Consistency proof of a feasible arithmetic inside a bounded arithmetic

1.
Consistency proof ofa feasible arithmetic inside a bounded arithmetic Consistency proof of a feasible arithmetic inside a bounded arithmetic Yoriyuki Yamagata Sendai Logic Seminar, Oct. 31, 2014

2.
Consistency proof ofa feasible arithmetic inside a bounded arithmetic Motivation Ultimate Goal Conjecutre S1 26= S2 where Si2 ; i = 1; 2; : : : are Buss's hierarchies of bounded arithmetics and S2 = S i=1;2;::: Si2

3.
Consistency proof ofa feasible arithmetic inside a bounded arithmetic Motivation Approach using consistency statement If S2 ` Con(S1 2 ), we have S1 26= S2. Theorem (Pudlak, 1990) S26` Con(S1 2 ) Question Can we

4.
nd a theoryT such that S2 ` Con(T) but S1 26` Con(T)?

5.
Consistency proof ofa feasible arithmetic inside a bounded arithmetic Motivation Unprovability Theorem (Buss and Ignjatovic 1995) S1 26` Con(PV) PV : Cook Urquhart's equational theory PV minus induction Buss and Ignjatovic enrich PV by propositional logic and BASICe-axioms

6.
Consistency proof ofa feasible arithmetic inside a bounded arithmetic Motivation Provability Theorem (Beckmann 2002) S1 2 ` Con(PV) if PV is formulated without substitution

7.
Consistency proof ofa feasible arithmetic inside a bounded arithmetic Main results Main result Theorem S1 2 ` Con(PV) Even if PV is formulated with substitution Our PV is equational

8.
Consistency proof ofa feasible arithmetic inside a bounded arithmetic Consistency of PV-Soundness theorem Cook Urquhart's PV : language We formulate PV as an equational theory of binary digits. 1. Constant : 2. Function symbols for all polynomial time functions s0; s1; n; projni ; : : :

9.
Consistency proof ofa feasible arithmetic inside a bounded arithmetic Consistency of PV-Soundness theorem Cook Urquhart's PV : axioms 1. Recursive de

10.
nition of functions 1.1 Composition 1.2 Recursion 2. Equational axioms 3. Substitution t(x) = u(x) t(s) = u(s) (1) 4. PIND for all formulas

11.
Consistency proof ofa feasible arithmetic inside a bounded arithmetic Consistency of PV-Soundness theorem Consistency proof of PV inside S1 2 Theorem (Consistency theorem) S1 2 ` Con(PV) (2)

12.
Consistency proof ofa feasible arithmetic inside a bounded arithmetic Consistency of PV-Soundness theorem Soundness theorem Theorem Let 1. `PV 0 = 1 2. r ` t = u be a sub-proof of 3. : sequence of substitution such that t is closed 4. ` ht; i # v; Then, 9; ` hu; i # v; (3)

13.
Consistency proof ofa feasible arithmetic inside a bounded arithmetic Consistency of PV-Soundness theorem Soundness theorem Theorem (S1 2 ) Let 1. `PV 0 = 1, U jjjj 2. r ` t = u be a sub-proof of 3. : sequence of substitution such that t is closed and jjjj U jjr jj 4. ` ht; i # v; and jjjjjj;2M() U jjr jj Then, 9; ` hu; i # v; (4) s.t. jjj jjj jjjjjj + jjr jj

14.
Consistency proof ofa feasible arithmetic inside a bounded arithmetic Consistency of PV-Soundness theorem Consistency of big-step semantics Lemma (S1 2 ) There is no computation which proves h; i # s1 Proof. Immediate from the form of inference rules

15.
Consistency proof ofa feasible arithmetic inside a bounded arithmetic Consistency of PV-Big- step semantics Inference rules for ht; i # v De

16.
nition (Substitution) ht;2i # v hx; 1[t=x]2i # v (5) where 1 does not contain a substitution to x.

17.

18.
nition (; s0;s1) h; i # (6) ht; i # v hsi t; i # siv (7) where i is either 0 or 1.

19.

20.
nition (Constant function) hn(t1; : : : ; tn); i # (8)

21.

22.
nition (Projection) hti; i # vi hprojni (t1; : : : ; tn); i # vi (9) for i = 1; ; n.

23.

