In this paper, we prove that S21 can prove consistency of PV−, the system obtained from Cook and Urquhart’s PV [3] by removing induction. This apparently contradicts Buss and Ignjatovi ́c [2], since they prove that PV ̸⊢ Con(PV−). However, what they actually prove is unprovability of consistency of the system which is obtained from PV− by addition of propositional logic and BASICe-axioms. On the other hand, our PV− is strictly equational and our proof relies on it.

1. Consistency proof of a 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 of a 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 of a 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 theory T such that S2 ` Con(T) but
S1
26` Con(T)?

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

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

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

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

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

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

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

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

24. nition (Composition)
If f 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. Consistency proof of a feasible arithmetic inside a bounded arithmetic
Consistency of PV-Big-
step semantics
Inference rules for ht; i # v
De

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

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

31. nition
Computation sequence A 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 of a 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 of a 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 of a 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 of a 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 of a 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 of a 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 of a 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 of a 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 of a 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 of a 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 of a 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 of a 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 of a feasible arithmetic inside a bounded arithmetic
Consistency of PV-Soundness
of de

47. ning axioms
Backward lemma
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. Consistency proof of a feasible arithmetic inside a bounded arithmetic
Consistency of PV-Soundness
of de

50. ning axioms
Forward lemma
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. Consistency proof of a feasible arithmetic inside a bounded arithmetic
Consistency of PV-Soundness
of de

52. ning axioms
Fusion lemma
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. Consistency proof of a feasible arithmetic inside a bounded arithmetic
Consistency of PV-Soundness
of de

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

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

56. ning axioms
Proof of 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. Consistency proof of a feasible arithmetic inside a bounded arithmetic
Consistency of PV-Soundness
of de

58. ning axioms
Proof of 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. Consistency proof of a feasible arithmetic inside a bounded arithmetic
Consistency of PV-Soundness
of de

62. ning axioms
Proof of 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 of a 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 of a 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 of a 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 of a 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 of a 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 of a 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 of a 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
Assume u1[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 of a 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 of a 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 of a 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 of a 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 of a 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 of a 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 of a 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 of a 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 of a 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 of a 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 of a 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