Derivatives of proofs in linear logic

Transcription

1 Derivatives pros in linear logic Daniel Murfet based on joint work with James Clift

2 Precis HK interpretation: intuitionistic pros give rise to functions Pros) Pros) Can se functions be differentiated? What would such ivatives be good for? 1 Efficient re)computation 2 Differentiable reasoning 3 Investigating logic vs physics ccording to rouwer-heyting-kolmogorov interpretation intuitionistic logic a pro -> is a construction transforming any pro into a pro This construction describes a function from set pros to set pros The subject this talk is following question: can such functions be differentiated? t first glance answer is obviously No or even worse that question itself is ill-posed To differentiate something means to vary input slightly and measure resulting variation in output ut input to such a function is a pro and pros are discrete syntactic things There is no reasonable sense in which we can vary a pro infinitesimally and have result still be a pro in usual sense Of course we could change meaning word pro and admit say all usual pros intuitionistic logic plus some formal things which stand for infinitesimal variations real pros Roughly speaking Ehrhard and Regnier showed that one can proceed in this manner to enlarge universe intuitionistic pros in such a way that resulting class has a natural notion ivatives at least for first-or logic) Then obvious question is: what good is it to differentiate a pro? I m not entirely clear on original motivations Ehrhard and Regnier but here are what I consi three best reasons to care about pro ivatives: a) un Curry-Howard correspondence this pro is a program taking inputs type and returning outputs type Suppose output is calculated for some particular input x and that it takes a week to complete calculation nd n suppose we want to know output for an input y which is close to x say that y is a 1 digit binary word differing in one digit from x) If we had computed ivatives program ahead time we could imagine using a Taylor expansion to compute output program on y from its output on x toger with its precomputed ivatives Obviously finding ivatives imposes some overhead cost but Taylor method will still be efficient for some use-cases One form this for programs computing real-valued functions is known as incremental computation and is widely used in applied computer science

3 Precis HK interpretation: intuitionistic pros give rise to functions Pros) Pros) Can se functions be differentiated? What would such ivatives be good for? 1 Efficient re)computation 2 Differentiable reasoning 3 Investigating logic vs physics a) Suppose re is a phenomena in world that I m trying to abstract in logic For concreteness say I m trying to represent a pattern I observe in nature by a sequence integers Perhaps at first my idea is in error in sense that pattern I have in my head is not consistent with all evidence In what direction should I vary my idea in or to bring it into better agreement with world? This kind smooth variation ideas is behind connectionist or neural network approach to machine learning and forms one widely subscribed model human reasoning It would be good to have a more principled way thinking about this b) There are many interesting speculations out re about connections between information ory logic and physics It s hard to know how seriously to take some m The most well-unstood lie in rmodynamics as I ve discussed in this seminar previously One very naive observation is following: physical instantiations computational processes are generally described by continuous physics at some level and so relevant processes admit ivatives To what extent are se ivatives just operational junk which bears no deep significance and to what extent are y shadow something meaningful in logic that we happen to have missed? We can t even ask this question until we have grappled with notion ivatives pros Let me now recontextualise se issues using backdrop Curry-Howard correspondence

4 Curry-Howard correspondence logic programming math formula type space pro program function cut-elimination execution contraction copying coalgebra?? calculus The first two columns in this table are standard Curry-Howard correspondence Formulas in logic correspond to types in programming and so on What I mean by third column is following: re are many semantics logic in which one assigns spaces to formulas and functions between spaces to pros On previous slide I was alluding to possibility expanding notion syntactic pros so as to have a pro-oretic notion ivative That s Step 1 ut given that we have some semantics logic in which denotation pros are functions between spaces Step 2 must be to make sure that se syntactic pro ivatives are consistent with usual calculus ivatives denotations The mechanism conciliation between syntactic and semantic ivatives is coalgebras as I will explain