24.
nition (Composition) Iff is de

25.
ned by g(h(t)), hg(w); i # v hh(v); i # w ht; i # v hf (t1; : : : ; tn); i # v (10) t : members of t1; : : : ; tn such that t are not numerals

26.

27.
nition (Recursion -) hg(v1; : : : ; vn); i # v ht; i # ht; i # v hf (t; t1; : : : ; tn); i # v (11) t : members ti of t1; : : : ; tn such that ti are not numerals

28.

29.
nition (Recursion -s0; s1) hgi (v0; w; v); i # v hf (v0; v); i # w fht; i # siv0g ht; i # v hf (t; t1; : : : ; tn); i # v (12) i = 0; 1 t : members ti of t1; : : : ; tn such that ti are not numerals The clause fht; i # siv0g is there if t is not a numeral

30.
Consistency proof ofa feasible arithmetic inside a bounded arithmetic Consistency of PV-Big- step semantics Computation (sequence) De

31.
nition Computation sequenceA DAG (or sequence with pointers) which is built by these inference rules Computation statement A form ht; i # v t : main term : environment v : value Conclusion A computation statement which are not used for assumptions of inferences

32.
Consistency proof ofa feasible arithmetic inside a bounded arithmetic Consistency of PV-Bounding Godel numbers by number of nodes Upper bound of Godel number of main terms Lemma (S1 2 ) M() : the maximal size of main terms in . M() max g2 f(jjgjj + 1) (C + jjgjj + jjjjjj + T)g = Base(t1; : : : ; tm; 1; : : : ; m) T = jjt1jj + + jjtmjj + jj1jj + + jjmjj t1; : : : ; tm : main terms of conclusions 1; : : : ; m : environments of conclusions .

33.
Consistency proof ofa feasible arithmetic inside a bounded arithmetic Consistency of PV-Bounding Godel numbers by number of nodes Size of environments Lemma (S1 2 ) Assume ` ht1; 1i # v1; : : : ; htm; mi # vm. If ht; i # v 2 then is a subsequence of one of the 1; : : : ; m.

34.
Consistency proof ofa feasible arithmetic inside a bounded arithmetic Consistency of PV-Bounding Godel numbers by number of nodes Size of value Lemma (S1 2 ) If contains ht; i # v, then jjvjj jjjjjj.

35.
Consistency proof ofa feasible arithmetic inside a bounded arithmetic Consistency of PV-Bounding Godel numbers by number of nodes Size of arguments Lemma (S1 2 ) contains ht; i # v and u is a subterm of t s.t. u is a numeral Then, jjujj maxfjjjjjj;Tg (13)

36.
Consistency proof ofa feasible arithmetic inside a bounded arithmetic Consistency of PV-Bounding Godel numbers by number of nodes Proof of bound for Godel numbers of main terms = 1; : : : n M(i ) be the size of the main term of i . Prove M(i ) max g2 f(jjgjj + 1) (C + jjgjj + jjjjjj + T)g (14) by induction on n i If i is a conclusion, M(i ) T Assume i is an assumption of j ; j i .

37.
Consistency proof ofa feasible arithmetic inside a bounded arithmetic Consistency of PV-Bounding Godel numbers by number of nodes Proof of bound for Godel numbers of main terms Assume that M(j ) satis

38.
es induction hypothesis We analyze dierent cases of the inference which derives j ht; i # v hx; 0[t=x]i # v (15) Since jjtjj jj0[t=x]jj T, we are done

39.
Consistency proof ofa feasible arithmetic inside a bounded arithmetic Consistency of PV-Bounding Godel numbers by number of nodes Proof of bound for Godel numbers of main terms hf1(v1); i # w1; : : : ; hfl1(v1;w); i # wl1 ; htk1 ; i # v2 1 ; : : : ; htkl2 ; i # v2 l2 hf (t); i # v (16) v1; v2; w; v are numerals tk1 ; : : : ; tkl2 are a subsequence t. f1; : : : ; fl1 2 the elements of v1 come from either v2 or t that are numerals.

