## Bounded Arithmetic and Propositional Proof Complexity (1995)

Venue: | in Logic of Computation |

Citations: | 11 - 0 self |

### BibTeX

@INPROCEEDINGS{Buss95boundedarithmetic,

author = {Samuel R. Buss},

title = {Bounded Arithmetic and Propositional Proof Complexity},

booktitle = {in Logic of Computation},

year = {1995},

pages = {67--122},

publisher = {Springer-Verlag}

}

### Years of Citing Articles

### OpenURL

### Abstract

This is a survey of basic facts about bounded arithmetic and about the relationships between bounded arithmetic and propositional proof complexity. We introduce the theories S 2 of bounded arithmetic and characterize their proof theoretic strength and their provably total functions in terms of the polynomial time hierarchy. We discuss other axiomatizations of bounded arithmetic, such as minimization axioms. It is shown that the bounded arithmetic hierarchy collapses if and only if bounded arithmetic proves that the polynomial hierarchy collapses. We discuss Frege and extended Frege proof length, and the two translations from bounded arithmetic proofs into propositional proofs. We present some theorems on bounding the lengths of propositional interpolants in terms of cut-free proof length and in terms of the lengths of resolution refutations. We then define the RazborovRudich notion of natural proofs of P NP and discuss Razborov's theorem that certain fragments of bounded arithmetic cannot prove superpolynomial lower bounds on circuit size, assuming a strong cryptographic conjecture. Finally, a complete presentation of a proof of the theorem of Razborov is given. 1 Review of Computational Complexity 1.1 Feasibility This article will be concerned with various "feasible" forms of computability and of provability. For something to be feasibly computable, it must be computable in practice in the real world, not merely e#ectively computable in the sense of being recursively computable.

### Citations

264 |
The intractability of resolution
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- 1985
(Show Context)
Citation Context ...,j means "pigeon i is in hole j ". Theorem 26 (Cook-Reckhow [18, 19]) There are polynomial size eF -proofs of PHP n . Theorem 27 ([6]) There are polynomial size F -proofs of PHP n . Theorem =-=28 (Haken [26]-=-) The shortest resolution proofs of PHP n are of exponential size. Cook and Reckhow had proposed PHP n as an example for showing that F could not simulate eF . However, Theorem 27 implies this is not ... |

264 |
The polynomial-time hierarchy
- Stockmeyer
- 1977
(Show Context)
Citation Context ...on of # b i - and # b i -formulas is the following theorem. Theorem 2 Fix k # 1 . A predicate Q is in # p k i# there is a # b k formula which defines it. This theorem is essentially due to Stockmeyer =-=[48]-=- and Wrathall [51]; Kent and Hodgson [31] were the first to prove a full version of this. Remarks: There are several reasons why the # function and sharply bounded quantifiers are natural choices for ... |

176 | Natural proofs
- Razborov, Rudich
- 1997
(Show Context)
Citation Context ...falsifies some B j (#p, #r) . Q.E.D. Claim and Theorem. 41 8 Natural Proofs, Interpolation and Bounded Arithmetic 8.1 Natural Proofs The notion of natural proofs was introduced by Razborov and Rudich =-=[44]-=-. In order for a proof to be a natural proof of P #= NP , it must give a suitably constructive way of proving that certain Boolean functions are not in P (for example, the Boolean function Sat which r... |

150 |
The intrinsic computational difficulty of functions
- Cobham
- 1964
(Show Context)
Citation Context ...s use the convention that functions and predicates operate on strings of symbols; but this is equivalent to operations on integers, if one identifies integers with their binary representation. Cobham =-=[15]-=- defined FP as the closure of some base functions under composition and limited iteration on notation. As base functions, we can take 0 , S (successor), b 1 2 xc , 2 \Delta x , xsy = ae 1 if xsy 0 oth... |

115 |
Feasibly constructive proofs and the propositional calculus
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- 1975
(Show Context)
Citation Context .... It is also possible to define PV 1 as being the natural first-order, purely universal theory which has function symbols from all polynomial time functions (this is the same as the theory PV of Cook =-=[16]-=-, except extended to first-order logic). Using PV i in place of T i 2 , and applying Herbrand's theorem to PV i , one obtains the following witnessing theorem. Theorem 22 (Krajcek-Pudlak-Takeuti [35])... |