5 Outline 1 History ivatives in logic 2 Derivatives in syntax 3 Relation to calculus via coalgebras based on arxiv: )

6 History ivatives in logic Leibniz s stepped reckoner 167s) abbage s difference engine 183s) Circuits and 2nd or differential equations utomatic differentiation real-valued programs Ehrhard-Regnier s differential lambda calculus 23) Differential linear logic I d like to give a very brief look at history logic and computing devices highlighting some strands that resonate with our me ivatives Perhaps very beginning such a history is Leibniz and his fascination with formal languages for reasoning and with devices for carrying out calculations in m The only device this kind that he actually built as far as I m aware was stepped reckoner which was a mechanical calculator for arithmetical operations The physical operation this machine is interesting from point view mon logic and its semantics This will be one example in this list that I will examine in detail so let me not say too much about it now I think most you will be familiar with abbage s difference engine which is usually given as one eminent ancestors today s computers It was designed to automatically compute values polynomial functions for example truncated Taylor expansion logarithm It did so using Newton s ory divided differences which is way that ivatives are dealt with within abstract algebra s students in my Calculus 2 class have just learned eleical circuits containing only capacitors inductors and resistors are modelled by second-or ordinary differential equations The most well developed link between logic and calculus that I'm aware is differential lambda calculus Ehrhard and Regnier and refinement its simplytyped variant in differential linear logic I'll go into detail about latter system later on in this talk

7 History: Leibniz s stepped reckoner The stepped reckoner was first true four-function calculator which formed basis for many calculator designs for next two hundred years Leibniz actually built several m I'm going to explain key innovation in mechanical design this calculator which is stepped drum that you see in animation I'll n explain how an idealised mamatical model this mechanism leads us to denotational semantics Church numerals

8 n History: Leibniz s stepped reckoner 2 1 fter a full rotation drum shaft rotates by nk = nk 2 = 2 4 Here is a top down view orange stepped drum and red gear and shaft There are gears on red shaft which I am not drawing here obviously Let ta and psi denote respectively angle rotation orange and red components We assume that integer n is fixed that is that we have selected position red gears along long axis stepped drum which is axis running out page such that after a single rotation orange drum red gears have encountered exactly n orange gear teeth The effect is to cause red shaft to ungo a total rotation nk \Delta \ta on 2\pi whenever \Delta \ta is a multiple 2 \pi Here k is quantity angle induced by a single gear tooth on orange drum

9 History: Leibniz s stepped reckoner n =3 fter a full rotation drum shaft rotates by nk = nk 2 = 2 4 This slide shows anor view same system This time long axis orange drum runs vertically on page and I'm showing horizontal axis dotted line) along which red gear runs when n is fixed to 3

10 History: Leibniz s stepped reckoner n =3 fter a full rotation drum shaft rotates by nk if we halve rotation caused by each tooth while doubling number) = nk 2 = 2 t moment this is not a very continuous model because an infinitesimal change in angle \ta does not induce an infinitesimal change in angle \psi To refine model into a continuous one we can double number vertical teeth on orange drum while halving angle rotation each individual tooth induces on red shaft In that case a full rotation \Delta \ta = 2\pi will still cause a rotation nk on red shaft ut a half rotation \Delta \ta = \pi will induce a rotation nk/2 That is we have arranged by this physical change to make formula relating \Delta \psi to \Delta \ta true for values \Delta \ta which are multiples \pi not just multiples 2\pi

11 History: Leibniz s stepped reckoner n =3 In limit infinitely many repetitions this group nine teeth d = nk d 2 = nk d k = k 2 d d = nk Obviously we can continue this adding more teeth and reducing induced angle so as to make formula true for in limit arbitrary values \Delta \ta In limit we get an idealised form stepped reckoner where d \psi is equal to n/k d\ta on 2\pi We may rewrite this as d \psi equals nk' d\ta where k' is k/2\pi That is rate change \psi with respect to \ta is nk' One way to physically implement this is to have an orange drum variable radius

