Asterix calculus - classical computation in detail

Transcription

1 Asterix calculus - classical computation in detail (denoted X ) Dragiša Žunić 1 Pierre Lescanne 2 1 CMU Qatar 2 ENS Lyon Logic and Applications - LAP th conference, September 19-23, Dubrovnik Croatia

2 Outline Sequent calculus, proofs and programs The X calculus : explicit erasure and duplication Strong normalisation property

3 Outline Sequent calculus, proofs and programs Implicit structural rules the X calculus Explicit structural rules the X calculus The X calculus : explicit erasure and duplication Logical setting From sequent proofs to terms The syntax and reduction rules Diagrammatic view Strong normalisation property

4 The framework Classical logic Sequent calculus of G.Gentzen (two important formalisms : sequent calculus and natural deduction calculus) The Curry-Howard correspondence (the relation between proof theory and the programming language theory) The goal is to study classical computation, by assigning computational interpretation(s) to classical logic represented in the sequent calculus.

5 The Curry-Howard correspondence The intuitionistic logic and simply typed λ-calculus Proofs Terms Propositions Types Normalization Reduction Classical logic : T. Griffin (1990) M. Parigot (1992) : λµ-calculus (natural deduction) P. L. Curien, H. Herbelin ( ) : λµ µ-calculus (sequent calculus)

6 Related work The two predecessors Classical logic : the X calculus C. Urban (2000) S. van Bakel, S.Lengrand, P. Lescanne (2005) Intuitionistic logic : the λlxr-calculus D. Kesner, S. Lengrand (2005) X λlxr X

7 G3 sequent system for classical logic X -calculus (axiom) Γ, A A, Γ A, Γ, B ( left) Γ, A B Γ A, Γ, A (cut) Γ, A B, ( right) Γ A B, Γ Contexts Γ, are sets Context-sharing style Structural rules implicit

8 G1 sequent system for classical logic X calculus (axiom) A A Γ A, Γ, Γ, A B, Γ, B ( left) Γ, A B, ( right) Γ A B, Γ A, Γ, Γ, Γ, A (cut) Γ (left weakening) Γ, A Γ (right weakening) Γ A, Γ, A, A (left contraction) Γ, A Γ A, A, (right contraction) Γ A,

9 Outline Sequent calculus, proofs and programs Implicit structural rules the X calculus Explicit structural rules the X calculus The X calculus : explicit erasure and duplication Logical setting From sequent proofs to terms The syntax and reduction rules Diagrammatic view Strong normalisation property

10 Computational interpretation of classical proofs Sequent calculus with explicit structural rules (weakening and contraction) 1) Terms in X -calculus are in fact annotations for proofs, 2) Computation in X -calculus corresponds to cut-elimination Weakening as an eraser / Contraction as a duplicator

11 Names The terms are built from names. Two categories of names : x, y, z... in-names α, β, γ... out-names Binders wear hats : x, ŷ, ẑ... α, β, γ... Examples : x.α x x.β β. α [P β γ >α

12 Name Variable In λ-calculus : variables β (λy.xyz)m xmz An arbitrary term M substitutes the variable y In X -calculus : names A name can never be substituted for a term A name can only be renamed

13 G1 sequent system for classical logic (axiom) A A Γ A, Γ, Γ, A B, Γ, B ( left) Γ, A B, ( right) Γ A B, Γ A, Γ, Γ, Γ, A (cut) Γ (left weakening) Γ, A Γ (right weakening) Γ A, Γ, A, A (left contraction) Γ, A Γ A, A, (right contraction) Γ A,

14 The terms correspond to proofs : (caps) x.α :. x : A α : A P :. Γ α : A, Q :. Γ, x : B P :. Γ, x : A α : B, (imp) (exp) P α [y] xq :. Γ, Γ, y : A B, x P α. β :. Γ β : A B, P :. Γ α : A, P α xq :. Γ, Γ, Q :. Γ, x : A (cut) P :. Γ (left eraser) x P :. Γ, x : A P :. Γ (right eraser) P α :. Γ α : A, P :. Γ, y : A, z : A (left dupl.) x < ŷ ẑ P] :. Γ, x : A P :. Γ β : A, γ : A, (right dupl.) [P β γ >α :. Γ α : A,