40.
Consistency proof ofa feasible arithmetic inside a bounded arithmetic Consistency of PV-Bounding Godel numbers by number of nodes Proof of bound for Godel numbers of main terms jjtk1 jj; : : : ; jjtkl2 jj jjf (t)jj (17)

41.
Consistency proof ofa feasible arithmetic inside a bounded arithmetic Consistency of PV-Bounding Godel numbers by number of nodes Proof of bound for Godel numbers of main terms jjfk (v1)jj jjfk jj + ar(fk ) (C + jjjjjj + T) (18) max g2 fjjgjj + ar(g) (C + jjjjjj + T)g (19) max g2 f(jjgjj + 1) (C + jjjjjj + T)g (20) where t1 are terms ti s.t. ti is a numeral

42.
Consistency proof ofa feasible arithmetic inside a bounded arithmetic Consistency of PV-Bounding Godel numbers by number of nodes Polynomial bound of number of symbols in computation sequence Corollary There is a polynomial p such that jjjj p(jjjjjj; jjjj) (21) where jjjj are conclusions of

43.
Consistency proof ofa feasible arithmetic inside a bounded arithmetic Consistency of PV-Bounding Godel numbers by number of nodes Proof of Corollary Proof. M() max g2 f(jjgjj + 1) (C + jjgjj + jjjjjj + T)g (22) (jjjj + 1) (C + jjjj + jjjjjj + T) (23)

44.
Consistency proof ofa feasible arithmetic inside a bounded arithmetic Consistency of PV-Bounding Godel numbers by number of nodes Polynomial bound of Godel number of computation sequence De

45.
nition `B ,def 9; jjjjjj B ^ ` (24) Corollary `B is a b1 -formula

46.
Consistency proof ofa feasible arithmetic inside a bounded arithmetic Consistency of PV-Soundness of de

47.
ning axioms Backwardlemma Lemma (S1 2 ) If f has a de

48.
ning axiom f(c(t); t) = tf ;c (t; f (t; t); t) (25) c(t) : , s0t, s1t, t `B hf (c(t); t); i # v; ) `B+jj(25)jj htf ;c (t; f (t; t); t); i # v; (26)

49.

50.
ning axioms Forwardlemma Lemma (S1 2 ) `B htf ;c (t; f (t; t); t); i # v; ) `B+jj(25)jj hf (c(t); t); i #B v; (27)

51.

52.
ning axioms Fusionlemma Lemma (S1 2 ) `n (28) hf (v); i # v; ht; i # v 2 (29) v are numerals Then 9 such that ` hf (t); i # v; (30) jn = (31) jjj jjj jjjjjj + 1 (32)

53.

54.
ning axioms Constructorlemma Lemma (S1 2 ) h; i # v 2 ) v hsi t; i # v 2 ) v siv0; ht; i # v0 2

55.

56.
ning axioms Proofof Backward lemma For the case that f = g(h1(x); : : : ; hn(x)) hg(w); i # v hh(v); i # w ht; i # v hf (t1; : : : ; tn); i # v By Fusion lemma, 1 ` hg(w); i # v; hh(t); i # w where jjj1jjj jjjjjj + #h By Fusion lemma ` hg(h(t)); i # v jjj jjj jjj1jjj + 1 jjjjjj + #h + 1 jjjjjj + jjg(h(t))jj

57.

58.
ning axioms Proofof Forward lemma ;

59.
1; ht0; i# v hh1(t); i # w1 ; : : : hg(h(t)); i # v is transformed to g(w);

60.
1 hh1(v); i# w1 ; : : : ; ht0; i # v hf (t); i # v:

61.

62.
ning axioms Proofof soundness theorem - de

63.
ning axioms f(c(t); t) = tf ;c (t; f (t; t); t) By Backward and Forward lemma, `B hf (c(t); t); i # v; )`B+jjr jj htf ;c (t; f (t; t); t); i # v; `B htf ;c (t; f (t; t); t); i # v; )`B+jjr jj hf (c(t); t); i # v;

64.
Consistency proof ofa feasible arithmetic inside a bounded arithmetic Consistency of PV-Substitution lemma Substitution I Lemma (S1 2 ) U; V : large integers ` ht1[u=x]; i # v1; : : : ; htm[u=x]; i # vm; (33) jjjjjj U jjt1[u=x]jj jjtm[u=x]jj (34) M() V (35) Then, 9; ` ht1; [u=x]i # v1 : : : ; htm; [u=x]i # vm; (36) jjj jjj jjjjjj + jjt1[u=x]jj + + jjtm[u=x]jj (37) M( ) M() (38)

