Stratifications and complexity in linear logic. Daniel Murfet

Transcription

1 Stratifications and complexity in linear logic Daniel Murfet

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3 Curry-Howard correspondence logic programming categories formula type objects sequent input/output spec proof program morphisms cut-elimination execution contraction copying coproducts stratification complexity? this talk

4 Outline 1 Turing machines 2 Sequent calculus of linear logic 3 Programs (Turing machines) as proofs 4 Stratification vs complexity

5 Turing machines w op L w R configuration is (w L,w R,a) 2 {0, 1} {0, 1} {1,,q} binary integer binary integer q-boolean Shown configuration: (1011, 101, 3)

6 Turing machines Turing machine T is a function T : {0, 1} {1,,q}! {0, 1} {1,,q} {L, R} read symbol current state write new state move to {0, 1} {0, 1} {1,,q}! {0, 1} {0, 1} {1,,q}

7 Linear logic Discovered by Girard in the 1980s, linear logic is a substructural logic with contraction and weakening available only for formulas marked with an exponential connective, written! The usual connectives of logic (eg conjunction, implication) are decomposed into! together with a linearised version of that connective (called resp tensor, linear implication) Under Curry-Howard, linear logic corresponds to a programming language with resource management and symmetric monoidal categories equipped with a special kind of comonad We will use second-order intuitionistic linear logic with additives (as expressive as polymorphic lambda calculus)

8 Linear logic variables:,,, formulas:!f, F F 0,F ( F 0,F&F 0, 8 F, constants int = 8!( ( ) ( ( ( ) bint = 8!( ( ) (!( ( ) ( ( ( )

9 for a list of ` may be used with an empty premise In the promotion rule,! stands ulas each of which is preceeded by an exponential modality, for example0!1,,!n 0, ` ` ` -R, ` ( (45) (Right diagrams on the right are ): string diagrams and should be ignored until Section 5 In,,, ` C C cut (Cut): Deduction logic 0,` -Lrules for linear! ` (46) (Left ):,, ` prom (49) (Promotion): cular they are not the trees associated to proofs,! `!, ` C! 0 (xiom): `,, ` (Left (): 0,, (, (Left ): `C (Cut): (L `C (Promotion):! `! `!,,, ` C,, 0 ` C,, ` (Dereliction):,,, ` C!, `` (Exchange): (Right ():,,,,, `,, ` (Exchange): cut C (48) (42) 0,, `! 0 der (R ` C `(Right ( ): -L prom C (Right ): ( ( 0 ` `, `,,,,,, `C `C 0 ( (49) C -R (46)! (410) 0! `,, ` C (43) ` (Left`(): -R 0,, (, ` C ( L (47), ` (! C ( 8 Definition 75 (Second-order linear logic) The Definition 75 (Second-order linear logic) The formulas of second-order linear logic, formula `,, ` recursively! ` are defined as follows: any pr prom,,, ` C weak of (Weakening): (49) der (Promotion): (410 are defined recursively as follows: any formula of (Left propositional linear logic is a formula, (Dereliction): -L ):,!, 0 `,!, `! `!, x, `so Care if is a variable formula then is 8x for any proposi and if is a formula then so is 8x for anyand propositional There two new C deduction rules: deduction rules:, ` (47) (R (Right ():!! ` ` `(8R, [/x] ` C ` 8R,!,!,, 8L (411) (Contraction): ` 8x ctr ` 8x,!, `, 8x,,!, ` C ` C -L (Left ):,, ` C is a sequence of formulas,8possibly empty, and in the right introduction rule we,8 ( where ( Here [/x] is a sequence of formulas, possibly empty, a require that x is not free in any formula of where means a formula with all, `!! ` (Left 1-L Here 1):necessary prom that x is not free in any formula of a sequent of is x replaced ` forbya asequence of(as formulae, where ` is the turnstile (Promotion): free occurrences formula require usual, there is some chicanery,!,!, `, 1, `! `! ctr (411 (Contraction): to avoid free variables in being captured, free but,we occurrences!,ignore ` this)of x replaced by a formula (as usu 9,, `

