polar equations and graphs
t7 polar equations and graphst7 polar equations and graphs
t7 polar equations and graphsmath260
 
3 graphs of second degree functions x
3 graphs of second degree functions x3 graphs of second degree functions x
3 graphs of second degree functions xTzenma
 
Yoneda lemma and string diagrams
Yoneda lemma and string diagramsYoneda lemma and string diagrams
Yoneda lemma and string diagramsRay Sameshima
 
10 rectangular coordinate system x
10 rectangular coordinate system x10 rectangular coordinate system x
10 rectangular coordinate system xmath260
 
Problems and solutions inmo-2012
Problems and solutions  inmo-2012Problems and solutions  inmo-2012
Problems and solutions inmo-2012askiitians
 
Computer Science and Information Science 4th semester (2012-June Question
Computer Science and Information Science 4th semester (2012-June Question Computer Science and Information Science 4th semester (2012-June Question
Computer Science and Information Science 4th semester (2012-June Question B G S Institute of Technolgy
 

Tendances (20)

Notes parabolas
Notes   parabolasNotes   parabolas
Notes parabolas
 
Afa 2020
Afa 2020Afa 2020
Afa 2020
 
Parabola
ParabolaParabola
Parabola
 
16 partial derivatives
16 partial derivatives16 partial derivatives
16 partial derivatives
 
12 graphs of second degree functions x
12 graphs of second degree functions x12 graphs of second degree functions x
12 graphs of second degree functions x
 
27 triple integrals in spherical and cylindrical coordinates
27 triple integrals in spherical and cylindrical coordinates27 triple integrals in spherical and cylindrical coordinates
27 triple integrals in spherical and cylindrical coordinates
 
4 areas in polar coordinates
4 areas in polar coordinates4 areas in polar coordinates
4 areas in polar coordinates
 
2 polar graphs
2 polar graphs2 polar graphs
2 polar graphs
 
1 polar coordinates
1 polar coordinates1 polar coordinates
1 polar coordinates
 
13 graphs of factorable polynomials x
13 graphs of factorable polynomials x13 graphs of factorable polynomials x
13 graphs of factorable polynomials x
 
11 graphs of first degree functions x
11 graphs of first degree functions x11 graphs of first degree functions x
11 graphs of first degree functions x
 
t7 polar equations and graphs
t7 polar equations and graphst7 polar equations and graphs
t7 polar equations and graphs
 
Math1.2
Math1.2Math1.2
Math1.2
 
1576 parabola
1576 parabola1576 parabola
1576 parabola
 
3 graphs of second degree functions x
3 graphs of second degree functions x3 graphs of second degree functions x
3 graphs of second degree functions x
 
Yoneda lemma and string diagrams
Yoneda lemma and string diagramsYoneda lemma and string diagrams
Yoneda lemma and string diagrams
 
10 rectangular coordinate system x
10 rectangular coordinate system x10 rectangular coordinate system x
10 rectangular coordinate system x
 
Problems and solutions inmo-2012
Problems and solutions  inmo-2012Problems and solutions  inmo-2012
Problems and solutions inmo-2012
 
Computer Science and Information Science 4th semester (2012-June Question
Computer Science and Information Science 4th semester (2012-June Question Computer Science and Information Science 4th semester (2012-June Question
Computer Science and Information Science 4th semester (2012-June Question
 
130553704223906250
130553704223906250130553704223906250
130553704223906250
 

En vedette

Jarrar: First Order Logic- Inference Methods
Jarrar: First Order Logic- Inference MethodsJarrar: First Order Logic- Inference Methods
Jarrar: First Order Logic- Inference MethodsMustafa Jarrar
 
Jarrar: First Order Logic
Jarrar: First Order LogicJarrar: First Order Logic
Jarrar: First Order LogicMustafa Jarrar
 
Class first order logic
Class first order logicClass first order logic
Class first order logicchandsek666
 
First order logic in knowledge representation
First order logic in knowledge representationFirst order logic in knowledge representation
First order logic in knowledge representationSabaragamuwa University
 
Using Controlled Natural Language and First Order Logic to improve e-consulta...
Using Controlled Natural Language and First Order Logic to improve e-consulta...Using Controlled Natural Language and First Order Logic to improve e-consulta...
Using Controlled Natural Language and First Order Logic to improve e-consulta...jodischneider
 
Artificial Intelligence: Knowledge Engineering
Artificial Intelligence: Knowledge EngineeringArtificial Intelligence: Knowledge Engineering
Artificial Intelligence: Knowledge EngineeringThe Integral Worm
 
Propositional And First-Order Logic
Propositional And First-Order LogicPropositional And First-Order Logic
Propositional And First-Order Logicankush_kumar
 

En vedette (9)

Jarrar: First Order Logic- Inference Methods
Jarrar: First Order Logic- Inference MethodsJarrar: First Order Logic- Inference Methods
Jarrar: First Order Logic- Inference Methods
 
Discrete Math Lecture 02: First Order Logic
Discrete Math Lecture 02: First Order LogicDiscrete Math Lecture 02: First Order Logic
Discrete Math Lecture 02: First Order Logic
 
Jarrar: First Order Logic
Jarrar: First Order LogicJarrar: First Order Logic
Jarrar: First Order Logic
 
Class first order logic
Class first order logicClass first order logic
Class first order logic
 
First order logic in knowledge representation
First order logic in knowledge representationFirst order logic in knowledge representation
First order logic in knowledge representation
 
First order logic
First order logicFirst order logic
First order logic
 
Using Controlled Natural Language and First Order Logic to improve e-consulta...
Using Controlled Natural Language and First Order Logic to improve e-consulta...Using Controlled Natural Language and First Order Logic to improve e-consulta...
Using Controlled Natural Language and First Order Logic to improve e-consulta...
 