12 History: Leibniz s stepped reckoner d d = n = n e i = e in =e i ) n U1) Upshot: The stepped reckoner gives a "physical semantics" Church numerals matching denotational semantics in vector spaces SO2) M 2 R) ) n ) n ) n = JnK U1) SO2) M 2 R) I'm not sure how seriously to take this example It might be for instance that many or mechanical aspects stepped reckoner as well as more complex systems like difference engine are just physical instantiations algorithms involving Church numerals It would be interesting to see if for example re is a relationship between way arithmetic operations on Church numerals are encoded into simply-typed lambda calculus or linear logic and physical mechanisms which implement arithmetic in se mechanical devices If that is case this example should be taken seriously ut in context this talk I want it to serve only to illustrate following point: algorithms in logic are discrete syntactic objects but y are represented in world as manipulations _continuous even differentiable_ systems In case we've just discussed algorithm is numeral n and we discussed two its representations: physical one in stepped reckoner and mamatical one in denotational semantics linear logic in vector spaces The continuous quantities in former case are angles and in latter case y are linear operators and se are related by fact that rotation by a given angle is a linear operator on vectors in plane If you buy that n claim that algorithms have meaningful ivatives doesn't seem as outrageous It is simply claim that ivatives _representation_ algorithm as a manipulation continuous objects) are so completely determined by algorithm itself that y are mselves representations something that properly belongs at level logic With that in mind we now move on to describe what se "somethings" are in logic

13 Derivatives in syntax Differential linear logic adds a new deduction rule which produces ivative a pro in a direction specified by a new linear) hyposis diff a 1 a 2 ) lim h a 1 + ha 2 ) a 1 ) h this is meaningless) In best formulation diff is ived from coeliction cocontraction and coweakening The idea extending linear logic with differential operators goes back right to genesis linear logic in Girard s exploration quantitative semantics lambda calculus and his paper on normal functors power series and lambda calculus in 1988 These ideas were furr developed in semantics papers by Ehrhard in early 2s and reached a kind fruition in a paper by Ehrhard and Regnier in which y developed a variant untyped lambda calculus with ivatives in 23 This was followed by development differential linear logic by same authors with important work on categorical semantics being done by lute Cockett and Seely

14 Dereliction): Deduction rules for intuitionistic first-or) linear logic 41) C -R Leaves are presented calculus notation with an empty numerator 45) Right in ): sequent Cut): cut Dereliction): C L Left): Definition 41 pro is a pre-pro toger with a compatible labelling vertices by C deduction rules The list deduction rules is given in first column 42) 414) labelling is compatible if at each vertex sequents labelling edges match incident Contraction): format displayed in deduction rule C 41) 44) 48) 411) In all deduction rules sets and may be empty and in particular promotion rule may be used with an empty premise In promotion rule stands for a list an C exponential modality weak 1 412) formulas each which -R Rightfor ):example n 46) 45) -L Left ):is preceeded by Weakening): C The diagrams on right are string diagrams and should be ignored until Section 5 In particular y are not trees associated to pros 411) 44) Contraction): cut Cut): C 1 C xiom): Promotion): cut 44) Cut): 42) prom C C Left ): -L48) 46) L Left ): 1-L 413) Left 1): C C C 1 1 -R 47) Right ): 45) R Right ): 1 C 43) Exchange): C Right ): -R 45) 1 1-R Right 1): 414) C Dereliction): 1 1 C C R 47) Right ): -L ): Example 42 For any Left formula let 2denote pro 24) from Section 2 which we46) C prom here for rea s convenience: 49) Promotion): repeat C -L 46) Left ): L C 8 L 9 R Contraction): 415) ) ) ) ) 47) R R Right ): int 41) Dereliction): 9 by reading above pro We also write 2 for pro ) obtained 1 constructed up Right to ): penultimate line For re is a pro n 47) int R each integer n along similar lines see [22 532] and [15 31] In what sense is this pro an avatar number 2? In context -calculus we appreciated relationship between term T and number 2 only after we saw how T interacted with or terms M by forming function application T M ) and n finding a normal form with respect to -equivalence The analogue function application in linear logic is cut rule The analogue 49) 41) 411)