10 inary integers bint = 8!( ( ) (!( ( ) ( ( ( ) S 2 {0, 1} 7! proof t S of ` bint ` ` ( L `, ( ` ( L `, (, ( ` ( L, (, (, ( ` (, (, ( ` ( der t 001!( ( ),!( ( ),!( ( ) ` ( ctr!( ( ),!( ( ) ` (!( ( ) `!( ( ) ( ( ( ) `!( ( ) (!( ( ) ( ( ( )

11 side on linear logic!, ` C

12 inary integers bint = 8!( ( ) (!( ( ) ( ( ( ) S 2 {0, 1} 7! proof t S of ` bint ` ` ( L `, ( ` ( L `, (, ( ` ( L, (, (, ( ` (, (, ( ` ( der t 001!( ( ),!( ( ),!( ( ) ` ( ctr!( ( ),!( ( ) ` (!( ( ) `!( ( ) ( ( ( ) `!( ( ) (!( ( ) ( ( ( )

13 inary integers bint = 8!( ( ) (!( ( ) ( ( ( ) S 2 {0, 1} 7! proof t S of ` bint input g input f output ffg = g f f 001 7! {(f,g) 7! ffg} 101 7! {(f,g) 7! gfg} ` ` ( L `, ( ` ( L `, (, ( ` ( L, (, (, ( ` (, (, ( ` ( der!( ( ),!( ( ),!( ( ) ` ( ctr!( ( ),!( ( ) ` (!( ( ) `!( ( ) ( ( ( ) `!( ( ) (!( ( ) ( ( ( ) t 001

14 Stratified Linear logic variables:,,, formulas:!f, F, F F, F ( F, F&F, 8 F, constants bint = 8!( ( ) (!( ( ) ( ( ( ) int = 8!( ( ) ( ( ( )

15 (Left ): (46) -L, an, empty ` C premise In the promotion rule,! stands for a list o rule may be used with 0 formulas each by an exponential modality, for example!1,,!n ` of which,, is` preceeded C ( L (Left (): 0 Deduction rules for stratified linear (48) logic until Section 5 I The diagrams,, on (the, right ` C are string diagrams and should be ignored!! ` particular they are not the trees associated to proofs prom (Promotion): same rules as! before eg `! (xiom): ( i 0,, ` (` cut (Cut): 0,, `! (410) i i! 0 `,, ` C ( L (Left (): 0,,C (, ` C i! i+1 0 `,, `i i der (Dereliction): i,!, ` i, ` (Right (): `( (R i i+1 (Promotion):! ` prom (Dereliction):! `!,,, ` C (Exchange): =1 (Contraction): i,,, i i,!,!, `,!, ` i `C ctr,, `,!, ` i (Right ): plus! i der C (47), ` (Weakening): (49),!, ` ` `, ` -R i+1, `, ` i (42) i 0 0 weak (43) i+1!, ` (411)!, ` i!, ` 1-L (Left 1): proof in the stratified sequent calculus is a proof the sense,! `in promusual, 1, ` (Promotion):,!,!, `! `! ctr (Contraction): together with a stratification, which is9an assignment of integers to all,(left!, ): `,,, ` C -L occurrences such that conclusions, are assigned 0 and the,, of` formulas,, ` C 10 der (410) (Dereliction): assignment changes across deduction rules are as shown in blue,!, ` 8!! C

16 inary integers bint = 8!( ( ) (!( ( ) ( ( ( ) S 2 {0, 1} 7! proof t S of ` bint ` ` ( L `, ( ` ( L `, (, ( ` ( L, (, (, ( ` (, (, ( ` ( der t 001!( ( ),!( ( ),!( ( ) ` ( ctr!( ( ),!( ( ) ` (!( ( ) `!( ( ) ( ( ( ) `!( ( ) (!( ( ) ( ( ( )

17 inary integers (stratified) bint = 8!( ( ) (!( ( ) ( ( ( ) S 2 {0, 1} 7! proof t S of ` bint ` ` ( L `, ( ` ( L `, (, ( ` ( L, (, (, ( ` (, (, ( ` ( der, t 001!( ( ),!( ( ),!( ( ) ` ( ( ) ctr!( ( ),!( ( ) ` ( ( )!( ( ) `!( ( ) ( ( ( ) `!( ( ) (!( ( ) ( ( ( )