Artificial Intelligence: Knowledge Engineering
Artificial Intelligence: Knowledge EngineeringArtificial Intelligence: Knowledge Engineering
Artificial Intelligence: Knowledge Engineering
 
Propositional And First-Order Logic
Propositional And First-Order LogicPropositional And First-Order Logic
Propositional And First-Order Logic
 

Similaire à Intuitionistic First-Order Logic: Categorical semantics via the Curry-Howard isomorphism

FINAL PROJECT, MATH 251, FALL 2015[The project is Due Mond.docx
FINAL PROJECT, MATH 251, FALL 2015[The project is Due Mond.docxFINAL PROJECT, MATH 251, FALL 2015[The project is Due Mond.docx
FINAL PROJECT, MATH 251, FALL 2015[The project is Due Mond.docxvoversbyobersby
 
Nbhm m. a. and m.sc. scholarship test 2006
Nbhm m. a. and m.sc. scholarship test 2006Nbhm m. a. and m.sc. scholarship test 2006
Nbhm m. a. and m.sc. scholarship test 2006MD Kutubuddin Sardar
 
00_1 - Slide Pelengkap (dari Buku Neuro Fuzzy and Soft Computing).ppt
00_1 - Slide Pelengkap (dari Buku Neuro Fuzzy and Soft Computing).ppt00_1 - Slide Pelengkap (dari Buku Neuro Fuzzy and Soft Computing).ppt
00_1 - Slide Pelengkap (dari Buku Neuro Fuzzy and Soft Computing).pptDediTriLaksono1
 
3.7 applications of tangent lines
3.7 applications of tangent lines3.7 applications of tangent lines
3.7 applications of tangent linesmath265
 
Andrei rusu-2013-amaa-workshop
Andrei rusu-2013-amaa-workshopAndrei rusu-2013-amaa-workshop
Andrei rusu-2013-amaa-workshopAndries Rusu
 
Probability and Entanglement
Probability and EntanglementProbability and Entanglement
Probability and EntanglementGunn Quznetsov
 
From moments to sparse representations, a geometric, algebraic and algorithmi...
From moments to sparse representations, a geometric, algebraic and algorithmi...From moments to sparse representations, a geometric, algebraic and algorithmi...
From moments to sparse representations, a geometric, algebraic and algorithmi...BernardMourrain
 
The Chase in Database Theory
The Chase in Database TheoryThe Chase in Database Theory
The Chase in Database TheoryJan Hidders
 
Spanos lecture+3-6334-estimation
Spanos lecture+3-6334-estimationSpanos lecture+3-6334-estimation
Spanos lecture+3-6334-estimationjemille6
 
Existence Theory for Second Order Nonlinear Functional Random Differential Eq...
Existence Theory for Second Order Nonlinear Functional Random Differential Eq...Existence Theory for Second Order Nonlinear Functional Random Differential Eq...
Existence Theory for Second Order Nonlinear Functional Random Differential Eq...IOSR Journals
 
Complete l fuzzy metric spaces and common fixed point theorems
Complete l fuzzy metric spaces and  common fixed point theoremsComplete l fuzzy metric spaces and  common fixed point theorems
Complete l fuzzy metric spaces and common fixed point theoremsAlexander Decker
 
Interval valued intuitionistic fuzzy homomorphism of bf algebras
Interval valued intuitionistic fuzzy homomorphism of bf algebrasInterval valued intuitionistic fuzzy homomorphism of bf algebras
Interval valued intuitionistic fuzzy homomorphism of bf algebrasAlexander Decker
 
conference_poster_5_UCSB
conference_poster_5_UCSBconference_poster_5_UCSB
conference_poster_5_UCSBXining Li
 
Hecke Operators on Jacobi Forms of Lattice Index and the Relation to Elliptic...
Hecke Operators on Jacobi Forms of Lattice Index and the Relation to Elliptic...Hecke Operators on Jacobi Forms of Lattice Index and the Relation to Elliptic...
Hecke Operators on Jacobi Forms of Lattice Index and the Relation to Elliptic...Ali Ajouz
 
Solution to schrodinger equation with dirac comb potential
Solution to schrodinger equation with dirac comb potential Solution to schrodinger equation with dirac comb potential
Solution to schrodinger equation with dirac comb potential slides
 
Definition ofvectorspace
Definition ofvectorspaceDefinition ofvectorspace
Definition ofvectorspaceTanuj Parikh
 
1.  Write an equation in standard form of the parabola that has th.docx
1.  Write an equation in standard form of the parabola that has th.docx1.  Write an equation in standard form of the parabola that has th.docx
1.  Write an equation in standard form of the parabola that has th.docxKiyokoSlagleis
 

Similaire à Intuitionistic First-Order Logic: Categorical semantics via the Curry-Howard isomorphism (20)

FINAL PROJECT, MATH 251, FALL 2015[The project is Due Mond.docx
FINAL PROJECT, MATH 251, FALL 2015[The project is Due Mond.docxFINAL PROJECT, MATH 251, FALL 2015[The project is Due Mond.docx
FINAL PROJECT, MATH 251, FALL 2015[The project is Due Mond.docx
 
Nbhm m. a. and m.sc. scholarship test 2006
Nbhm m. a. and m.sc. scholarship test 2006Nbhm m. a. and m.sc. scholarship test 2006
Nbhm m. a. and m.sc. scholarship test 2006
 
00_1 - Slide Pelengkap (dari Buku Neuro Fuzzy and Soft Computing).ppt
00_1 - Slide Pelengkap (dari Buku Neuro Fuzzy and Soft Computing).ppt00_1 - Slide Pelengkap (dari Buku Neuro Fuzzy and Soft Computing).ppt
00_1 - Slide Pelengkap (dari Buku Neuro Fuzzy and Soft Computing).ppt
 
3.7 applications of tangent lines
3.7 applications of tangent lines3.7 applications of tangent lines
3.7 applications of tangent lines
 
Andrei rusu-2013-amaa-workshop
Andrei rusu-2013-amaa-workshopAndrei rusu-2013-amaa-workshop
Andrei rusu-2013-amaa-workshop
 
Probability and Entanglement
Probability and EntanglementProbability and Entanglement
Probability and Entanglement
 
From moments to sparse representations, a geometric, algebraic and algorithmi...
From moments to sparse representations, a geometric, algebraic and algorithmi...From moments to sparse representations, a geometric, algebraic and algorithmi...
From moments to sparse representations, a geometric, algebraic and algorithmi...
 