65.
Consistency proof ofa feasible arithmetic inside a bounded arithmetic Consistency of PV-Substitution lemma Substitution II Lemma (S1 2 ) U; V : large integers ` ht1; [u=x]i # v1; : : : ; htm; [u=x]i # vm; (39) jjjjjj U jjt1[u=x]jj jjtm[u=x]jj (40) M() V (41) Then, 9; ` ht1[u=x]; i # v1 : : : ; htm[u=x]; i # vm; (42) jjj jjj jjjjjj + jjt1[u=x]jj + + jjtm[u=x]jj (43) M( ) M() jjujj (44)

66.
Consistency proof ofa feasible arithmetic inside a bounded arithmetic Consistency of PV-Substitution lemma Proof of Substitution I By induction on jjt1[u=x]jj + + jjtm[u=x]jj Sub-induction on the length of Case analysis of the last inference of

67.
Consistency proof ofa feasible arithmetic inside a bounded arithmetic Consistency of PV-Substitution lemma Proof of Substitution I hs; 1i # v hy; 0[s=y]1i # v R Two possibility: 1. t1 x and u y 2. t1 y 2 is impossible

68.
Consistency proof ofa feasible arithmetic inside a bounded arithmetic Consistency of PV-Substitution lemma Proof of Substitution I .... 1 hs; 1i # v hx[y=x]; 0[s=y]1i # v R M(1) M(), we can apply IH to 1

69.
Consistency proof ofa feasible arithmetic inside a bounded arithmetic Consistency of PV-Substitution lemma Proof of Substitution I Let be ....1 hs; 1i # v hy; [s=y]1i # v hx; [y=x][s=y]1i # v jjj jjj jjj1jjj + 2 jjj1jjj + jjt2[u=x]jj + + jjtm[u=x]jj + 2 jjj1jjj + jjt1[u=x]jj + + jjtm[u=x]jj + 1 jjjjjj + jjt1[u=x]jj + + jjtm[u=x]jj

70.
Consistency proof ofa feasible arithmetic inside a bounded arithmetic Consistency of PV-Substitution lemma Proof of Substitution I The case that R is not a substitution We consider two cases 1. t1[u=x] f (u1[u=x]; : : : ; ul [u=x]) 2. t1[u=x] u We only consider the

71.
rst case Assumeu1[u=x]; : : : ; ul [u=x] is partitioned into u1 1[u=x]; : : : ; u1 l1 [u=x] which are not numeral, and numerals u2 1[u=x]; : : : ; u2 l2 [u=x]

72.
Consistency proof ofa feasible arithmetic inside a bounded arithmetic Consistency of PV-Substitution lemma Proof of Substitution I

73.
; hu1 1[u=x];i # v1 1 ; : : : ; hu1 l1 [u=x]; i # v1 l1 ht1[u=x]; i # v R jjj0jjj jjjjjj 1 + l 1 U jjt1[u=x]jj jjtm[u=x]jj 1 + l 1 U jju1 1[u=x]jj jju1 l1 [u=x]jj jjt2[u=x]jj jjtm[u=x]jj 0 : The computation which is obtained by removing the last element of and make all huj [u=x]; i # v1 j conclusions

74.
Consistency proof ofa feasible arithmetic inside a bounded arithmetic Consistency of PV-Substitution lemma Proof of Substitution I 0 ` hu1 1; [u=x]i # v1 1 ; : : : ; hu1 l1 ; [u=x]i # v1 l1 ; : : : (45) jjj0jjj jjj0jjj + jju1 1[u=x]jj + + jju1 l1 [u=x]jj+ jjt2[u=x]jj + + jjtm[u=x]jj (46)

75.
Consistency proof ofa feasible arithmetic inside a bounded arithmetic Consistency of PV-Substitution lemma Proof of Substitution I We add the proof of hu2 k2 ; [u=x]i # v1 k2 for all k2 = 1; : : : ; l 2 whose numbers of nodes are bounded by jju2 k2 [u=x]jj + 1 1 ` hu1; [u=x]i # v1; : : : ; hul ; [u=x]i # vl ; : : : (47) jjj1jjj jjj0jjj + jju1[u=x]jj + + jjul [u=x]jj + l 2 + jjt2[u=x]jj + + jjtm[u=x]jj (48)