15 Dereliction): 2 R 41) bint final right R introduction rules second copy ) associated with Deduction rulesin for intuitionistic first-or) linear logic n l = C 1 s in S) is moved across turnstile first If S is empty sequence and pro is a pair weakenings on left followed by R introduction rules For rest this section is fixed and we write S for S notation Leaves are presented calculus with an empty numerator Right in ): sequent 45) 1 : Cut): bint = ) ) -R cut )) 41) Dereliction): Example 31 The pro 1 is C L Left): Definition 41 pro is a pre-pro toger with a compatible labelling vertices by C L deduction rules The list deduction rules is given in first column 42) 414) labelling is compatible if at each vertex L sequents labelling incident edges match C 411) formatcontraction): displayed in deduction rule L R In all deduction rules sets and may be empty and in particular promotion 3 rule may be used with an empty premise In promotion rule stands for a list ) ) ) C formulas each which by an weak exponential modality -R Weakening): Rightfor ):example 1 n 412) -L Left ):is preceeded ) ) 46) diagrams C The diagrams on right are string and should be ignored until Section 5 In 2 R bint particular y are not trees associated to pros Contraction): ) are411) copies where contracted Using 18) cut colouring indicates Cut):which C 1 " " 3 48) Right 1): 1 1-R 1 13 Dereliction): 414) C C R Right ): -L Left ): Example 42 For any formula let 2 denote pro 24) 2 which we Section from C repeat herepromotion): for rea s convenience: prom 49) C -L Left ): CL 8 L 9 R ) ) ) Contraction): 415) C 45) 44) 21) 1" γ = comppromotion): = δ γ γc 44) δ cut γ " γ δ prom 42) Cut): Generalising calculation Section 31 we now describe ivatives binary integers The general formula computes for S { 1} linear operator C C -L Left ): "48) L Left 1-L C 413) β 1 C Left): 1): CS" α1 αr γ β Endk V ) s δ 1 1 is described byinserting γ for and δ for 1 in reversal ) this operator Informally -R 47) Right ): R Right ): replacing r γ s in this composite with αi s ways and n summing over all 1 C S 43) Exchange): t δ s with βj s Let InjP Q) denote set injective functions P Q C and -R Right 45) s} and write ): [s] = {1 1 xiom): 44) 49) 46) 45) 41) 46) ) Right ): R R int 41) Dereliction): 9 We also write 2 for pro ) obtained by reading above pro R is a pro n int constructed up to penultimate line For each): integer n re Right along similar lines see [22 532] and [15 31] In what sense is this pro an avatar number 2? In context -calculus we appreciated relationship between term T and number 2 only after we saw how T interacted with or terms M by forming function application T M ) and n finding a normal form with respect to -equivalence The analogue function in linear logic is cut rule The analogue application 47) 46) 411) 1 47) 47)