15 The terms correspond to proofs : (caps) x.α :. x : A α : A P :. Γ α : A, Q :. Γ, x : B P :. Γ, x : A α : B, (imp) (exp) P α [y] xq :. Γ, Γ, y : A B, x P α. β :. Γ β : A B, P :. Γ α : A, P α xq :. Γ, Γ, Q :. Γ, x : A (cut) P :. Γ (left eraser) x P :. Γ, x : A P :. Γ (right eraser) P α :. Γ α : A, P :. Γ, y : A, z : A (left dupl.) x < ŷ ẑ P] :. Γ, x : A P :. Γ β : A, γ : A, (right dupl.) [P β γ >α :. Γ α : A,

16 The terms correspond to proofs : (caps) x.α :. x : A α : A P :. Γ α : A, Q :. Γ, x : B P :. Γ, x : A α : B, (imp) (exp) P α [y] xq :. Γ, Γ, y : A B, x P α. β :. Γ β : A B, P :. Γ α : A, P α xq :. Γ, Γ, Q :. Γ, x : A (cut) P :. Γ (left eraser) x P :. Γ, x : A P :. Γ (right eraser) P α :. Γ α : A, P :. Γ, y : A, z : A (left dupl.) x < ŷ ẑ P] :. Γ, x : A P :. Γ β : A, γ : A, (right dupl.) [P β γ >α :. Γ α : A,

17 The terms correspond to proofs : (caps) x.α :. x : A α : A P :. Γ α : A, Q :. Γ, x : B P :. Γ, x : A α : B, (imp) (exp) P α [y] xq :. Γ, Γ, y : A B, x P α. β :. Γ β : A B, P :. Γ α : A, P α xq :. Γ, Γ, Q :. Γ, x : A (cut) P :. Γ (left eraser) x P :. Γ, x : A P :. Γ (right eraser) P α :. Γ α : A, P :. Γ, y : A, z : A (left dupl.) x < ŷ ẑ P] :. Γ, x : A P :. Γ β : A, γ : A, (right dupl.) [P β γ >α :. Γ α : A,

18 The terms correspond to proofs : (caps) x.α :. x : A α : A P :. Γ α : A, Q :. Γ, x : B P :. Γ, x : A α : B, (imp) (exp) P α [y] xq :. Γ, Γ, y : A B, x P α. β :. Γ β : A B, P :. Γ α : A, P α xq :. Γ, Γ, Q :. Γ, x : A (cut) P :. Γ (left eraser) x P :. Γ, x : A P :. Γ (right eraser) P α :. Γ α : A, P :. Γ, y : A, z : A (left dupl.) x < ŷ ẑ P] :. Γ, x : A P :. Γ β : A, γ : A, (right dupl.) [P β γ >α :. Γ α : A,

19 The X calculus The syntax P, Q ::= x.α capsule x P β. α exporter P α [y] xq importer P α xq cut x P left-eraser P α right-eraser x < ŷ ẑ P] [P β γ >α left-duplicator right-duplicator

20 Linearity In X -calculus only linear terms are considered : every free name occurs only once every binder does bind an occurrence of a free name (and therefore only one) Examples of non-linear terms : x y.β β. α and x.α α Every non-linear term has a linear representation : x (x y.β ) β. α and [ x.α 1 α 2 α 1 α 2 >α

21 Linearity In X -calculus only linear terms are considered : every free name occurs only once every binder does bind an occurrence of a free name (and therefore only one) Examples of non-linear terms : x y.β β. α and x.α α Every non-linear term has a linear representation : x (x y.β ) β. α and [ x.α 1 α 2 α 1 α 2 >α