The Chase in Database Theory
The Chase in Database TheoryThe Chase in Database Theory
The Chase in Database Theory
 
Spanos lecture+3-6334-estimation
Spanos lecture+3-6334-estimationSpanos lecture+3-6334-estimation
Spanos lecture+3-6334-estimation
 
Existence Theory for Second Order Nonlinear Functional Random Differential Eq...
Existence Theory for Second Order Nonlinear Functional Random Differential Eq...Existence Theory for Second Order Nonlinear Functional Random Differential Eq...
Existence Theory for Second Order Nonlinear Functional Random Differential Eq...
 
Complete l fuzzy metric spaces and common fixed point theorems
Complete l fuzzy metric spaces and  common fixed point theoremsComplete l fuzzy metric spaces and  common fixed point theorems
Complete l fuzzy metric spaces and common fixed point theorems
 
cswiercz-general-presentation
cswiercz-general-presentationcswiercz-general-presentation
cswiercz-general-presentation
 
Interval valued intuitionistic fuzzy homomorphism of bf algebras
Interval valued intuitionistic fuzzy homomorphism of bf algebrasInterval valued intuitionistic fuzzy homomorphism of bf algebras
Interval valued intuitionistic fuzzy homomorphism of bf algebras
 
conference_poster_5_UCSB
conference_poster_5_UCSBconference_poster_5_UCSB
conference_poster_5_UCSB
 
Hecke Operators on Jacobi Forms of Lattice Index and the Relation to Elliptic...
Hecke Operators on Jacobi Forms of Lattice Index and the Relation to Elliptic...Hecke Operators on Jacobi Forms of Lattice Index and the Relation to Elliptic...
Hecke Operators on Jacobi Forms of Lattice Index and the Relation to Elliptic...
 
Solution to schrodinger equation with dirac comb potential
Solution to schrodinger equation with dirac comb potential Solution to schrodinger equation with dirac comb potential
Solution to schrodinger equation with dirac comb potential
 
Matrix calculus
Matrix calculusMatrix calculus
Matrix calculus
 
Definition ofvectorspace
Definition ofvectorspaceDefinition ofvectorspace
Definition ofvectorspace
 
1.  Write an equation in standard form of the parabola that has th.docx
1.  Write an equation in standard form of the parabola that has th.docx1.  Write an equation in standard form of the parabola that has th.docx
1.  Write an equation in standard form of the parabola that has th.docx
 
7_AJMS_246_20.pdf
7_AJMS_246_20.pdf7_AJMS_246_20.pdf
7_AJMS_246_20.pdf
 

Plus de Marco Benini

Point-free semantics of dependent type theories
Point-free semantics of dependent type theoriesPoint-free semantics of dependent type theories
Point-free semantics of dependent type theoriesMarco Benini
 
The Graph Minor Theorem: a walk on the wild side of graphs
The Graph Minor Theorem: a walk on the wild side of graphsThe Graph Minor Theorem: a walk on the wild side of graphs
The Graph Minor Theorem: a walk on the wild side of graphsMarco Benini
 
Explaining the Kruskal Tree Theore
Explaining the Kruskal Tree TheoreExplaining the Kruskal Tree Theore
Explaining the Kruskal Tree TheoreMarco Benini
 
The Graph Minor Theorem: a walk on the wild side of graphs
The Graph Minor Theorem: a walk on the wild side of graphsThe Graph Minor Theorem: a walk on the wild side of graphs
The Graph Minor Theorem: a walk on the wild side of graphsMarco Benini
 
Dealing with negative results
Dealing with negative resultsDealing with negative results
Dealing with negative resultsMarco Benini
 
Variations on the Higman's Lemma
Variations on the Higman's LemmaVariations on the Higman's Lemma
Variations on the Higman's LemmaMarco Benini
 
Dealing with negative results
Dealing with negative resultsDealing with negative results
Dealing with negative resultsMarco Benini
 
Well Quasi Orders in a Categorical Setting
Well Quasi Orders in a Categorical SettingWell Quasi Orders in a Categorical Setting
Well Quasi Orders in a Categorical SettingMarco Benini
 
Proof-Theoretic Semantics: Point-free meaninig of first-order systems
Proof-Theoretic Semantics: Point-free meaninig of first-order systemsProof-Theoretic Semantics: Point-free meaninig of first-order systems
Proof-Theoretic Semantics: Point-free meaninig of first-order systemsMarco Benini
 
Point-free foundation of Mathematics
Point-free foundation of MathematicsPoint-free foundation of Mathematics
Point-free foundation of MathematicsMarco Benini
 
Fondazione point-free della matematica
Fondazione point-free della matematicaFondazione point-free della matematica
Fondazione point-free della matematicaMarco Benini
 
Numerical Analysis and Epistemology of Information
Numerical Analysis and Epistemology of InformationNumerical Analysis and Epistemology of Information
Numerical Analysis and Epistemology of InformationMarco Benini
 
L'occhio del biologo: elementi di fotografia
L'occhio del biologo: elementi di fotografiaL'occhio del biologo: elementi di fotografia
L'occhio del biologo: elementi di fotografiaMarco Benini
 