16 Dereliction): P P 41) P It seems more convenient to model differentiation syntactically using coeliction cocontraction andfor coweakening rar than iving transformation D itself maps Deduction rules differential linear logic C We briefly sketch how this works following [6] In sequent calculus forlinear logic one introduces three new deduction rules dual to eliction contraction and weakening: calculus Leaves are presented notation with an empty numerator Γ Right in ): sequent -R 45) cut co Cut): Coeliction): 41) Dereliction): Γ C L Left): Definition 41 pro is a pre-pro toger with a compatible labelling vertices by C deduction rules The list deduction rules is given in first column 42) 414) 8 labelling is compatible if at eachvertex labelling incident edges Γ match sequents co Cocontraction): 411) Contraction): format displayed in deduction rule C Γ Γ 44) co Cocontraction): In all deduction rules sets and may be empty and in particular promotion Γ Γ coweak Coweakening): for Γ rule may be used with an empty premise In promotion rule stands a list C by Γ weak 412) Weakening): coweak formulas each which is preceeded an exponential modality for example Coweakening): -R Right ): 1 n 45) -L Left ): Γ 5 In46) 143] rules toger string diagrams with C new The diagrams on right are andcut-elimination should be ignored until[6 Section with cut-elimination [6 143] particular y are not toger trees associated to pros new Given a pro πrules Definition 21 in linear logic ivative π is pro 411) Contraction): cut 44) Cut): 21 C in Definition Given a pro pro π linear logic ivative π is 1 π C xiom): 42) prom Promotion): cut 44) Cut): π 12) C co C L -L48) Left ): Left ): 12) 46) 413) Left 1): 1-L C C co 1 C co 1 ): co -R 47) Right 45) R Right ): 1 whose C denotation is by our earlier remark composite 43) Exchange): C Right ): -R 45) whose denotation is by earlier remark composite π" our " " " " 1 D 13) 1-R Right 1): π" 414) DDereliction): " " C 1 13) " " Remark 211 Given π as above we1 have function [26 Definition 51] Remark 211 Given π as above we have function [26 Definition 51] C " C PR % π" P π" " 14)47) nl : ): Right ): -L Left Example 42 For any formula let 2 denote pro 24) from Section 2 which we 46) C prom π"nl : " " π" P P % 49) 14) repeatpromotion): here for rea s P νconvenience: " we interpret and for vector C 46) Left C -L and for P we interpret vector ν ): " L 8 = 15) π"d π" ν " ν) P P L 15) π"d π" ν P Pν)in= P " as ivative π"9nl at point direction ν Here we implicitly identify R tangent space Contraction): Tpoint " direction Tπ"nl P ) " with tangent space with This and 415) P " ν Here we implicitly as " ivative at in identify π" P nl ) interpretation is justified by following elaboration remarks in Introduction tangent space tangent space Tπ"nl P ) " This " with P " ) ) T and " with π whichinhas Let promπ) denote by pro which is promotion its denotation interpretation is justified following elaboration remarks forintroduction ) morphism coalgebras " with = π" promπ)" : " d for promπ)" 47) R Right which R): is promotion Letunique promπ) denote pro which has its denotation π 2 int Ψ : k[ε]/ε ) be promπ)" morphism :" coalgebras as with in 27) to morphism " Let unique " coalgebras promπ)" = π" d corresponding 41) Dereliction): " vector at a" point Thenby morphism 9 coalgebras We also write Let 2tangent for pro ) be P obtained reading above pro 2 Ψ : k[ε]/ε ) ν to morphism coalgebras as in 27) corresponding 1 47) up to penultimate line For each integer n re is a pro n int constructed R Right ): 2 tangent vector ν at a point morphism coalgebras ) " Ppromπ)" " Then Ψ : k[ε]/ε 16) along similar lines see [22 532] and [15 31] Ψ : k[ε]/ε2 ) " promπ)" has following values writing Pcontext ) we have by -calculus [25 Theorem 222] Q= In what sense is this pro an avatar number 2? π" In nl we appreciated relationship termqt =and P number 2 onlybyafter saw has following between values writing ) we have [25we Theorem 222] nl π" promπ)"ψ1) promπ)" = = P QM ) and how T interacted with or terms M by forming function application T " = promπ)" = n finding a normal form with respect to -equivalence promπ)"ψ1) promπ)"ψε ) = promπ)" ν = Qπ" ν P P analogue " Q The analogue function application in linear logic is cut rule PThe 16) 48) 49) 41) 411)