111 |
Recherches sur la Theorie de la Demonstration, Travaux de la Societe des Sciences et de Lettres de Varsovie
- Herbrand
- 1930
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Citation Context ... free T 1 2 -proof can be transformed into a PLS problem. # 4.3 Herbrand's Theorem and the KPT Witnessing Theorem The following theorem is a version of Herbrand's Theorem from Herbrand's dissertation =-=[27, 30, 28]-=-. A proof of (a strengthening of) this version of Herbrand's theorem can be found in [10]. Theorem 21 Let T be a theory axiomatized by purely universal axioms. Let A(x, y, z) be quantifier-free. Suppo... |

107 |
Metamathematics of First-Order Arithmetic
- Hájek, Pudlák
- 1998
(Show Context)
Citation Context ...trapping can be found in Buss-Ignjatović [13] and a proof outline can be found in [7]. Alternative approaches to bootstrapping in different settings are given by Wilkie-Paris [50] and in Hájek-Pudlák =-=[25]-=-. A large part of the importance of Σb 1 -definable functions and ∆b 1 - predicates comes from the following theorem: Theorem 5 [3, 4] Let i ≥ 1. Any Σ b 1-definable function or ∆ b 1 -definable predi... |

103 |
Linear reasoning. A new form of the Herbrand-Gentzen theorem
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- 1957
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Citation Context ...rems for Propositional Logic 7.1 Craig's Theorem The interpolation theorem is one of the fundamental theorems of mathematical logic; this was first proved in the setting of first-order logic by Craig =-=[20]-=-. For propositional logic, the interpolation is much simpler: Theorem 40 Let A(#p, #q) and B(#p, #r) be propositional formulas involving only the indicated variables. Suppose A(#p, #q) # B(#p, #r) is ... |

88 | Interpolation theorems, lower bounds for proof systems and independence results for bounded arithmetic
- Krajíček
- 1997
(Show Context)
Citation Context ...r),...,¬Bℓ(�p, �r) The next theorem gives bounds on the size or computational complexity of the interpolant, in terms of the number of inferences in the resolution refutation. 39sTheorem 45 (Krajíček =-=[32]-=-) Let {Ai(�p, �q)} i ∪{Bj(�p, �r)} j have a refutation R of n resolution inferences. Then an interpolant, C(�p), can be chosen with O(n) symbols in dag representation. If R is tree-like, then C(�p) is... |

87 |
Propositional proof systems, the consistency of first order theories and the complexity of computations
- Krajíček, Pudlák
- 1989
(Show Context)
Citation Context ...en question of whether P is equal to alternating logarithmic time (ALOGTIME). Open Question: Is there a maximal proof system which simulates all other propositional proof systems? Krajíček and Pudlák =-=[33]-=- have shown that if NEXP (non-deterministic exponential time) is closed under complements then the answer is “Yes”. 5.3 The Propositional Pigeonhole Principle We now introduce tautologies that express... |

83 |
Complete sets and the polynomial-time hierarchy
- Wrathall
- 1976
(Show Context)
Citation Context ...# b i -formulas is the following theorem. Theorem 2 Fix k # 1 . A predicate Q is in # p k i# there is a # b k formula which defines it. This theorem is essentially due to Stockmeyer [48] and Wrathall =-=[51]-=-; Kent and Hodgson [31] were the first to prove a full version of this. Remarks: There are several reasons why the # function and sharply bounded quantifiers are natural choices for inclusion in the l... |

81 |
Existence and feasibility in arithmetic
- Parikh
- 1971
(Show Context)
Citation Context ...inson's Q since it has to be used with weaker induction axioms. (2) The # b i -IND axioms. T -1 2 has no induction axioms. T 2 is the union of the T i 2 's. T 2 is equivalent to I# 0 +# 1 (see Parikh =-=[40]-=- and Wilkie and Paris [50]) modulo di#erences in the nonlogical language. Definition Let i # 0 . S i 2 is the first-order theory with language 0 , S , + , , # 1 2 x# , |x| , # and # and axioms: 8 (1) ... |

68 | Bounded arithmetic
- Buss
- 1986
(Show Context)
Citation Context ... (" x log x is total") for similar reasons, since it gives similar growth rate. The original use of polynomially bounded quantifiers was by Bennett [1]; they were first defined in form given=-= above by [3, 4]. 2.2 Induction Axio-=-ms for Bounded Arithmetic The IND axioms are the usual induction axioms. The PIND and LIND axioms are "polynomial" and "length" induction axioms that are intended to be feasibly e#... |

