## How to Lie Without Being (easily) Convicted and the Lengths of Proofs in Propositional Calculus (0)

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Citations: | 17 - 1 self |

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@MISC{Pudlák_howto,

author = {Pavel Pudlák and Samuel R. Buss},

title = {How to Lie Without Being (easily) Convicted and the Lengths of Proofs in Propositional Calculus},

year = {}

}

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### Abstract

We shall describe two general methods for proving lower bounds on the lengths of proofs in propositional calculus and give examples of such lower bounds. One of the methods is based on interactive proofs where one player is claiming that he has a falsifying assignment for a tautology and the second player is trying to convict him of a lie.

### Citations

355 | The relative efficiency of propositional proof systems - Cook, Reckhow - 1979 |

238 | Bounded Arithmetic, Propositional Logic and Computational Complexity - Kraj́ıček - 1995 |

151 |
The synthesis of two-terminal switching circuits
- Shannon
- 1949
(Show Context)
Citation Context ...es.) We do not construct the formulas explicitly, but use a counting argument to show that they exist. Although counting arguments sometimes easily give exponential lower bounds in circuit complexity =-=[19]-=-, it seems that for the propositional calculus we cannot get such strong bounds. We consider the following formulas sn,π =df p π (1) ∧ ...∧p π(n) →p1 ∧...∧pn, where π is a permutation of {1,...,n}. Th... |

126 |
Exponential lower bounds for the pigeonhole principle
- Beame, Impagliazzo, et al.
- 1992
(Show Context)
Citation Context ...equent calculi with the cut rule. For their associated theories of bounded arithmetic, see [5, 3, 13]. So far, superpolynomial lower bounds have been proved only for more restricted systems, see e.g. =-=[1, 2]-=-. In this paper we introduce two frameworks for proving lower bounds for Frege systems and their restricted versions. First we shall define an interactive way of proving propositional tautologies. Thi... |

126 |
Bounded Arithmetic. Bibliopolis
- Buss
- 1986
(Show Context)
Citation Context ...easure: every two systems polynomially simulate each other. Moreover they are equivalent in this sense with sequent calculi with the cut rule. For their associated theories of bounded arithmetic, see =-=[5, 3, 13]-=-. So far, superpolynomial lower bounds have been proved only for more restricted systems, see e.g. [1, 2]. In this paper we introduce two frameworks for proving lower bounds for Frege systems and thei... |

122 |
Feasibly constructive proofs and the propositional calculus
- Cook
- 1975
(Show Context)
Citation Context ...h the famous open problem of “NP=?coNP”, since a proof system for propositional calculus can be thought of as a nondeterministic procedure for the coNP-complete set of propositional tautologies, Cook =-=[5]-=-. Thus proving superpolynomial lower bounds on the lengths of proofs in increasingly stronger proof systems parallels in a sense an approach to the problem P =?NP, where for restricted classes of circ... |

116 |
The complexity of the pigeonhole principle
- Ajtai
- 1988
(Show Context)
Citation Context ...equent calculi with the cut rule. For their associated theories of bounded arithmetic, see [5, 3, 13]. So far, superpolynomial lower bounds have been proved only for more restricted systems, see e.g. =-=[1, 2]-=-. In this paper we introduce two frameworks for proving lower bounds for Frege systems and their restricted versions. First we shall define an interactive way of proving propositional tautologies. Thi... |

97 |
Propositional proof systems, the consistency of first order theories and the complexity of computations
- Kraj́ıček, Pudlák
- 1989
(Show Context)
Citation Context ...easure: every two systems polynomially simulate each other. Moreover they are equivalent in this sense with sequent calculi with the cut rule. For their associated theories of bounded arithmetic, see =-=[5, 3, 13]-=-. So far, superpolynomial lower bounds have been proved only for more restricted systems, see e.g. [1, 2]. In this paper we introduce two frameworks for proving lower bounds for Frege systems and thei... |

32 | Quantified propositional calculi and fragments of bounded arithmetic. Zeitschrift für mathematische Logik und Grundlagen der Mathematik - Kraj́ıček, Pudlák - 1990 |

30 |
On the Lengths of Proofs in the Propositional Calculus (Preliminary Version
- Cook, Reckhow
- 1974
(Show Context)
Citation Context ... this approach to the problem NP=?coNP may have interesting consequences. The most important class of propositional proof systems is called Frege systems. This concept was defined by Cook and Reckhow =-=[7, 6]-=- and was intended to capture the properties of the most common propositional proof systems. Formally, a Frege system is determined by a complete finite basis of connectives and a finite set of rules ϕ... |

27 |
Some results on the length of proofs
- Parikh
- 1973
(Show Context)
Citation Context ...hat every tree-like proof of sn,π has at least εn log n steps. The proof of Theorem 2 is very similar to the proof of Theorem 1, and, in the setting of proofs, is a well-known technique due to Parikh =-=[16]-=-. For a Frege proof P we define the skeleton of P to be the labeled graph whose vertices correspond to the formulas, the label of a vertex v corresponding to a formula ϕ determines the rule by which ϕ... |

19 |
Propositional representations of arithmetic proofs
- Dowd
- 1979
(Show Context)
Citation Context .... The fragments that we have in mind are often referred to by a generic name Bounded Arithmetic. For many theories R of bounded arithmetic one can find an associated propositional proof system R prop =-=[5, 8, 14]-=-. For a given theory R of arithmetic, R prop is the strongest system system whose soundness is provable in the theory R and which simulates provability in R. The simulation means that for a certain cl... |