Constructive Adpositional Grammars, Formally
Constructive Adpositional Grammars, FormallyConstructive Adpositional Grammars, Formally
Constructive Adpositional Grammars, FormallyMarco Benini
 
Marie Skłodowska Curie Intra-European Fellowship
Marie Skłodowska Curie Intra-European FellowshipMarie Skłodowska Curie Intra-European Fellowship
Marie Skłodowska Curie Intra-European FellowshipMarco Benini
 
Programming modulo representations
Programming modulo representationsProgramming modulo representations
Programming modulo representationsMarco Benini
 
Algorithms and Their Explanations
Algorithms and Their ExplanationsAlgorithms and Their Explanations
Algorithms and Their ExplanationsMarco Benini
 
Programming modulo representations
Programming modulo representationsProgramming modulo representations
Programming modulo representationsMarco Benini
 
June 22nd 2014: Seminar at JAIST
June 22nd 2014: Seminar at JAISTJune 22nd 2014: Seminar at JAIST
June 22nd 2014: Seminar at JAISTMarco Benini
 
CORCON2014: Does programming really need data structures?
CORCON2014: Does programming really need data structures?CORCON2014: Does programming really need data structures?
CORCON2014: Does programming really need data structures?Marco Benini
 

Plus de Marco Benini (20)

Point-free semantics of dependent type theories
Point-free semantics of dependent type theoriesPoint-free semantics of dependent type theories
Point-free semantics of dependent type theories
 
The Graph Minor Theorem: a walk on the wild side of graphs
The Graph Minor Theorem: a walk on the wild side of graphsThe Graph Minor Theorem: a walk on the wild side of graphs
The Graph Minor Theorem: a walk on the wild side of graphs
 
Explaining the Kruskal Tree Theore
Explaining the Kruskal Tree TheoreExplaining the Kruskal Tree Theore
Explaining the Kruskal Tree Theore
 
The Graph Minor Theorem: a walk on the wild side of graphs
The Graph Minor Theorem: a walk on the wild side of graphsThe Graph Minor Theorem: a walk on the wild side of graphs
The Graph Minor Theorem: a walk on the wild side of graphs
 
Dealing with negative results
Dealing with negative resultsDealing with negative results
Dealing with negative results
 
Variations on the Higman's Lemma
Variations on the Higman's LemmaVariations on the Higman's Lemma
Variations on the Higman's Lemma
 
Dealing with negative results
Dealing with negative resultsDealing with negative results
Dealing with negative results
 
Well Quasi Orders in a Categorical Setting
Well Quasi Orders in a Categorical SettingWell Quasi Orders in a Categorical Setting
Well Quasi Orders in a Categorical Setting
 
Proof-Theoretic Semantics: Point-free meaninig of first-order systems
Proof-Theoretic Semantics: Point-free meaninig of first-order systemsProof-Theoretic Semantics: Point-free meaninig of first-order systems
Proof-Theoretic Semantics: Point-free meaninig of first-order systems
 
Point-free foundation of Mathematics
Point-free foundation of MathematicsPoint-free foundation of Mathematics
Point-free foundation of Mathematics
 
Fondazione point-free della matematica
Fondazione point-free della matematicaFondazione point-free della matematica
Fondazione point-free della matematica
 
Numerical Analysis and Epistemology of Information
Numerical Analysis and Epistemology of InformationNumerical Analysis and Epistemology of Information
Numerical Analysis and Epistemology of Information
 
L'occhio del biologo: elementi di fotografia
L'occhio del biologo: elementi di fotografiaL'occhio del biologo: elementi di fotografia
L'occhio del biologo: elementi di fotografia
 
Constructive Adpositional Grammars, Formally
Constructive Adpositional Grammars, FormallyConstructive Adpositional Grammars, Formally
Constructive Adpositional Grammars, Formally
 
Marie Skłodowska Curie Intra-European Fellowship
Marie Skłodowska Curie Intra-European FellowshipMarie Skłodowska Curie Intra-European Fellowship
Marie Skłodowska Curie Intra-European Fellowship
 
Programming modulo representations
Programming modulo representationsProgramming modulo representations
Programming modulo representations
 
Algorithms and Their Explanations
Algorithms and Their ExplanationsAlgorithms and Their Explanations
Algorithms and Their Explanations
 
Programming modulo representations
Programming modulo representationsProgramming modulo representations
Programming modulo representations
 
June 22nd 2014: Seminar at JAIST
June 22nd 2014: Seminar at JAISTJune 22nd 2014: Seminar at JAIST
June 22nd 2014: Seminar at JAIST
 
CORCON2014: Does programming really need data structures?
CORCON2014: Does programming really need data structures?CORCON2014: Does programming really need data structures?
CORCON2014: Does programming really need data structures?
 

Intuitionistic First-Order Logic: Categorical semantics via the Curry-Howard isomorphism