61 |
The circuit value problem is log space complete for P
- Ladner
- 1975
(Show Context)
Citation Context ...nomial time function of the T -proof. Open Question: Does F simulate eF ? This open question is related to the question of whether Boolean circuits have equivalent polynomial size formulas. By Ladner =-=[36]-=- and Buss [5] this is a non-uniform version of the open question of whether P is equal to alternating logarithmic time (ALOGTIME). Open Question: Is there a maximal proof system which simulates all ot... |

58 | Notes on polynomially bounded arithmetic
- Zambella
- 1996
(Show Context)
Citation Context ... theories of bounded arithmetic collapses if and only if bounded arithmetic is able to prove that the polynomial time hierarchy is proper. Theorem 23 (Krajcek-Pudlak-Takeuti [35], Buss [11], Zambella =-=[52]-=-) If T i 2 = S i+1 2 , then the polynomial time hierarchy collapses, provably in T i 2 . In fact, in this case, T i 2 proves that every # p i+3 predicate is (a) equivalent to a Boolean combination of ... |

57 |
Bounded arithmetic and the polynomial hierarchy
- Krajíček, Pudlák, et al.
- 1991
(Show Context)
Citation Context ...ok [16], except extended to first-order logic). Using PVi in place of T i 2 , and applying Herbrand’s theorem to PVi, one obtains the following witnessing theorem. Theorem 22 (Krajíček-Pudlák-Takeuti =-=[35]-=-) Suppose A ∈ Σ b i+2 and T i 2 proves (∀x)(∃y)(∀z)A(x, y, z). Then there are k > 0 and functions fi(x, z1, ..., zi−1) so that (1) Each fi is Σ b i+1 -defined by T i 2 . 21s(2) T i 2 proves (∀x)[(∀z1)... |

55 | kn application of Boolean complexity to separation problems in bounded arithmetic
- Buss, Krajicek
- 1994
(Show Context)
Citation Context ...lds; and if F (s, x) , then |s|sp(|x|) for p some polynomial. A solution to the PLS problem is a (multivalued) function f , s.t., for all x , c(N(f(x), s), s) = c(f(x), x) and F (f(x), x). Theorem 19 =-=[14]-=- Suppose T 1 2 proves (#x)(#y)A(x, y) where A # # b 1 . Then there is a PLS function f(x) = y and a polynomial time function # such that T 1 2 # (#x)A(x, # # f(x)). Furthermore, every PLS function (an... |

49 |
Die Widerspruchsfreiheit der reinen Zahlentheorie
- Gentzen
- 1936
(Show Context)
Citation Context ...roof system for first-order logic. It is probably the most elegant way of formulating first-order logic; and a primary factor in its elegance is the following fundamental theorem: Theorem 12 (Gentzen =-=[23]-=-) . LK is complete. . LK without the Cut inference is complete. In particular, if P is an LK-proof of ### then there is a cut-free proof P # of ### . There is an e#ective (but not feasible) procedure ... |

45 |
On the scheme of induction for bounded arithmetic formulas
- Wilkie, Paris
- 1987
(Show Context)
Citation Context ... was by E. Nelson [38]; his primary reason for its introduction was that the growth rate of # allows a smooth treatment of sequence coding and of the metamathematics of substitution. Wilkie and Paris =-=[50] independe-=-ntly introduced the axiom # 1 (" x log x is total") for similar reasons, since it gives similar growth rate. The original use of polynomially bounded quantifiers was by Bennett [1]; they wer... |

44 | On provably disjoint NP-pairs
- Razborov
- 1994
(Show Context)
Citation Context ...e resulting PLS algorithm to construct a kind of interpolant which serves as a natural proof. He later gave a second, more direct proof based on translating T 1 2 (#)-proofs into propositional proofs =-=[42]-=-. We give below a variation of this second proof based on propositional interpolation theorems, using a translation of T 1 2 (#) proofs into limited extension resolution refutations, generalizing a co... |

42 |
On the lengths of proofs in the propositional calculus
- Cook, Reckhow
- 1974
(Show Context)
Citation Context ...atman [47] and Cook-Reckhow [18, 19]. Statman proved that the number of symbols in an extended Frege proof can be polynomially bounded in terms on the number of steps in the proof. Theorem 24 Reckhow =-=[45]-=- The choice of axiom schemas or of logical language does not a#ect the lengths of F - or eF -proofs by more than a polynomial amount. 5.2 Abstract Proof Systems The following definition of proposition... |

