Proof Nets and Boolean Circuits

Transcription

1 Proof Nets and Boolean Circuits Kazushige Terui National Institute of Informatics, Tokyo 14/07/04, Turku.1/44

2 Motivation (1) Proofs-as-Programs (Curry-Howard) corresondence: Proofs = Programs Cut-Elimination = Comutation (Normalization) Usually interested in infinite, uniform, seuential comutation such as functional rograms. Can be extended to finite, nonuniform, arallel comutation such as boolean circuits? 14/07/04, Turku.2/44

3 Motivation (2) Our goal: Proofs = Circuits Cut-Elimination = Evaluation What system of Proofs? Cut-elimination in Classical/Intuitionistic Logics: Non-elementary time. Too much! 14/07/04, Turku.3/44

4 Motivation (3) Linear Logic (Girard 87): a decomosition of Classical/Intuitionistic Logics: Multilicatives Classical Logic Additives Exonentials MLL: Classical Logic without weakening nor contraction. Has a nice arallel syntax: Proof Nets (Girard 87). Quadratic time cut-elimination rocedure. 14/07/04, Turku.4/44

5 Motivation (4) Our secific goal: corresondence between MLL roof nets and circuits. (Mairson & Terui 2003) Encoding of circuits by linear size MLL roof nets = P-comleteness of cut-elimination in MLL. Proof nets can reresent all finite functions as SIZE-efficient as circuits. What about the DEPTH-efficiency? What about the converse? 14/07/04, Turku.5/44

6 Parallel comutation Effective arallelization: Achieve a dramatic seed-u by use of a reasonable number of rocessors. Addition: School method (O(n) time) = Parallel algorithm: (constant time) In boolean circuits, time = deth num of rocessors = size (num of gates) Fundamental uestion: Are all feasible algorithms effectively arallelizable? NC = P roblem: Are all roblems in P solvable by olynomial size oly-log deth circuits? 14/07/04, Turku.6/44

7 Outline Unbounded fan-in boolean circuits. Proof nets for MLLu and arallel cut-elimination rocedure. Simulation of circuits by roof nets Simulation of roof nets by circuits Proof net comlexity classes 14/07/04, Turku.7/44

8 Boolean circuits (1) An unbounded fan-in circuit with n inuts (and 1 outut): a directed acyclic grah made of - inut nodes x 1,...,x n - boolean gates,, (and ossibly more). (there is a distinguished node for outut.) and may have an arbitrary number of inuts. size = the number of gates deth = the length of the longest ath 14/07/04, Turku.8/44

9 Boolean circuits (2) A circuit C accets w = i 1 i n {0,1} n if C[x 1 := i 1,...,x n := i n ] evaluates to 1. C accets X {0,1} n if C accets w w X. 14/07/04, Turku.9/44

10 Formulas of MLLu. Formulas: α,α Literals n (A 1,...,A n ), n 2 n-ary Conjunction n. (A 1,...,A n ), n 2 n-ary Disjunction Negation: defined by ( n (A 1,...,A n )) n (A n,...,a 1 ) (. n (A 1,...,A n )) n (A n,...,a 1 )... Notation: A B 2 (A,B) A B 2 (A,B) A B A B 14/07/04, Turku.10/44

11 Seuent calculus for MLLu Seuents: Γ, where Γ is a multiset of formulas. Inference Rules: A,A (Axiom) Γ,C,C Γ, (Cut) Γ 1,A 1 Γ n,a n Γ 1,...,Γ n, n (A 1,...,A n ) n Γ,A 1,...,A n Γ,. n (A 1,...,A n ). n Exchange is imlicit. No weakening, no contraction. 14/07/04, Turku.11/44

12 Proof nets for MLLu Links: n 0 n 0 1 Each link has several orts. Princial ort(s) numbered 0. Cut: an edge connecting two rincial orts. r i n n 0 n n 1 0 Proof nets are obtained from seuent roofs by extracting their structures (forgetting about formulas). 14/07/04, Turku.12/44

13 Examle: Negation α. α,α α α,α α,α 3 (α. α,α,α),α,α,α α 3 (α. α,α,α), 3. (α,α,α α) = r 14/07/04, Turku.13/44

14 Examle: Booleans B α α α α. 3 (α,α,α α). α,α α,α α,α,α α 3 (α,α,α α) 2. 3 = b 1. α,α α,α α,α,α α 3 (α,α,α α) 2. 3 = b 0 14/07/04, Turku.14/44

15 Reduction rules Axiom reduction: r 0 0 i 0 r i 0 Multilicative reduction: 1 n n 0 n n n n /07/04, Turku.15/44

16 Examle: Comuting Negation of True 14/07/04, Turku.16/44

17 Examle: Comuting Negation of True 14/07/04, Turku.17/44

18 Examle: Comuting Negation of True 14/07/04, Turku.18/44

19 Seuential cut elimination Size P : number of links. 14/07/04, Turku.19/44

20 Seuential cut elimination Size P : number of links. Theorem (Girard 87): Every roof net P reduces to a cut-free roof net in P stes. 14/07/04, Turku.19/44