  • 1. Intuitionistic First-Order Logic Categorical semantics via the Curry-Howard isomorphism Marco Benini M.Benini@leeds.ac.uk Department of Pure Mathematics University of Leeds 14th November 2012
  • 2. Introduction An observation: in part D of P. Johnstone’s Sketches of an Elephant, there is a categorical semantics for the simply typed λ-calculus. In the very same class of models, one can give a semantics to the corresponding fragment of propositional logic. The problem: is it possible to do the same for full first-order intuitionistic logic? 2 of 32
  • 3. Introduction Johstone’s account comes from Lambek and Scott’s Introduction to Higher-Order Categorical Logic. The considered propositional logic is minimal logic limited to conjunction and implication. In Taylor, Practical Foundation of Mathematics, one finds that the treatment of disjunction requires distributive categories in order to follow the same pattern as the previous works. To my knowledge, no categorical semantics appears in literature which models the full first-order intuitionistic logic AND the corresponding λ-calculus. 3 of 32
  • 4. The λ-calculus Definition 1 (Lambda signature) A λ-signature Σ = 〈S , F , R , Ax〉 is a structure where 1. 〈S , F , R 〉 is a logical signature, i.e., 1.1 a set S of sort symbols; 1.2 a set F of function symbols, each one decorated as f : s1 × · · · × sn → s0 , with s0 , . . . , sn ∈ S; 1.3 a set R of relation symbols, each one decorated as r : s1 × · · · × sn , with s1 , . . . , sn ∈ S; 2. Ax is the set of axiom symbols, each one decorated as a : A → B where A, B ∈ λTypes(Σ) and FV(A → B ) = . We call LTerms(Σ) the collection of logical terms constructed from the signature Σ, assuming to have a denumerable set of variables Vs for each s ∈ S. 4 of 32
  • 5. The λ-calculus Definition 2 (Lambda type) Fixed a λ-signature Σ, the λ-types on Σ are inductively defined along with their free variables as follows: 1. 0, 1 ∈ λTypes(Σ) and FV(0) = FV(1) = ; 2. if p : s1 × · · · × sn ∈ R and t1 : s1 , . . . , tn : sn ∈ LTerms(Σ), then p(t1 , . . . , tn ) ∈ λTypes(Σ) and FV(p(t1 , . . . , tn )) = n=1 FV(ti : si ); i 3. if A, B ∈ λTypes(Σ) then A × B , A + B , A → B ∈ λTypes(Σ) and FV(A × B ) = FV(A + B ) = FV(A → B ) = FV(A) ∪ FV(B ); 4. if x ∈ Vs and A ∈ λTypes(Σ) then ∀x : s. A, ∃x : s. A ∈ λTypes(Σ) and FV(∀x : s. A) = FV(∃x : s. A) = FV(A) {x : s}. 5 of 32
  • 6. The λ-calculus Definition 3 (Lambda term) Fixed a λ-signature Σ = 〈S , F , R , Ax〉, for each type t ∈ λTypes(Σ), we assume there is a denumerable set Wt of (typed) variables. A λ-term is inductively defined together with its free variables as: 1. if x ∈ Wt then x : t ∈ λTerms(Σ) and FV(x : t ) = {x : t }; 2. if f : A → B ∈ Ax and t : A ∈ λTerms(Σ) then f (t ): B ∈ λTerms(Σ) and FV(f (t ): B ) = FV(t : A); 3. if s : A, t : B ∈ λTerms(Σ) then 〈s, t 〉 : A × B ∈ λTerms(Σ) and FV(〈s, t 〉 : A × B ) = FV(s : A) ∪ FV(t : B ); 4. if t : A × B ∈ λTypes(Σ) then fst(t ): A ∈ λTerms(Σ), snd(t ): B ∈ λTerms(Σ) and FV(fst(t ): A) = FV(snd(t ): B ) = FV(t : A × B ); → 6 of 32
  • 7. The λ-calculus → (Lambda term) 5. if t : A ∈ λTerms(Σ) then inlB (t ): A + B ∈ λTerms(Σ), inrB (t ): B + A ∈ λTerms(Σ) and FV(inlB (t ): A + B ) = FV(inrB (t ): B + A) = FV(t : A); 6. if s : A + B , t : A → C , r : B → C ∈ λTerms(Σ) then when(s, t , r ): C ∈ λTerms(Σ) and FV(when(s, t , r ): C ) = FV(s : A + B ) ∪ FV(t : A → C ) ∪ FV(r : B → C ); 7. if x ∈ WA and t : B ∈ λTerms(Σ) then (λx : A. t ): A → B ∈ λTerms(Σ) and FV((λx : A. t ): A → B ) = FV(t : B ) {x : A}; 8. if s : A → B , t : A ∈ λTerms(Σ) then s · t : B ∈ λTerms(Σ) and FV(s · t : B ) = FV(s : A → B ) ∪ FV(t : A); 9. ∗ : 1 ∈ λTerms(Σ) and FV(∗ : 1) = ; 10. FA : 0 → A ∈ λTerms(Σ) and FV(FA : 0 → A) = ; → 7 of 32
  • 8. The λ-calculus → (Lambda term) 11. if x ∈ Vs and t : A ∈ λTerms(Σ) where x : s ∈ FV∗ (t : A), then allI(λx : s. t ):(∀x : s. A) ∈ λTerms(Σ) and FV(allI(λx : s. t ):(∀x : s. A)) = FV(t : A); 12. if t :(∀x : s. A) ∈ λTerms(Σ) and r : s ∈ LTerms(Σ) then allE(t , r ):(A[r /x ]) ∈ λTerms(Σ) and FV(allE(t , r ):(A[r /x ])) = FV(t :(∀x : s. A)); 13. if x ∈ Vs , r : s ∈ LTerms(Σ) and t :(A[r /x ]) ∈ λTerms(Σ) then exIx (t ):(∃x : s. A) ∈ λTerms(Σ) and FV(exIx (t ):(∃x : s. A)) = FV(t :(A[r /x ])); 14. if t :(∃x : s. A), ∈ λTerms(Σ) and r : A → B ∈ λTerms(Σ) where x : s ∈ FV∗ (r : A → B ), then exE(t , (λx : s. r )): B ∈ λTerms(Σ) and FV(exE(t , (λx : s. r )): B ) = FV(t :(∃x : s. A)) ∪ FV(r : A → B ). In the previous definition, x : s ∈ FV∗ (t : A) if and only if there is r ∈ λTypes(Σ) and y ∈ Wr such that x : s ∈ FV(r ) and y : r ∈ FV(t : A). 8 of 32
