Notes on proof-nets. Daniel Murfet. 8/4/2016 therisingsea.org/post/seminar-proofnets/

Transcription

1 Notes on proof-nets Daniel Murfet 8/4/2016 therisingseaorg/post/seminar-proofnets/

2 Why proof-nets Girard Proof-nets: the parallel syntax for proof-theory

3 The formulas of second order unit-free multiplicative exponential linear logic (mell) aregeneratedbythefollowinggrammar,wherex, X range over a denumerable set of propositional variables:, B ::= X X B Linear negation is defined through De Morgan laws: B! X X (X) = X (X ) = X ( B) = B ( B) = B (!) = () =! ( X) = X ( X) = X ( ) = Two connectives exchanged by negation are said to be dual Note that the self-dual paragraph modality is not present in the standard definition of mell [Girard, 1987]; we include it here for convenience lso observe that full linear logic has a further pair of dual binary connectives, called additive (denoted by and ), which we shall briefly discuss in Sect 5 They are not strictly needed for our purposes, hence we restrict to mell in the paper Linear implication is defined as B = B Multisets of formulas will be ranged over by Γ,, For technical reasons, it is also useful to consider discharged formulas, which will be denoted by, where is a formula Baillot, Mazza Linear logic by levels and bounded time complexity, 2009

4 Sequent calculus, xiom Γ,, Γ, Cut Γ,,B Γ,,B Γ,, B Tensor Γ, B Par Γ, Γ, X For all (X not free in Γ) Γ,[B/X] Γ, X Exists Γ, Γ,! Promotion Γ, Γ, Dereliction Γ Γ,, Γ, Weakening Γ, Contraction Daimon Γ Γ, Mix

5 Proof-net links ax axiom B B [B/X] [Z/X] B B X X pax tensor par exists for all flat pax why not of course paragraph!! π B 1 B n pax pax B n! B 1 promotion! dereliction, weakening Baillot, Mazza Linear logic by levels and bounded time complexity, 2009

6 Example of a proof-net titiosnot_ttototosotf_artotott8hothotto_ctri3cot3kot3t81loitot7c@ Church numeral 2 ) t8 2 in one-sided sequent calculus :

7 Exercise: what is this 1 ax ax ax ax z 1 s 1 s

8 Figure 4: xiom step ax B B B B B B B B B B B Figure 4: xiom step Figure 5: Multiplicative step Figure 5: Multiplicative step [B/X] [Z/X] X X [B/X] [B/Z] Figure 6: Quantifier step; the substitution is performed on the whole net 1 n 1 n π 0 π 0 π 0 π 0 1 n Γ 1 Γ n π 0 π 0 Γ Γ 1 Γ n π 0 π 0 pax! Γ! π Γ 0 Γ 1 Γ n pax! Γ! Γ pax! Γ! Γ Γ Γ Figure Γ Γ 7: Exponential step; Γ is a multiset of discharged formulas, so one pax prom/der, prom/weak link, why not link, or wire in the picture may Γ in some case stand for several (including zero) pax links, why not links, or wires Cut-elimination Figure 7: Exponential step; Γ is a multiset of discharged formulas, so one pax link, why not link, or wire in the 13 picture may in some case stand for several (including zero) pax links, why not links, or wires Figure 7: Exponential step; Γ is a multiset of discharged formulas, so one pax link, why not link, or wire in the picture may in some case stand for severa (including zero) pax links, why not links, or wires

9 Cut-elimination statement of strong normalisation (from Pagani-Tortora de Falco Strong Normalization Property for Second Order Linear Logic )

10 Cut-elimination: explosion (-elimination introduces new s at higher depth) π! pax pax! pax pax! pax pax!

11 From sequent calculus proofs to proof-nets 1 2 Γ 1 2 B Γ B tensor Γ par B B [B/X] Γ X exists Γ paragraph [Z/X] Γ X for all (X not free in Γ) Γ weakening Γ dereliction Figure 3: Rules for building sequentializable nets

12 From sequent calculus proofs to proof-nets Definition 5 (Sequentializable net) We define the set of sequentializable nets inductively: the empty net and the net consisting of a single axiom link are sequentializable (daimon and axiom); the juxtaposition of two sequentializable nets is sequentializable (mix); if, 1, 2 are sequentializable nets of suitable conclusions, the nets of Fig 3 are sequentializable; if Γ B 1 B 1 B n B n B 1 B n is a sequentializable net, then the net is a sequentializable net, then the net B 1 is sequentializable (promotion); if B 1 B n B n pax pax pax pax! B 1 B 1 B n B n B 1 B n! is sequentializable (contraction) Γ

13 From sequent calculus proofs to proof-nets sequent calculus proofs with the same proof-net (from Davoren Lazy Logician s Guide to Linear Logic p140, p156, p157)

14 From sequent calculus proofs to proof-nets definition of switching (note Baillot-Mazza is wrong, see Pagani-Tortora de Falco and Girard Proof-nets: the parallel syntax for proof-theory )

15 Example of switching acyclicity ) t8 :

16 Example of switching cyclicity (non proof-nets) a(9 0) at atom!t*3!

17 From sequent calculus proofs to proof-nets statement of sequentialisation theorem (from Girard)