22 Linearity In X -calculus only linear terms are considered : every free name occurs only once every binder does bind an occurrence of a free name (and therefore only one) Examples of non-linear terms : x y.β β. α and x.α α Every non-linear term has a linear representation : x (x y.β ) β. α and [ x.α 1 α 2 α 1 α 2 >α

23 Linearity In X -calculus only linear terms are considered : every free name occurs only once every binder does bind an occurrence of a free name (and therefore only one) Examples of non-linear terms : x y.β β. α and x.α α Every non-linear term has a linear representation : x (x y.β ) β. α and [ x.α 1 α 2 α 1 α 2 >α

24 Linearity In X -calculus only linear terms are considered : every free name occurs only once every binder does bind an occurrence of a free name (and therefore only one) Examples of non-linear terms : x y.β β. α and x.α α Every non-linear term has a linear representation : x (x y.β ) β. α and [ x.α 1 α 2 α 1 α 2 >α

25 The X calculus Grouping the rules : Logical rules (L-principal names involved) Structural rules (S-principal names involved) (Activation rules) (Deactivation rules) Propagation rules

26 The X calculus The syntax P, Q ::= x.α capsule x P β. α exporter P α [y] xq importer P α xq cut x P left-eraser P α right-eraser x < ŷ ẑ P] [P β γ >α left-duplicator right-duplicator The notion of principal name of a term. 1. L-principal name 2. S principal name

27 The X calculus The syntax P, Q ::= x.α capsule x P β. α exporter P α [y] xq importer P α xq cut x P left-eraser P α right-eraser x < ŷ ẑ P] [P β γ >α left-duplicator right-duplicator The notion of principal name of a term. 1. L principal name 2. S-principal name

28 X : logical rules Renaming : (ren-l) : y.α α xq Q{y/x} (ren-r) : P α x x.β P{β/α} Diagrammatically : (ren-l) : y α x Q y Q α x β (ren-r) : P P β

29 X : logical rules Inserting (ei-insert) : (ŷ P β. α) α x(q γ [x] ẑr) either { (Q γ ŷp) β ẑr Q γ ŷ(p β ẑr) Diagrammatically : y P β Q γ I z R E α x y β Q γ z P R Notice : logical rules define reducing when a cut binds L-principal names

30 X : structural rules Left erasure : ( -eras) : (P α) α xq I Q P O Q, where I Q = I (Q) \x, O Q = O(Q) Diagrammatically : I Q { α x P Q Q } O I Q { O P } Q

31 X : structural rules Left duplication ( -dupl) : ([P α1 α 2 >α) α xq I Q < I Q 1 I Q 2 (P α 1 x 1 Q 1 ) α 2 Ô Q 1 x 2 Q 2 > O Q, O Q 2 Diagrammatically : I Q { P β γ α x Q Q } O I Q { P β γ x Q 2 x Q 1 Q } O

32 Symmetry... The -erasure } O P I P { P α x Q I P { Q } O P An illustration of the symmetry...

33 Symmetry... The -duplication P I { P α x y z Q } O P P I { P 1 α P 2 α y z Q } O P An illustration of the symmetry...

34 Duplication, informally Classical Logic Intuitionistic Logic Component duplicated, interface preserved. Similarly for erasure

35 X : propagation rules (left subgroup) (exp prop) : ( x P γ. α) β ŷr x (P β ŷr) γ. α (imp prop 1 ) : (P α [x] ẑq) β ŷr (P β ŷr) α [x] ẑq, β O(P) (imp prop 2 ) : (P α [x] ẑq) β ŷr P α [x] ẑ(q β ŷr), β O(Q) (cut(c) prop) : (P α x x.β ) β ŷr P α ŷr (cut prop 1 ) : (P α xq) β ŷr (P β ŷr) α xq, β O(P), Q x.β (cut prop 2 ) : (P α xq) β ŷr P α x(q β ŷr), β O(Q), Q x.β (L eras prop) : (x M) β ŷr x (M β ŷr) (R eras prop) : (M α) β ŷr (M β ŷr) α, α β (L dupl prop) : (x< x 1 x2 M]) β ŷr x< x 1 x2 M β ŷr] (R dupl prop) : ([M α 1 >α) β α 2 ŷr [M β ŷr α 1 >α, α 2 α β Propagation rules do not have a diagrammatic representation