76.
Consistency proof ofa feasible arithmetic inside a bounded arithmetic Consistency of PV-Substitution lemma Proof of Substitution I ` ht1; [u=x]i # v1; : : : ; htm; [u=x]i # vm; (49) jjj jjj jjj1jjj + 1 jjj0jjj + jju1[u=x]jj + + jjuk [u=x]jj+ l 2 + jjt2[u=x]jj + + jjtm[u=x]jj + 1 jjjjjj + l 1 + jju1[u=x]jj + + jjul [u=x]jj+ l 2 + jjt2[u=x]jj + + jjtm[u=x]jj + 1 jjjjjj + jjt1[u=x]jj + + jjtm[u=x]jj

77.
Consistency proof ofa feasible arithmetic inside a bounded arithmetic Consistency of PV-Substitution lemma Proof of Substitution II Similar to the proof of Substitution I

78.
Consistency proof ofa feasible arithmetic inside a bounded arithmetic Consistency of PV-Substitution lemma Proof of soundness theorem - substitution .... r1 t = u t[t0=x] = u[t0=x] ht[t0=x]; i #B v (Assumption) ht; [t0=x]i #B+jjt[t0=x]jj v (Substitution I) Since jj[t0=x]jj U jjr jj + jj[t0=x]jj U jjr1jj hu; [t0=x]i #B+jjt[t0=x]jj+jjr1jj v (IH) hu[t0=x]; i #B+jjt[t0=x]jj+jjr1jj+jju[t0=x]jj v (Substitution II) Since B + jjt[t0=x]jj + jjr1jj + jju[t0=x]jj B + jjr jj

79.
Consistency proof ofa feasible arithmetic inside a bounded arithmetic Consistency of PV-Equality axioms Equality axioms - transitivity .... r1 t = u .... r2 u = w t = w By induction hypothesis, `B ht; i # v ) 9; `B+jjr1jj hu; i # v (50) Since jjjj U jjr jj U jjr2jj jjj jjj U jjr jj + jjr1jj U jjr2jj By induction hypothesis, 9; `B+jjr1jj+jjr2jj hw; i # v

80.
Consistency proof ofa feasible arithmetic inside a bounded arithmetic Consistency of PV-Equality axioms Equality axioms - substitution .... r1 t = u f (t) = f (u) : computation of hf (t); i # v. We only consider that the case t is not a numeral

81.
Consistency proof ofa feasible arithmetic inside a bounded arithmetic Consistency of PV-Equality axioms Proof of soundness of equality axioms has a form hf1(v0; v); i # w1 ; : : : ; ht; i # v0; htk1 ; i # v1; : : : hf (t; t2; : : : ; tk ); i # v 91, s.t. 1 ` ht; i # v0; hf (t; t2; : : : ; tk ); i # v; jjj1jjj jjjjjj + 1 (51) U jjr jj + 1 (52) U jjr1jj (53) and jjf (t; t2; : : : ; tk )jj + 2M() U jjr jj + jjf (t; t2; : : : ; tk )jj U jjr1jj (54)

82.
Consistency proof ofa feasible arithmetic inside a bounded arithmetic Consistency of PV-Equality axioms Proof of soundness of equality axioms By IH, 91 s.t. 1 ` hu; i # v0; hf (t; t2; : : : ; tk ); i # v; and jjj1jjj jjj1jjj + jjr1jj hf1(v0; v); i # w1 ; : : : ; ht; i # v0; htk1 ; i # v1; : : : hf (t; t2; : : : ; tk ); i # v

83.
Consistency proof ofa feasible arithmetic inside a bounded arithmetic Consistency of PV-Equality axioms Proof of soundness of equality axioms hf1(v0; v); i # w1 ; : : : ; hu; i # v0; htk1 ; i # v1; : : : hf (u; t2; : : : ; tk ); i # v Let this derivation be jjj jjj jjj1jjj + jjr1jj + 1 jjjjjj + jjr1jj + 2 jjjjjj + jjr jj

Consistency proof of a feasible arithmetic inside a bounded arithmetic

More Related Content

What's hot

Viewers also liked

Similar to Consistency proof of a feasible arithmetic inside a bounded arithmetic

More from Yamagata Yoriyuki

Recently uploaded

Consistency proof of a feasible arithmetic inside a bounded arithmetic