21 Seuential cut elimination Size P : number of links. Theorem (Girard 87): Every roof net P reduces to a cut-free roof net in P stes. Too slow! 14/07/04, Turku.19/44

22 Parallel cut elimination (1) Alying two axiom reductions in arallel may conflict: = Global axiom reduction: 1 2 n... r r 14/07/04, Turku.20/44

23 Parallel cut elimination (2) Parallel multilicative reduction: 1 n n 0 n n n n 0 n n n n 0 n n 1 0 = 1 n n n n n n Parallel axiom reduction: n r n r n r = r r r P 1 = P 2 if P 2 is obtained from P 1 either by arallel m-reduction or by arallel a-reduction. 14/07/04, Turku.21/44

24 Parallel cut elimination (3). What controls the runtime of arallel cut-elimination? Deth d(p): Maximal deth of cut formulas in it. (P is assumed to be tyed by rincial tyes (most general tyes)) Deth of formulas: d(α)=d(α ) = 1 d( n (A 1,...,A n )) = d( n (A 1,...,A n )) = max(d(a 1 ),...,d(a n )) /07/04, Turku.22/44

25 Parallel cut elimination (4) Theorem: Every roof net P reduces to a normal form in 2 d(p) arallel reduction stes. Proof: By alying arallel a-reduction, every cut becomes multilicative. By alying arallel m-reduction, the deth decreases by 1. 14/07/04, Turku.23/44

26 Limitation on arallelization P n : n...{ P n and d(p n ) are linear in n. Parallel cut-elimination takes almost as long time as seuential cut-elimination. 14/07/04, Turku.24/44

27 Reresenting circuits by roof nets Idea: Reresent Circuits by Proof nets Boolean values by Proof nets (b 1, b 0 ) Assignment by Cuts Evaluation by Cut elimination 14/07/04, Turku.25/44

28 Reresentation of Parity Parity: PARITY n (x 1,...,x n )=1 iff the sum of x 1,...,x n is odd. CANNOT be reresented by (oly-size) circuits of constant deth The conclusion is B,...,B,B }{{} n times 14/07/04, Turku.26/44

29 Correctness of arity There are exactly 2 crossings in the drawing. Multilicative reduction does not change the num of crossings. Axiom reduction reserves the arity of the num of crossings: = = Therefore, the arity is 0 after cut-elimination. 14/07/04, Turku.27/44

30 Exressivity of flat roof nets Would break down if there were a self-crossing: = But self-crossing can be avoided. As far as roof nets of conclusion B,...,B,B are }{{} n times concerned, all what matters is the arity of the num of crossings. Theorem: Every roof net with the above conclusion reresents either PARITY n or PARITY n. 14/07/04, Turku.28/44

31 Boolean roof nets Boolean roof net: a roof net with conclusion B[A 1 /α],...,b[a n /α], m (B, G) for some A 1,...,A n and G (garbage). Given w = i 1 i n {0,1} n, define P(w) to be: b.... in bi P n P accets w {0,1} n if P(w) reduces to (b 1, P G ) for some roof nets P G with conclusions G. 14/07/04, Turku.29/44

32 Conditional and Disjunction if then P 1 else P 2 P2 2 P1 1 r Disjunction: or(,) if then b 1 else Disjunctions of arbitrary arity can be reresented by roof nets of constant deth. 14/07/04, Turku.30/44

33 Comosition Comosition: P Q r s The deth may increase:. P 1 Γ,B. P 2 Γ[A/α],B[A/α] B [A/α],,B Γ[A/α],,B 14/07/04, Turku.31/44

34 Proof Net for Majority (1) MAJ n (x 1 x n )=1 if at least half of x i s are 1. Let id,sh : {0,1} n+1 {0,1} n+1 be: id(i 1 i n+1 ) = i 1 i n+1 sh(i 1 i n+1 ) = i 2 i n+1 i 1 F: higher order functional F(0) = id F(1) = sh 14/07/04, Turku.32/44

35 Proof Net for Majority (2) MAJ n given by MAJ n (x 1 x n )=FstBit(F(x 1 ) F(x n )(0 0 }{{} 1 1 }{{})) n/2 n/2+1 Examle: MAJ 6 (101101) = FstBit(F(1)F(0)F(1)F(1)F(0)F(1)( )) = FstBit(sh id sh sh id sh( )) = FstBit( ) = 1 Reresented by roof nets of constant deth. (CANNOT be reresented by (oly-size) circuits of constant deth.) 14/07/04, Turku.33/44

36 St-connectivity 2 - Inut: an undirected grah of degree 2 (i.e., nonbranching grah) with vertices {1,...,n}. Assume it is coded by a n n boolean matrix. - Outut: is 1 if vertices 1 and n are connected. 1 n 14/07/04, Turku.34/44