17 Dereliction): P P 41) P It seems more convenient to model differentiation syntactically using coeliction cocontraction andfor coweakening maps rar than iving transformation D itself Deduction rules differential linear logic We briefly sketch how this works following [6] In sequent calculus for linear logic one introduces three new deduction rules dual to eliction contraction and weakening: Dereliction): Coeliction): Γ Γ co 41) 8 co Cocontraction): Γ co Cocontraction): coco Cocontraction): 411) Contraction): Cocontraction): Γ Γ coweak Cocontraction): Γ co Coeakening): Γ coweak Coweakening): coweak Coeakening): Coeakening): coweak Γ Γ weak toger with new cut-elimination rules [6 143] 412) Weakening): coweak Coweakening): with new cut-elimination rules 143] toger [6 Γ toger with new cut-elimination rules [6 143] toger with new cut-elimination rules [6 143] Definition 21 Given a pro toger in linear logic [6 is 143] pro with rules new cut-elimination Given π in linear logic Definition 21 a pro ivative π is pro 411) Contraction): Definition 21 Given a pro in linear logic is pro Definition 21 Given a pro Definition in linear logic is pro 21 Given pro π linear logic ivative π is pro 1 π in π 12) 12) co co 12) 12) co 1-L 413) Left 1): 12) co 1 co is defined to be diff co co co 1 co co 1 denotation is by our earlier remark composite whose denotation is by our earlier remark whose composite whose denotation isis bybyour whose denotation ourearlier earlierremark remark composite composite π" D 1 " " J K " " 13) whose denotation is by our earlier remark composite D / / JK JK Right 1):JK 1-R JK 13) J K DD π" / JK / 414) " " JK JK JK 13) 1 " 13) " J K Remark 211 Given as above we have function [26 Definition 51] π 1 D / JK JKwe have JK 13) Remark 211 Given JK as above function [26 / Definition 51] Remark 211 Given πasasabove Remark 211 Given abovewewehave have function function[26 [26 Definition Definition 51] 51] π" : " " P % π" nl P we Example 42 For any formula let 2 denote pro 24) from Section 2 which JK function P 7 [26 J K ;i : JK Remark 211 Given J K as above have Definition 51] nl : JK P nl π" " JK J K repeatwe here for rea s convenience: nl : " and for P ν " we interpret vector 14) π" PP J K ;i PP7 % and for P 2 JK we interpret vector J Knl : JK and JK 7 J K ;i 14) interpret interpret P vector and we vector ν JK " for for P P P 2 we L = π"d π" ν ν) P P " L J KD ;i 15)P 2 JK π" ν " P 2 JK J KD ;i π"d )ν)==j K i P ) = J K i and for P 2 JK we interpret vector P 14) 14) 14) 15) 15) 15) as ivative π"nl at point ν Here we implicitly identify P in direction P P R " Tpoint " direction Tπ"nl P ) " with tangent space with tangent This and 415) P " ivative at Herespace we implicitly implicitly identify π" ν Here as ivative J Knl at J KD ;i point as P as in direction Here implicitly identify ivative J K at point PPinin direction we identify ) we nl nl ) = J K i 2 JK 15) interpretation is justified by following elaboration remarks in Introduction P P with space and with tangent tangent space space TTJ K " " This P " π"nl P )JK JK with tangent space TP JK and JK with tangent space TJ K This JK" with tangent TPTpro JK and JK with This ) JK tangent ) space ) nl P P its ) denotation nl π Let promπ) denote which is promotion which has for interpretation is justified by following elaboration remarks in Introduction interpretation is justified by point following inimplicitly promπ)" Introduction interpretation is justified following elaboration remarks ind Introduction by we as ivative J Knl at P inelaboration unique direction ) remarks Here identify morphism coalgebras : " " with = π" promπ)" which R Let promπ) denote pro is promotion which has for its denotation π 2 Let prom ) denote pro which is promotion which has for its denotation Let prom ) denote pro which is promotion which has for its denotation int JK with tangent space TP JK and JK with tangent space TJ K JK :" This Let Ψ : k[ε]/ε ) " be morphism coalgebras as with in 27) to nl P ) unique morphism coalgebras coalgebras " promπ)" promπ)" = π" d corresponding unique morphism Jprom )K : JK JK with d Jprom )K = J K unique morphism coalgebras Jprom )K : JK JK with d Jprom )K = J K " tangent vector at a point Thenby morphism coalgebras 2 interpretation is justified by following remarks in Introduction We also write Let 2elaboration for pro ν ) P obtained reading above pro Ψ : k[ε]/ε 2 ) " be morphism coalgebras as in 27) corresponding to Let : k["]/" ) JK be morphism coalgebras as in 27) corresponding to LetLet:prom ) k["]/"2 )denote JK be morphism coalgebras as in 27) corresponding to up towhich penultimate line Forν each re is adenotation pro n constructed pro istangent promotion a integer which has for Then its vector at point morphism coalgebras " Pnpromπ)" 2 int Ψ : k[ε]/ε ) " 16) tangent vector at a and point 2 JK Then morphism coalgebras tangent vector at a point Palong 2 JK Then morphism coalgebras similar lines see [22 532] [15P 31] unique morphism coalgebras Jprom )K : JK JK with d Jprom )K = J K 2 promπ)" =Ψπ" : k[ε]/ε " 2 we ) has following values writing P ) have by -calculus [25 Theorem 222] Q 2 Let : k["]/"2 ) JK Jprom )K be morphism coalgebras as in 27) corresponding to nl In what sense is this pro number 2? :In context JK k["]/" ) : k["]/" ) an avatar JK Jprom )K 16) relationship termqt =and P number 2 onlybyafter saw tangent vector at a point Pwe2appreciated JK Then morphism coalgebras π" has following between values writing ) we have [25we Theorem 222] nl promπ)"ψ1) promπ)" = = P by [25 QM has following values writing Q = J K P ) we have Theorem 222] how T interacted with or terms M by forming function application T ) and nl has following values writing Q = J Knl P ) we have by [25 Theorem 222] " 2 with respect to -equivalence = promπ)" n finding a normal form promπ)"ψ1) = Jprom )K : k["]/" ) JK 16) promπ)"ψε ) = promπ)" ν P P = Qπ" ν P Qanalogue " Q Jprom )K 1) logic = Jprom )K ;i ;i function application in linear is cut rule P =The Jprom )K The 1) analogue = Jprom )K ;i = ;i P Q E 16) 16)