36 The X calculus Propagation rules α ^ α ^ x Q

37 The X calculus Propagation rules α ^ α ^ x Q

38 The X calculus Propagation rules α ^ α ^ x Q

39 The X calculus Propagation rules α ^ α ^ x Q

40 The X calculus Propagation rules ^ α α ^ x Q

41 The X calculus Propagation rules ^ α α ^ x Q We may think of α xq as an explicit substitution

42 X : the basic properties 1. Linearity preservation If P is linear and P Q then Q is linear 2. Free names ( interface ) preservation If P Q then N(P) = N(Q) 3. Type preservation computation can be (has to be) seen as proof-transformation If P :. Γ and P P, then P :. Γ 4. Strong normalisation (termination of reduction)

43 Peirce s law in X x.α 1 :. x : A α 1 : A (capsule) x.α 1 β :. x : A α 1 : A, β : B x ( x.α 1 β) β. γ :. α 1 : A, γ : A B (R eraser) (exporter) y.α 2 :. y : A α 2 : A ( x ( x.α 1 β) β. γ) γ [z] ŷ y.α 2 :. z : (A B) A α 1 : A, α 2 : A [( x ( x.α 1 β) β. γ) γ [z] ŷ y.α 2 α 1 α 2 >α :. z : (A B) A α : A ẑ ([( x ( x.α 1 β) β. γ) γ [z] ŷ y.α 2 α 1 α 2 >α) α. δ :. δ : ((A B) A) A (capsule) (importer) (R duplicator) (exporter) (axiom) A A (R weakening) A A, B ( R) A, A B (axiom) A A ( L) (A B) A A, A (R contraction) (A B) A A ( R) ((A B) A) A

44 *X : (axiom) A A (R weakening) A A, B ( R) A, A B (axiom) A A ( L) (A B) A A, A (R contraction) (A B) A A ( R) ((A B) A) A X : (axiom) A A, B ( R) A, A B (axiom) A A ( L) (A B) A A ( R) ((A B) A) A

45 The terms of X are convenient for two-dimensional representation, due to the presence of erasers and duplicators but also the linearity constraints

46 Diagrammatic calculus Syntax P, Q ::= x α x P β α y P Q I E x α P α x Q P α β γ z x y P P α x P

47 Outline Sequent calculus, proofs and programs Implicit structural rules the X calculus Explicit structural rules the X calculus The X calculus : explicit erasure and duplication Logical setting From sequent proofs to terms The syntax and reduction rules Diagrammatic view Strong normalisation property

48 Encoding X in X calculus

49 Encoding preserves types

50 Simulating X reduction

51 Strong normalisation of X -calculus

52 Future work define equivalent terms, the c X calculus concurrency - which process calculus corresponds to this classical cut-elimination? a 3D computational model...

53 Some references S. Ghilezan, P. Lescanne, D. Žunić Comp. interpretation of classical logic with expl. struct. rules, draft. S. Ghilezan, J. Ivetic, P. Lescanne, D. Žunić Intuitionistic sequent-style calculus with explicit struct. rules, P. L. Curien, H. Herbelin The duality of computation, D. Kesner and S. Lengrand Explicit operators for λ-calculus, C. Urban and G. M. Bierman Strong normalisation of cut-elimination in classical logic, P. Wadler Propositions as sessions, E. Robinson Proof nets for classical logic, 2005.

54 Thanks..! Homepage: Dragisa Zunic