37 St-connectivity 2 - Inut: an undirected grah of degree 2 (i.e., nonbranching grah) with vertices {1,...,n}. Assume it is coded by a n n boolean matrix. - Outut: is 1 if vertices 1 and n are connected. 1 n Can simulate MAJ in constant deth. Reresented by roof nets of constant deth. 14/07/04, Turku.35/44

38 From Circuits to Proof nets Theorem: For every circuit C of size s and deth d (ossibly euied with st CONN 2 ), there is a boolean roof net P C of size O(s 5 ) and deth O(d) which accets the same set as C. 14/07/04, Turku.36/44

39 From Circuits to Proof nets Theorem: For every circuit C of size s and deth d (ossibly euied with st CONN 2 ), there is a boolean roof net P C of size O(s 5 ) and deth O(d) which accets the same set as C. Circuit C size s, deth d = Proof Net P C size O(s 5 ), deth O(d) 14/07/04, Turku.36/44

40 From Circuits to Proof nets Theorem: For every circuit C of size s and deth d (ossibly euied with st CONN 2 ), there is a boolean roof net P C of size O(s 5 ) and deth O(d) which accets the same set as C. Circuit C size s, deth d = Proof Net P C size O(s 5 ), deth O(d) Proof:, n, n, st CONN n 2 reresented by a roof net of size O(n 4 ) and of constant deth. Comosition increases the deth linearly. 14/07/04, Turku.36/44

41 Reresenting roof nets by circuits Idea: Reresent A roof net P by a set of boolean values One-ste reduction by constant deth circuit 2 d(p) reduction stes by O(d)-deth circuit 14/07/04, Turku.37/44

42 Coding roof nets by boolean values P : a roof net with links L. Con f (P): consists of the following boolean values: For, L,. alive() is a link of P sort(,s) link is of sort s {,, } edge(,i,, j) there is an edge between ort i of link and ort j of link Build a circuit C s.t. if P 1 = P 2 then C comutes Con f (P 2 ) from Con f (P 1 ). 14/07/04, Turku.38/44

43 Multilicative Reduction 1 n n 0 P n n n P n Easily imlemented by a constant deth circuit: E.g, edge ( i,0, i,0) = edge( i,0, i,0), L(edge(,0,,0) j edge( i,0,, j) edge( i,0,, j)) 14/07/04, Turku.39/44

44 Global Axiom Reduction P 1 2 n... P r r Naive attemt to build a constant deth circuit leads to exonential size. Use st CONN 2 gates. Aly to the axiom-cut subgrah of P, that is of degree 2. 14/07/04, Turku.40/44

45 From Proof nets to Circuits Theorem: For every boolean roof net P of size s and deth d, there is a circuit C (with st CONN 2 gates) of size O(s 4 ) and deth O(d) which accets the same set as P. 14/07/04, Turku.41/44

46 From Proof nets to Circuits Theorem: For every boolean roof net P of size s and deth d, there is a circuit C (with st CONN 2 gates) of size O(s 4 ) and deth O(d) which accets the same set as P. Proof Net P size s, deth d = Circuit C P size O(s 4 ), deth O(d) 14/07/04, Turku.41/44

47 From Proof nets to Circuits Theorem: For every boolean roof net P of size s and deth d, there is a circuit C (with st CONN 2 gates) of size O(s 4 ) and deth O(d) which accets the same set as P. Proof Net P size s, deth d = Circuit C P size O(s 4 ), deth O(d) Proof: Cut-elimination of P reuires of 2 d stes. Each ste simulated by a constant deth circuit (with st CONN 2 gates). Initialization and accetance checking also by constant deth circuits. 14/07/04, Turku.41/44

48 Proof net comlexity classes (1) For X {0,1}, X AC i (F ) X APN i def def X is acceted by a olynomial size log i -deth family of unbounded fan-in circuits with additional gates from F. X is acceted by a olynomial size log i -deth family of roof nets. Theorem: For every i 0, APN i = AC i (st CONN 2 ). 14/07/04, Turku.42/44

49 Proof net comlexity classes (2) APN = i APN i. Theorem APN = NC. Proof: AC i AC i (st CONN 2 )=APN i AC i+1 and NC = i AC i. P/oly : nonuniform version of P. Theorem P/oly = the class of languages X {0,1} acceted by olynomial size families of roof nets. Note AC 0 (st CONN 2 )=L/oly (nonuniform logsace). Hence, AC 0 NC 1 L/oly = APN 0 AC 1 14/07/04, Turku.43/44

50 Conclusion Extended the Proofs-as-Programs aradigm to a finite, nonuniform, arallel setting. Characterized roof net comlexity by circuit comlexity. Proof nets reresent higher order gates. Future Work NC i+1 APN i? Comarison with other aroaches to roofs-circuits corresondence (Proositional Proof Systems and Bounded Arithmetic). Study the bounded fan-in case. Imlicit arallel comlexity. 14/07/04, Turku.44/44