18 Product rule as cut-elimination rule ax D t t diff D IT : % Da : t t t t cut : t t t T my + D IT : cut Pros in differential linear logic are formal linear sums pro trees : tcut t t T

19 bint = ) ) )) E = repeat :bint bint comp 2 E E E E EE E L E E E E EE E L E E E E bint E E L E E E bint E E E L E E E E bint bint E E E E bint bint E E E bint bint E 2 R bint bint bint 2 bint bint bint bint bint J K

20 " " repeat J K J bint K J bint K J K diff bint J K J bint K J bint K J K J K J K ST) 7 ST + TS S + "T )S + "T )=SS + "ST + TS)+" 2 TT

21 Relation to calculus via coalgebras Following Ehrhard-Regnier we have defined ivatives in syntax via new deduction rules and cut-elimination rules Do se syntactic ivatives capture logical content lying behind semantic ivatives? In particular are y consistent with role Church numerals in Leibniz s stepped reckoner? Yes: because coalgebras So far I have sketched how to add new deduction rules and cut-elimination rules to intuitionistic first-or linear logic in such a way that you can differentiate pros nd I've shown that in at least one example ivative you get has some reasonable interpretation ut question remains: do se syntactic ivatives capture logical content lying behind ivatives representations se algorithms? For example does ivative Church numeral according to se rules match up with ivatives map which sends a matrix to its nth power? If this were case at least in this example syntactic ivatives really would capture what is going on with calculus involved in stepped reckoner The answer to both this particular case and broa questions is Yes To cut a long story short reason is that nonlinearities in linear logic are built on connective which gets interpreted as a cree coalgebra Derivatives polynomial functions can be computed using morphisms out coalgebra which is dual dual numbers So some standard mamatics about coalgebras and ir relation to calculus plus fact that coalgebras are built into foundations linear logic reconciles syntactic and semantic ivatives