32 |
Predicative Arithmetic
- Nelson
- 1986
(Show Context)
Citation Context ...antifier Exchange Principle holds: (#x # |a|)(#y # b)A(x, y) # # (#y # (2a + 1)#(4(2b + 1) 2 ))(#x # |a|) [A(x, #(x + 1, y)) # #(x + 1, y) # b] 7 . The original use of the # function was by E. Nelson =-=[38]-=-; his primary reason for its introduction was that the growth rate of # allows a smooth treatment of sequence coding and of the metamathematics of substitution. Wilkie and Paris [50] independently int... |

30 |
On the lengths of proofs in the propositional calculus (preliminary version
- Cook, Reckhow
- 1974
(Show Context)
Citation Context ...xtension rule can apparently make proofs logarithmically smaller by reducing the formula size. Tsetin [49] first used the extension rule, for resolution proofs. See also Statman [47] and Cook-Reckhow =-=[18, 19]-=-. Statman proved that the number of symbols in an extended Frege proof can be polynomially bounded in terms on the number of steps in the proof. Theorem 24 Reckhow [45] The choice of axiom schemas or ... |

29 |
On graph-theoretic lemmata and complexity classes (extended abstract), in
- Papadimitriou
- 1990
(Show Context)
Citation Context ... is: Theorem 18 [8] The # b i+1 -definable theories of T i 2 are precisely the p i+1 - functions. 4.2 Witnessing Theorem for T 1 2 The class Polynomial Local Search, PLS, was defined by Papadimitriou =-=[39]-=- to capture a common kind of search problem. Two representative examples of PLS problems are (1) linear programming and (2) simulated annealing. Of course, it is known that there are polynomial time a... |

25 |
On Spectra
- Bennett
- 1962
(Show Context)
Citation Context ...nd Paris [50] independently introduced the axiom # 1 (" x log x is total") for similar reasons, since it gives similar growth rate. The original use of polynomially bounded quantifiers was b=-=y Bennett [1]; they were fir-=-st defined in form given above by [3, 4]. 2.2 Induction Axioms for Bounded Arithmetic The IND axioms are the usual induction axioms. The PIND and LIND axioms are "polynomial" and "lengt... |

24 |
The intrinsic computational diculty of functions
- Cobham
- 1964
(Show Context)
Citation Context ...s use the convention that functions and predicates operate on strings of symbols; but this is equivalent to operations on integers, if one identifies integers with their binary representation. Cobham =-=[15]-=- defined FP as the closure of some base functions under composition and limited iteration on notation. As base functions, we can take 0 , S (successor), # 1 2 x# , 2 x , x # y = 1 if x # y 0 otherwise... |

20 | Are there hard examples for Frege systems
- Bonet, Buss, et al.
- 1994
(Show Context)
Citation Context ...wing that F could not simulate eF . However, Theorem 27 implies this is not the case. Presently there are no very good candidates of combinatorial principles that might separate F from eF ; the paper =-=[2]-=- describes some attempts to find such combinatorial principles. 25 5.4 PHP n has Polysize eF -Proofs The pigeonhole principle provides a nice example of how extended Frege (eF ) proofs can grow expone... |

20 |
conservation results for fragments of bounded arithmetic, in Logic and Computation, proceedings of a Workshop held Carnegie-Mellon
- Axiomatizations
- 1987
(Show Context)
Citation Context ...(B(x, 0) # B(x, x)). By the assumption, (#x)B(x, 0) ; hence (#x)B(x, x) , from whence (#x)(A(0) # A(x)) # The second result is concerns the conservation results between T i 2 and S i+1 2 : Theorem 17 =-=[8]-=- Fix i # 1 . S i+1 2 is conservative over T i 2 with respect to # b i+1 -formulas, and hence with respect to ### b i+1 -sentences. This means that any ### b i+1 -formula which is S i+1 2 -provable is ... |

19 |
Propositional representation of arithmetic proofs
- Dowd
- 1979
(Show Context)
Citation Context ...n is a polynomial size propositional formula, (2) A n says that A(x) is true whenever |x| # n , (3) A n has polynomial size eF -proofs. (Some generalizations of these results have been proved by Dowd =-=[21]-=- for PSPACE and Krajcek and Pudlak [34] for various of the theories of bounded arithmetic.) In these notes, we shall prove the version of Cook's theorem for S 1 2 and # b 2 -formulas A . (Our proof be... |

19 |
Complexity of derivations from quantifier-free Horn formulae, mechanical introduction of explicit definitions, and refinement of completeness theorems
- Statman
- 1977
(Show Context)
Citation Context ...f extension rule the extension rule can apparently make proofs logarithmically smaller by reducing the formula size. Tsetin [49] first used the extension rule, for resolution proofs. See also Statman =-=[47]-=- and Cook-Reckhow [18, 19]. Statman proved that the number of symbols in an extended Frege proof can be polynomially bounded in terms on the number of steps in the proof. Theorem 24 Reckhow [45] The c... |