  • 9. The λ-calculus Definition 4 (Lambda calculus) A derivation is inductively defined by the following inference rules, whose antecedents and consequents are equalities-in-context within a fixed λ-signature Σ: (eq0 ) x : A. s =C t y : B . s[r1 /x1 , . . . , rn /xn ] =C t [r1 /x1 , . . . , rn /xn ] where, for any 1 ≤ i ≤ n, y : B . ri : Ai is a term-in-context;  (x : A. s1 =B1 t1 )   . . (eq1 ) . x : A. r [s/y ] =C r [t /y ];  (x : A. sm =Bm tm )  (eq2 ) x : A. x =A x; (eq3 ) x : A, y : A. x =A y x : A, y : A. y =A x; (x : A, y : A, z : A. x =A y ) (eq4 ) x : A, y : A, z : A. x =A z; (x : A, y : A, z : A. y =A z ) → 9 of 32
  • 10. The λ-calculus → (Lambda calculus) (eq5 ) x : A. s =C t x : A. (λy : B . s) =B →C (λy : B . t ); (eq6 ) x : A. r =C t x : A. allI(λy : s. r ) =(∀y : s. C ) allI(λy : s. t ); (eq7 ) x : A. u =C v x : A. exE(t , (λy : s. u )) =C exE(t , (λy : s. v )); (×0 ) x : 1. x =1 ∗; (×1 ) x : A, y : B . fst(〈x , y 〉) =A x; (×2 ) x : A, y : B . snd(〈x , y 〉) =B y ; (×3 ) z : A × B . 〈fst(z ), snd(z )〉 =A×B z; (+0 ) x : A. when(inlB (a), t , s) =C t · a; (+1 ) x : A. when(inrD (b), t , s) =C s · b; → 10 of 32
  • 11. The λ-calculus → (Lambda calculus) (+2 ) when y : A1 ∈ FV(x1 : A1 + A2 ) ∪ FV(x3 : B1 → C ) ∪ FV(x4 : B2 → C ) and y : A2 ∈ FV(x2 : A1 + A2 ) ∪ FV(x3 : B1 → C ) ∪ FV(x4 : B2 → C ) x0 : A1 + A2 , x1 : A1 → (B1 + B2 ), x2 : A2 → (B1 + B2 ), x3 : B1 → C , x4 : B2 → C . when(when(x0 , x1 , x2 ), x3 , x4 ) =C =C when(x0 , (λy : A1 . when(x1 · y , x3 , x4 )), (λy : A2 . when(x2 · y , x3 , x4 ))) ; (+3 ) x : A, y : 0. FA ·y =A x; (→0 ) x : A. (λy : C . s) · t =B s[t /y ]; (→1 ) x : A. (λy : C . t · y ) =C →B t where y : C ∈ FV(t : C → B ); (∀0 ) x : A. allE(allI(λz : s. t ), r ) =B [r /z ] t [r /z ]; (∀1 ) x : A. allE(u , r ) =B allE(v , r ) r : s∈LTerms(Σ) x : A. u =(∀z : s. B ) v ; → 11 of 32
  • 12. The λ-calculus → (Lambda calculus) (∃0 ) x : A. exE(exIz (t ), (λz : s. v )) =B (v [r /z ]) · t; (∃1 ) x : A. exE(u , (λz : s. r )) =B exE(u , (λz : s. t )) x : A. r =C →B t where FV(r : C → B ) = FV(t : C → B ); (∃2 ) v :(∃y : s. A). w =B exE(v , (λy : s. (λz : A. w [exIy (z )/v ]))) with z : A ∈ FV(w : B ); (∃3 ) x : A. exE(exE(a, (λy : s. (λz : D . b))), (λy : s. c )) =C =C exE(a, (λy : s. (λz : D . exE(b, (λy : s. c ))))); (∃4 ) x : A. exE(a, (λy : s. (λz : C . b[exIy (z )/w ]))) =B b[a/w ] with z : C ∈ FV(b : B ). 12 of 32
  • 13. Semantics Definition 5 (Logically distributive category) Fixed a λ-signature Σ = 〈S , F , R , Ax〉, a category C together with a map M : λTypes(Σ) → Obj C is said to be logically distributive if it satisfies the following seven conditions: 1. C has finite products; 2. C has finite co-products; 3. C has exponentiation; 4. C is distributive, i.e., for every A, B , C ∈ Obj C, the arrow ∆ = [1A × ι1 , 1A × ι2 ]: (A × B ) + (A × C ) → A × (B + C ) has an inverse, where [_, _] is the co-universal arrow of the (A × B ) + (A × C ) co-product, _ × _ is the product arrow, 1A is the identity arrow on A, and ι1 : B → B + C, ι2 : C → B + C are the canonical injections of the B + C co-product. → 13 of 32
  • 14. Semantics → (Logically distributive category) For every s ∈ S, A ∈ λTypes(Σ), and x ∈ Vs , let ΣA (x : s): LTerms(Σ)(s) → C be the functor from the discrete category LTerms(Σ)(s) = t : s | t : s ∈ LTerms(Σ) to C defined by t : s → M (A[t /x ]). Also, for every s ∈ S, A ∈ λTypes(Σ), and x ∈ Vs , let C(∀x : s. A) be the subcategory of C whose objects are the vertices of the cones on ΣA (x : s) such that they are of the form MB for some B ∈ λTypes(Σ) and x : s ∈ FV(B ). Moreover, the arrows of C(∀x : s. A) , apart identities, are the arrows in the category of cones over ΣA (x : s) having the objects of C(∀x : s. A) as domain and M (∀x : s. A) as co-domain. → 14 of 32
  • 15. Semantics → (Logically distributive category) Finally, for every s ∈ S, A ∈ λTypes(Σ), and x ∈ Vs , let C(∃x : s. A) be the subcategory of C whose objects are the vertices of the co-cones on ΣA (x : s) such that they are of the form MB for some B ∈ λTypes(Σ) and x : s ∈ FV(B ). Moreover, the arrows of C(∃x : s. A) , apart identities, are the arrows in the category of co-cones over ΣA (x : s) having the objects of C(∃x : s. A) as co-domain and M (∃x : s. A) as domain. 5. All the subcategories C(∀x : s. A) have terminal objects, and all the subcategories C(∃x : s. A) have initial objects; → 15 of 32