11 |
Weak formal systems and connections to computational complexity. Lecture Notes for a Topics Course
- Buss
- 1988
(Show Context)
Citation Context ...he following formula tn ¬¬···¬¬ � �� � (p ∨¬p). n Note that t2n is always a tautology; this is a well-known example for which one can prove a linear lower bound on the number of steps in Frege proofs =-=[4, 11]-=-. We shall show an Ω(log n) lower bound for the number of rounds in the game. This, of course, follows from the cited result and Proposition 2. Still the proof is interesting, because it is different,... |

11 | i cek, Lower bounds to the size of constant-depth propositional proofs - Kraj' |

10 |
Independence in Bounded Arithmetic
- Riis
- 1993
(Show Context)
Citation Context ...the restriction means that we use only the De Morgan basis and the number of alternations of different connectives is bounded by a constant; e.g. CNF’s and DNF’s are of depth ≤ 3.) Ajtai [1] and Riis =-=[18]-=- use in fact a different approach, an approach based on forcing, but their results can be interpreted using the boolean values method. In model theory we use boolean values to prove independence resul... |

10 | cek and P. Pudl ak, Propositional proof systems, the consistency of first-order theories and the complexity of computations - Kraj - 1989 |

8 |
Bounded Arithmetic, Propositional Calculus and Complexity Theory
- Kraj́ıček
- 1995
(Show Context)
Citation Context ... [2], and other proofs can be interpreted in such a way. For the reader who wishes to get a deeper knowledge about lower bounds in propositional calculus we recommend the forthcoming book by Krajíček =-=[9]-=- and a forthcoming survey by the first author [17]. (1)s2 Interactive proofs of tautologies We shall introduce a game using a real life situation as an example. Suppose you are a prosecutor who wishes... |

5 |
propositional calculi and fragments of bounded arithmetic, Zeitschrift für mathematische Logik und Grundlagen der
- Quantified
- 1990
(Show Context)
Citation Context .... The fragments that we have in mind are often referred to by a generic name Bounded Arithmetic. For many theories R of bounded arithmetic one can find an associated propositional proof system R prop =-=[5, 8, 14]-=-. For a given theory R of arithmetic, R prop is the strongest system system whose soundness is provable in the theory R and which simulates provability in R. The simulation means that for a certain cl... |

5 | cek, Bounded Arithmetic, Propositional Calculus and Complexity Theory - Kraj - 1995 |

2 |
bounds to the size of constant-depth Frege proofs
- Kraj'icek, Lower
- 1994
(Show Context)
Citation Context ...e that the proof constructed from the game is in a tree form, except possibly for constant size pieces at the leaves, which can be easily changed into such a form. Thus we get: Corollary 1. (Krajíček =-=[10]-=-) A Frege proof can be transformed into a tree-like Frege proof with at most polynomial increase of size. Proof. Let an arbitrary Frege proof of size n be given. First transform it into the Prover-Adv... |

2 |
for propositional Frege systems via generalizations of proofs
- Speed-up
- 1989
(Show Context)
Citation Context ...he following formula tn ¬¬···¬¬ � �� � (p ∨¬p). n Note that t2n is always a tautology; this is a well-known example for which one can prove a linear lower bound on the number of steps in Frege proofs =-=[4, 11]-=-. We shall show an Ω(log n) lower bound for the number of rounds in the game. This, of course, follows from the cited result and Proposition 2. Still the proof is interesting, because it is different,... |

2 |
On lower bounds on the lengths of proofs in propositional logic (russian
- Orevkov
- 1980
(Show Context)
Citation Context ...ere π is a permutation of {1,...,n}. The distribution of parentheses is not important; for definiteness let us assume that we group the conjuncts to the left. These formulas have been used by Orevkov =-=[15]-=- to prove a speedup from Ω(n log n) toO(n) of the sequence-like proofs vs. tree-like proofs (this speedup was rediscovered later by the authors, and we sketch its proof below). Theorem 1 does not foll... |

1 |
The lengths of proofs. To appear
- Pudlák
(Show Context)
Citation Context ... a way. For the reader who wishes to get a deeper knowledge about lower bounds in propositional calculus we recommend the forthcoming book by Krajíček [9] and a forthcoming survey by the first author =-=[17]-=-. (1)s2 Interactive proofs of tautologies We shall introduce a game using a real life situation as an example. Suppose you are a prosecutor who wishes to convict someone at a trial. What is he saying ... |

1 | Speed-up for propositional FregeFrege, G. systems via generalizations of proofs - Kraj'icek - 1989 |

1 | AND ET AL., Weak formal systems and connections to computational complexity. Student-written Lecture Notes for a Topics Course at - BUSS - 1988 |

1 | Bounded Arithmetic, Propositional 6'alculus and 6'omplexity Theory - KRAJOEK |

1 | Lower bounds to the size of constant-depth Frege proofs - KRAJOEK |

1 | Propositionalproofsystems, the consistency offirstorder theories and the complexity of computations - KRAJEK, PUDLK |

1 | The lengths of proofs. To appear - PUDLK |

1 | Independence in BoundedArithmetic - PdIS - 1993 |

1 | ak, The lengths of proofs. To appear - Pudl |