12 |
Tautologies with a unique Craig interpolant, uniform vs. nonuniform complexity
- Mundici
- 1984
(Show Context)
Citation Context ...is prime # ##rB(#p, #r) # ##qA(#p, #q). and A(#p, #q) # B(#p, #r) is a tautology. An interpolant C(#p) must express " # p codes a composite". Generalizing the above example gives: Theorem 41=-= (Mundici [37]-=- If there is a polynomial upper bound on the circuit size of interpolants in propositional logic, then NP/poly # coNP/poly = P/poly Proof Let ##qA(#p, #q) express an NP/poly property R(#p) and ##rB(#p... |

11 |
Bounded arithmetic and the polynomial hierarchy
- cek, ak, et al.
- 1991
(Show Context)
Citation Context ... [16], except extended to first-order logic). Using PV i in place of T i 2 , and applying Herbrand's theorem to PV i , one obtains the following witnessing theorem. Theorem 22 (Krajcek-Pudlak-Takeuti =-=[35]-=-) Suppose A # # b i+2 and T i 2 proves (#x)(#y)(#z)A(x, y, z) . Then there are k > 0 and functions f i (x, z 1 , ..., z i-1 ) so that (1) Each f i is # b i+1 -defined by T i 2 . 21 (2) T i 2 proves (#... |

11 |
0 sets and induction
- Paris, Wilkie
- 1981
(Show Context)
Citation Context ... logic. The first, due to Cook [16], is a translation from PV-proofs (or more-or-less equivalently, from S 1 2 -proofs) into polynomial size extended Frege proofs. The second, due to Paris and Wilkie =-=[41]-=-, is a translation from proofs in the theory I# 0 or I# 0 + # - 1 or S 2 = T 2 into constant depth, polynomial size Frege proofs. 26 6.1 S 1 2 and Polysize eF Proofs PV is an equational theory of poly... |

10 |
Weak Formal Systems and Connections to Computational Complexity. Lecture notes
- Buss
- 1988
(Show Context)
Citation Context ... Pudlak [34] for various of the theories of bounded arithmetic.) In these notes, we shall prove the version of Cook's theorem for S 1 2 and # b 2 -formulas A . (Our proof below follows the version in =-=[12]-=-). This version is completely analogous to Cook's original theorem, in view of the ## b 1 -conservativity of S 1 2 over PV 1 . Definition Let t(#a) be a term. The bounding polynomial of t is a polynom... |

10 |
cek and P. Pudl ak, Propositional proof systems, the consistency of first-order theories and the complexity of computations
- Kraj
- 1989
(Show Context)
Citation Context ...pen question of whether P is equal to alternating logarithmic time (ALOGTIME). Open Question: Is there a maximal proof system which simulates all other propositional proof systems? Krajcek and Pudlak =-=[33] have show-=-n that if NEXP (non-deterministic exponential time) is closed under complements then the answer is "Yes". 5.3 The Propositional Pigeonhole Principle We now introduce tautologies that express... |

9 |
size proofs of the propositional pigeonhole principle, The
- Polynomial
- 1987
(Show Context)
Citation Context ...# p m,j ) states that n + 1 pigeons can't fit singly into n holes. p i,j means "pigeon i is in hole j ". Theorem 26 (Cook-Reckhow [18, 19]) There are polynomial size eF -proofs of PHP n . Th=-=eorem 27 ([6]-=-) There are polynomial size F -proofs of PHP n . Theorem 28 (Haken [26]) The shortest resolution proofs of PHP n are of exponential size. Cook and Reckhow had proposed PHP n as an example for showing ... |

7 |
i cek, Interpolation theorems, lower bounds for proof systems and independence results for bounded arithmetic
- Kraj'
- 1994
(Show Context)
Citation Context ...p, #r) are valid. # The next theorem gives bounds on the size or computational complexity of the interpolant, in terms of the number of inferences in the resolution refutation. 39 Theorem 45 (Krajcek =-=[32]-=-) Let {A i (#p, #q)} i # {B j (#p, #r)} j have a refutation R of n resolution inferences. Then an interpolant, C(#p) , can be chosen with O(n) symbols in dag representation. If R is tree-like, then C(... |