  • 16. Semantics → (Logically distributive category) 6. The M map is such that 6.1 M (0) = 0, the initial object of C; 6.2 M (1) = 1, the terminal object of C; 6.3 M (A × B ) = MA × MB, the binary product in C; 6.4 M (A + B ) = MA + MB, the binary co-prooduct in C; 6.5 M (A → B ) = MB MA , the exponential object in C; 6.6 M (∀x : s. A) is the terminal object in the subcategory C(∀x : s. A) ; 6.7 M (∃x : s. A) is the initial object in the subcategory C(∃x : s. A) ; 7. For every x ∈ Vs , A, B ∈ λTypes(Σ) with x : s ∈ FV(A), MA × M (∃x : s. B ) is an object of C(∃x : s. A×B ) since, if M (∃x : s. B ), {δt }t : s∈LTerms(Σ) is a co-cone over ΣB (x : s), and there is one by condition (5), then MA × M (∃x : s. B ), {1MA ×δt }t : s∈LTerms(Σ) is a co-cone over ΣA×B (x : s). Thus, there is a unique arrow !: M (∃x : s. A × B ) → MA × M (∃x : s. B ) in C(∃x : s. A×B ) . Our last condition requires that the arrow ! has an inverse. 16 of 32
  • 17. Semantics Definition 6 (Σ-structure) Given a λ-signature Σ = 〈S , F , R , Ax〉, a Σ-structure is a triple 〈C, M , MAx 〉 such that C together with M forms a logically distributive category and MAx is a map from Ax such that MAx (a : A → B ) ∈ HomC (MA, MB ). 17 of 32
  • 18. Semantics Definition 7 (λ-terms semantics) Fixed a Σ-structure 〈C, M , MAx 〉, let A ≡ A1 × · · · An , and let x ≡ x1 : A1 , . . . , xn : An be a context. The semantics of a term-in-context x . t : B, notation x . t : B , is an arrow in HomC (MA, MB ) inductively defined as follows: 1. x . xi : Ai = πi , the i-th projector of the product MA = MA1 × · · · × MAn ; 2. if a : C → B ∈ Ax then x . a(t ): B = MAx a ◦ x . t : C ; 3. x . 〈s, t 〉 : B × C = ( x . s : B , x . t : C ) where (_, _) is the universal arrow of the product MB × MC; 4. x . fst(t ): B = π1 ◦ x . t : B × C where π1 is the first canonical projector of the product MA × MB; 5. x . snd(t ): C = π2 ◦ x . t : B × C where π2 is the second canonical projector of the product MA × MB; → 18 of 32
  • 19. Semantics → (λ-terms semantics) 6. x . (λz : C . t ): C → B is the exponential transpose of x , z : C . t : B : MA × MC → MB; 7. x . s · t : B = ev ◦ ( x . s : C → B , x . t : C ) where ev is the exponential evaluation arrow; 8. x . inlB (t ): C + B = ι1 ◦ x . t : C with ι1 the first canonical injection of the co-product MC + MB; 9. x . inrC (t ): C + B = ι2 ◦ x . t : B with ι2 the second canonical injection of the co-product MC + MB; → 19 of 32
  • 20. Semantics → (λ-terms semantics) 10. calling [_, _] the co-universal arrow of (MA × MC1 ) + (MA × MC2 ), (_, _) the universal arrow of MA × (MC1 + MC2 ), and noticing that the arrow ∆ : (MA × MC1 ) + (MA × MC2 ) → MA × (MC1 + MC2 ) has an inverse because C with M is logically distributive x . when(t , u , v ): B = [ev ◦ ( x . u : C1 → B × 1MC1 ) , ev ◦ ( x . v : C2 → B × 1MC2 )] ◦ ◦ ∆−1 ◦ (1MA , x . t : C1 + C2 ) ; 11. x . ∗ : 1 =!: MA → 1, the universal arrow of the terminal object; 12. x . FB : 0 → B is the exponential transpose of (!: 0 → MB ) ◦ (πn+1 : MA × 0 → 0); → 20 of 32
  • 21. Semantics → (λ-terms semantics) 13. x . allI(λz : s. t ):(∀z : s. B ) = β ◦ α where α ≡ 1MAi1 × · · · × 1MAik : MA → MA with A ≡ Ai1 × · · · × Aik , where x ≡ {xi1 : Ai1 , . . . , xik : Aik } = FV(t : B ), and β : MA → M (∀z : s. B ) is the universal arrow from MA to the terminal object in C∀z : s. B ; 14. x . allE(t , r ): B [r /z ] = pr ◦ x . t :(∀z : s. B ) where pr : M (∀z : s. B ) → M (B [r /z ]) is the r -th projector of the unique cone on ΣB (z : s) whose vertex is M (∀z : s. B ). It is worth noticing that pr = w :(∀z : s. B ). allE(w , r ): B [r /z ] ; 15. x . exIz (t ):(∃z : s. B ) = jr ◦ x . t : B [r /z ] where jr : M (B [r /z ]) → M (∃z : s. B ) is the r -th injection of the unique co-cone on ΣB (z : s) whose vertex is M (∃z : s. B ). It is worth noticing that jr = w : B [r /z ]. exIz (w ):(∃z : s. B ) ; → 21 of 32
  • 22. Semantics → (λ-terms semantics) 16. x . exE(t , (λz : s. r )): B = γ ◦ β−1 ◦ (α, x . t :(∃z : s. C ) ) where 16.1 α ≡ 1MAi1 × · · · × 1MAik : MA → MA with A ≡ Ai1 × · · · × Aik , where x ≡ {xi1 : Ai1 , . . . , xik : Aik } = FV(t :(∃z : s. C )) ∪ FV(r : C → B ); 16.2 β : M (∃z : s. A × C ) → MA × M (∃z : s. C ) is the co-universal arrow in the subcategory C∃z : s. A ×C ; 16.3 γ : M (∃z : s. A × C ) → MB is the co-universal arrow in C∃z : s. A ×C . 22 of 32