22 o o lgebras over a field k multiplication m : u : k unit m 1 / 1 m associativity m / m = / k = / k 1 left unit m u 1 1 right unit m 1 u efore I can explain role coalgebras in differential linear logic I have to give a brief introduction to coalgebras and define cree coalgebra

23 o O o o o O O Coalgebras over a field k comultiplication : c : k counit 1 1 coassociativity 1 O = left counit k O c 1 1 O = right counit k 1 c / /

24 Examples polynomial algebra k[x 1 x n ] ring dual numbers k["]/" 2 )=k 1 k " " 2 = polynomial coalgebra k[x 1 x n ] nx x n )= x i x n i= i dual ring dual numbers k["]/" 2 )) = k 1 k " 1) = 1 1 " )=1 " + " 1 For us two most important examples coalgebras are cree coalgebra on a vector space which I ll introduce in a moment and coalgebra dual to ring dual numbers The ring dual numbers encodes algebraically idea an infinitesimal and it is standard in algebraic geometry to think tangent vectors at a point a space in terms functions to space from space with one point and an infinitesimal tangent vector which is speum ring dual numbers Without getting too far into that I d like to sketch why algebra morphisms into ring dual numbers are synonymous with tangent vectors since this will be crucial to explaining how to think about ivatives using just coalgebras and morphisms coalgebras

25 Consi a morphism k-algebras ' k[x 1 x n ] k["]/" 2 )=k 1 k " 'x i )= i + µ i " it is straightforward to see that for any polynomial f 'f) =f 1 n)+ X i i x= ~ " this gives rise to a bijection k-algebra morphisms with pairs 1:1 Hom k lg k[x 1 x n ]k["]/" 2 )) k n k n ' ~ ~µ) point tangent vector)

26 O 7 Universal coalgebra The cree coalgebra CV ) over a vector space V is a coalgebra toger with a linear map d : CV ) V which is universal in sense that for any coalgebra C and linear : C V re unique morphism coalgebras such that d = CV ) d / V C Theorem: CV ) is space distributions with finite support on V ie all ivatives Dirac distributions

27 Sweedler semantics J K : LL Vect J K = Hom k JK JK) J K = JK JK JK = CJK) eliction = universal linear map JK JK contraction = comultiplication JK JK JK weakening = counit JK k promotion = lifting JK JK to JK JK

28 = = Sweedler semantics J K : LL Vect J K = Hom k JK JK) J K = JK JK JK = CJK) diff The Sweedler semantics is also a semantics differential linear logic as follows: JK JK CJK) JK D D JK CJK) J K JK The important intuition here is that in Sweedler semantics to feed an input x to an algorithm \pi what you actually do is feed Dirac distribution at point x into denotation \pi To differentiate \pi in some direction at x you feed into [[\pi]] ivative Dirac distribution in that direction

29 V V point tangent vector) µ) V = k n Hom k lg k[x 1 x n ]k["]/" 2 )) = = Hom k SymV )k["]/" 2 )) = Hom k V k["]/" 2 )) = Hom k k["]/" 2 )) V) Hom k Coalg k["]/" 2 )) CV )) = 1 7 Dirac " µ Dirac

30 How to differentiate a pro denotation Given : : so that J K J K 2 JK CJK) = JK J K JK J K J K) k["]/" 2 )) J )K ) JK = CJK) The directional ivative at in direction

31 Conciliation: syntax vs semantics The semantics intuitionistic first-or) linear logic in vector spaces uses cree coalgebras to model contraction weakening and eliction Since cree coalgebra is made up Dirac distributions and ir ivatives this semantics is naturally a model differential linear logic Linear logic secretly wants to be differentiated

32 Conclusion/Questions Derivatives are natural in linear) logic Examples like stepped reckoner suggest use calculus in logic is justified re re more convincing mechanical examples this kind? The Sweedler semantics is a step in direction more interesting algebra and geometry What is logical content distributions with more general support? Differential linear logic forms basis for one approach to integrating symbolic reasoning with neural networks work in progress with H Hu)