5 |
consistency proofs, Annals of Pure and
- Propositional
- 1991
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Citation Context ...em 32 If S 1 2 #NP=coNP then eF is super. Theorem 33 eF has polynomial size proofs of the propositional formulas Con eF (n) which assert that there is no eF -proof of p # p of length # n . Theorem 34 =-=[9]-=-. F has polynomial size proofs of the self-consistency formulas ConF (n) . 31 Proof of Theorem 31 from Theorem 33: (Idea) Suppose there is a G proof P of a tautology # . A polynomial size eF proof of ... |

4 | Unprovability of consistency statements in fragments of bounded arithmetic
- Buss, A
- 1995
(Show Context)
Citation Context ...oduce polynomial time properties. We will omit the proof of this theorem here: for details, the reader can refer to Buss [3, 4], some improvements to the bootstrapping can be found in Buss-Ignjatovic =-=[13]-=- and a proof outline can be found in [7]. Alternative approaches to bootstrapping in di#erent settings are given by Wilkie-Paris [50] and in Hajek-Pudlak [25]. A large part of the importance of # b 1 ... |

4 | Papers of Gerhard Gentzen, North-Holland - Collected - 1969 |

4 |
propositional calculi and fragments of bounded arithmetic, Zeitschrift fur Mathematische Logik und Grundlagen der
- Quantified
- 1990
(Show Context)
Citation Context ...rmula, (2) A n says that A(x) is true whenever |x| # n , (3) A n has polynomial size eF -proofs. (Some generalizations of these results have been proved by Dowd [21] for PSPACE and Krajcek and Pudlak =-=[34]-=- for various of the theories of bounded arithmetic.) In these notes, we shall prove the version of Cook's theorem for S 1 2 and # b 2 -formulas A . (Our proof below follows the version in [12]). This ... |

4 |
On the complexity of derivation in propositional logic
- Tsejtin
- 1968
(Show Context)
Citation Context ...r q # #. This allows us to use q as abbreviation for # . By iterating uses of extension rule the extension rule can apparently make proofs logarithmically smaller by reducing the formula size. Tsetin =-=[49]-=- first used the extension rule, for resolution proofs. See also Statman [47] and Cook-Reckhow [18, 19]. Statman proved that the number of symbols in an extended Frege proof can be polynomially bounded... |

3 |
ajek and P. Pudl' ak, Metamathematics of First-order Arithmetic
- H'
- 1993
(Show Context)
Citation Context ...strapping can be found in Buss-Ignjatovic [13] and a proof outline can be found in [7]. Alternative approaches to bootstrapping in di#erent settings are given by Wilkie-Paris [50] and in Hajek-Pudlak =-=[25]-=-. A large part of the importance of # b 1 -definable functions and # b 1 - predicates comes from the following theorem: Theorem 5 [3, 4] Let i # 1 . Any # b 1 -definable function or # b 1 -definable p... |

3 |
of lower bounds on the circuit size in certain fragments of bounded arithmetic
- Unprovability
- 1995
(Show Context)
Citation Context ...greater than t . In other words, proving LB(t, S, #) proves that t is a lower bound on the size of any circuit computing S . (We give a more complete definition of LB() in the next section.) Razborov =-=[43]-=- proved the following theorem: Theorem 49 Let t denote any superpolynomial function and let S be any bounded formula. If S 2 2 (#) proves LB(t, S, #) , then the SPRNG conjecture is false. Razborov's f... |

2 | logique, Presses Universitaires de - Ecrits - 1968 |

2 |
An arithmetic characterization
- Kent, Hodgson
- 1982
(Show Context)
Citation Context ...following theorem. Theorem 2 Fix k # 1 . A predicate Q is in # p k i# there is a # b k formula which defines it. This theorem is essentially due to Stockmeyer [48] and Wrathall [51]; Kent and Hodgson =-=[31]-=- were the first to prove a full version of this. Remarks: There are several reasons why the # function and sharply bounded quantifiers are natural choices for inclusion in the language of bounded arit... |

2 |
0 sets and induction, inOpenDays in Model Theory and Set
- Paris, Wilkie
- 1981
(Show Context)
Citation Context ... logic. The first, due to Cook [16], is a translation from PV-proofs (or more-or-less equivalently, from S 1 2 -proofs) into polynomial size extended Frege proofs. The second, due to Paris and Wilkie =-=[41]-=-, is a translation from proofs in the theory I∆0 or I∆0 +Ω−1 or S2 = T2 into constant depth, polynomial size Frege proofs. 26s6.1 S 1 2 and Polysize eF Proofs PV is an equational theory of polynomial ... |