  • 23. Soundness Definition 8 (Validity) An equality-in-context x . s =A t is valid in the λ-theory T , a set of equalities-in-context, when, in every logically distributive category C, each model M of T is also a model of x . s =A t. A Σ-structure M in C is a model of a theory T when it is a model of each φ in T . Finally, M is a model of an equality-in-context x . t =A s if x . t : A = x . s : A . Theorem 9 (Soundness) If an equation-in-context x . s =A t is derivable from a λ-theory T , then x . s =A t is valid in each model of T in every logically distributive category. 23 of 32
  • 24. Completeness Definition 10 (Syntactical equivalence) Given a λ-theory T , the syntactical equivalence of two terms-in-context is defined by fixing the generated equivalence classes. Precisely, the equivalence class [x : A. t : B ] is defined as the minimal set, composed by terms-in-context, such that 1. x : A. t : B ∈ [x : A. t : B ]—reflexivity; 2. if T y : D . s =C r , where y : D . s =C r is an equality-in-context, and y : D . s : C ∈ [x : A. t : B ], then y : D . r : C ∈ [x : A. t : B ]—closure under provable equivalence; → 24 of 32
  • 25. Completeness → (Syntactical equivalence) 3. if y : D . s : C is a term-in-context and, for some 1 ≤ i < m and z : Di × Di +1 ∈ FV(s : C ) ∪ y1 : D1 , . . . , ym : Dm , it happens that y1 : D1 , . . . , yi −1 : Di −1 , z : Di × Di +1 , yi +1 : Di +2 , . . . , ym : Dm . s[fst(z )/yi ][snd(z )/yi +1 ]: C ∈ [x : A. t : B ] , then y : D . s : C ∈ [x : A. t : B ]—closure under associativity in contexts; 4. if y : D . s : C is a term-in-context and, for some 1 ≤ i < m and z : Di +1 × Di ∈ FV(s : C ) ∪ y1 : D1 , . . . , ym : Dm , it happens that y1 : D1 , . . . , yi −1 : Di −1 , z : Di +1 × Di , yi +1 : Di +2 , . . . , ym : Dm . s[snd(z )/yi ][fst(z )/yi +1 ]: C ∈ [x : A. t : B ] , then y : D . s : C ∈ [x : A. t : B ]—closure under commutativity in contexts; → 25 of 32
  • 26. Completeness → (Syntactical equivalence) 5. if y : D . s : C ∈ [x : A. y : B ] and z : Di ∈ FV(s : C ) ∪ y1 : D1 , . . . , ym : Dm for some 1 ≤ i ≤ m, then y1 : D1 , . . . , yi −1 : Di −1 , z : Di , yi +1 : Di +1 , . . . , ym : Dm . s[z /yi ]: C is in [x : A. t : B ]—closure under α-renaming in contexts. 26 of 32
  • 27. Completeness Definition 11 (Syntactical category) Given a λ-theory T , the syntactical category CT has λTypes(Σ) as objects, where Σ is the λ-signature of T , and the equivalence classes [x : A. t : B ]: A → B as arrows. Identities are given by the classes [x : A. x : A]: A → A for each λ-type A, and composition is given by substitution: [y : B . s : C ] ◦ [x : A. t : B ] = [x : A. s[t /y ]: C ] . Moreover, the map MT : λTypes(Σ) → Obj CT is defined as MT A = A. 27 of 32
  • 28. Completeness Proposition 12 The CT category is logically distributive. Proposition 13 Given a λ-theory T on the Σ signature, the Σ-structure 〈CT , MT , MAx 〉 on the corresponding syntactical category is defined by MAx which maps f : A → B ∈ Ax to [x : A. f (x ): B ]. This Σ-structure is a model for T and, moreover, it satisfies exactly those equalities-in-context which are provable in T . 28 of 32
  • 29. Completeness Proposition 14 For every logically distributive category C, there is a biijection between equivalence classes, modulo natural equivalences, of structure-preserving functors CT → C and equivalence classes, modulo isomorphisms, of models of T in C, induced by the map F → F (MT ). Theorem 15 (Completeness) If x . s =A t is an equality-in-context valid in every model for T in each logically distributive category, then T x . s =A t. 29 of 32
  • 30. Soundness and Completeness in Logic Definition 16 (Valid type) A λ-type A is valid in the model N = 〈N, N , NAx 〉 when there exists an arrow 1 → NA in N. A λ-type A is a logical consequence in the model N of the λ-types B1 , . . . , Bn when there exists N (B1 × · · · × Bn ) → NA in N. A λ-type A is a logical consequence of B1 , . . . , Bn when it is a logical consequence of B1 , . . . , Bn in every model in every logically distributive category. 30 of 32
  • 31. Soundness and Completeness in Logic Proposition 17 A λ-type A is a logical consequence of B1 , . . . , Bn if and only if there exists a term-in-context x : B1 × · · · × Bn . t : A. Corollary 18 A λ-type A is a logical consequence of B1 , . . . , Bn if and only if there is a proof of A from the hypotheses B1 , . . . , Bn , when λ-types are interpreted as logical formulae and λ-terms as logical proofs, according to the Curry-Howard isomorphism. 31 of 32
  • 32. The end Questions? 32 of 32