18 inary integers (stratified) bint = 8!( ( ) (!( ( ) ( ( ( ) S 2 {0, 1} 7! proof t S of ` bint 1 ` 1 1 ` 1 ( L 1 ` 1 1, ( 1 ` 1 ( L 1 ` 1 1, ( 1, ( 1 ` 1 ( L 1, ( 1, ( 1, ( 1 ` 1 ( 1, ( 1, ( 1 ` ( 1 der,!( ( 0 ),!( ( 0 ),!( ( 0 ) ` ( ( 0 ) ctr!( ( 0 ),!( ( 0 ) ` ( ( 0 )!( ( 0 ) `!( ( ) ( 0 ( ( ) `!( ( ) (!( ( 0 ) ( ( ( ) t 001

19 Integers int = 8!( ( ) ( ( ( ) 8n 2 N there is a proof n of ` int ddition is a proof of int, int ` int Multiplication is a proof of int, int ` int polynomial of degree k is a proof of int ` int

20 Integers (stratified) int = 8!( ( ) ( ( ( ) 8n 2 N there is a proof n of ` int (note that!( ( ) ( ( ( ) is not provable) ddition is a proof of int, int ` int Multiplication is a proof of int, int ` int polynomial of degree k is a proof of int ` k int

21 Turing machines as proofs w op L w R Tur = bint bint bool q Configuration (w L,w R,q) of Turing machine 7! proof of ` Tur Instructions for a Turing machine T 7! proof of ` Tur ( Tur

22 Running a Turing machine (Identify a Turing machine T with a proof of ` Tur ( Tur) T prepare initial state ` Tur ( Tur `!(Tur ( Tur) prom bint ` Tur Tur ` Tur bint, Tur ( Tur ` Tur bint,!(tur ( Tur) ( (Tur ( Tur) ` Tur bint, int ` Tur 8 L ( L ( L input binary integer number of steps n to run config of Turing machine after n steps

23 Running a Turing machine (stratified) Tur = bint bint bool q prepare initial state T bint ` Tur Tur ` Tur ( L ` Tur ( Tur `!(Tur ( Tur ) prom bint, Tur ( Tur ` Tur bint, (Tur ( Tur ) ` Tur bint,!(tur ( Tur ) ( (Tur ( Tur ) ` Tur bint, int ` Tur 8 L ( L

24 Theorem (Girard) function {0, 1}! {0, 1} is polytime if and only if it can be typed as a proof of bint ` bint which admits a stratification bint ` k+2 bint stratifies bint ` bint

25 Theorem (Girard) function {0, 1}! {0, 1} is polytime if and only if it can be typed as a proof of bint ` bint which admits a stratification f : {0, 1}! {0, 1} computed by a Turing machine T with polyclock P length P copy bint ` bint bint bint ` bint bint ` bint int bint ` int bint ` int bint bint ` bint int int ` int cut bint ` Tur R, L cut iterate T bint, int ` Tur bint int ` Tur

26 Theorem (Girard) function {0, 1}! {0, 1} is polytime if and only if it can be typed as a proof of bint ` bint which admits a stratification f : {0, 1}! {0, 1} computed by a Turing machine T with polyclock P length P copy bint ` bint bint bint ` bint bint ` bint int bint ` int bint ` int bint bint ` bint int int ` int cut R, L cut iterate T bint, int ` Tur bint int ` Tur read off output ` bint ` Tur Tur ` bint bint ` bint Upshot: computes f

27 Theorem (Girard) function {0, 1}! {0, 1} is polytime if and only if it can be typed as a proof of bint ` bint which admits a stratification f : {0, 1}! {0, 1} computed by a Turing machine T with polyclock P copy bint ` (bint bint ) P int ` k int iterate T bint, int ` Tur bint ` k+2 bint stratifies bint ` bint

28 Summary There is a notion of stratification for proofs Turing machines can be encoded into linear logic If a Turing machine is polytime, the stratification of the clock polynomial gives a stratification of the corresponding proof in linear logic Theorem: a function of binary integers is polytime iff it admits a stratification

29 References JY Girard, Light linear logic, Information and Computation 14, 1995 P aillot, D Mazza Linear logic by levels and bounded time complexity, Theoretical Computer Science 4112, 2010 P oudes, D Mazza, and L Tortora de Falco, n abstract approach to stratification in linear logic, Information and Computation 241, 2015 D Murfet, Logic and linear algebra: an introduction, arxiv: Slides of this lecture available at therisingseaorg