J. Krajcek. Bounded arithmetic, propositional logic and complexity theory. Cambridge University Press, 1995.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....when it comes to proof techniques. Given this diversity, as well as quite a substantial amount of material gathered during recent years, it is fairly impossible even to touch all aspects of propositional proof complexity in one lecture. I will not even try that; instead let me refer to the sources [Urq95, Kra95, Raz96, BP98, Pud98] that serve various tastes and, taken together, cover virtually everything which is currently known about the complexity of propositional proofs (see, however, the remark at the end of Section 4) 2. De nitions The framework underlying propositional proof complexity was developed in the seminal ....

J. Krajcek. Bounded arithmetic, propositional logic and complexity theory. Cambridge University Press, 1995.

....paper are to generalize the natural notion of space complexity to other propositional proof systems and complexity measures and to research its properties. The rst question arising is how to measure the memory content at any given moment of time for a speci ed proof system. Recall (see, e.g. [Kra95]) that the most customary measures for size complexity of propositional proofs are the bit size and the number of lines. Of these two, the bit size is by far more important, and can be de ned analogously, and naturally, in the context of space complexity. The only simpli cation we allowed ....

J. Kraj   cek, Bounded Arithmetic, Propositional Logic and Complexity Theory, Cambridge University Press, Cambridge, UK, 1995.

....F , and an assignment, then there is a subsequence of Rd that is a regular resolution derivation of Cd from F d . Proof. This is a straightforward induction on the length of the derivation from F . Every regular resolution refutation can be represented by a read once branching program [12]. Although we prefer to speak in terms of refutations rather than branching programs, we nevertheless need some ideas from the latter framework. A path in a resolution refutation can be considered as determined by the answers to a series of queries. That is to say, starting at the root of the ....

J. Krajcek. Bounded Arithmetic, Propositional Logic and Complexity Theory. Cambridge University Press, 1995.

....of propositional calculus and many questions studied in theoretical computer science are related to it too. The purpose of this paper is to survey some results which should give an idea to an outsider of what is going on in this eld and explain motivations for the studied problems. We recommend [3, 5, 15, 11, 34] to those who want to learn more about this subject. 2 Basic concepts 2.1 The basic theory used in logic for studying natural numbers is the so called Robinson s arithmetic, or Q, which is axiomatized by the following axioms: S(x) 6= 0; S(x) S(y) x = y; x 6= 0 9y(x = S(y) x 0 = x; x ....

....4 has an easy corollary which can be interpreted as an independence result. Corollary 5. If it is provable in S 1 2 for a class X that X 2 NP coNP , then X 2 P , so NP coNP 6= P is not provable in S 1 2 . It has also been shown that there exists a model of PV in which NP coNP 6= P [15]. We cannot deduce much for complexity theory from this result, since as soon as we make the theory only slightly stronger, the proof breaks down. Still it is a nice result and we would like to get more results like this. 5.2 The second result concerns the problem of the hierarchy of the ....

J. Kraj  cek, Bounded Arithmetic, Propositional Logic and Complexity Theory, Cambridge University Press 1995.

.... Gamma be a monotone sequent with n variables. If Sigma Gamma has an LK proof of size S, then Sigma Gamma has a tree like MLK proof with S O(1) lines and size S O(1) Delta n O(log n) Proof : By Theorem 2 and the well known result that tree like LK polynomially simulates LK [7], it will be sufficient to simulate tree like LK De Morgan proofs by tree like MLK proofs. Let P be a treelike LK De Morgan proof of Sigma Gamma of size S. By the previous lemma and Theorem 2, for each k 2 f0; ng we obtain tree like MLK proofs of the sequents th n k (x 1 ; xn ....

....MLK proofs. Theorem 5 FPHP n 1 n and OPHP n 1 n have tree like MLK proofs of size polynomial in n. Proof : Buss proved that PHP n 1 n has a Frege proof of size polynomial in n, and therefore, so do FPHP n 1 n and OPHP n 1 n . Since tree like LK polynomially simulates any Frege system [7], they also have polynomial size tree like LK proofs. We first consider FPHP n 1 n . For every i 2 f1; n 1g and j 2 f1; ng, let ij be the formula W j 0 6=j p i;j 0 where j 0 ranges over f1; ng. Let LFPHP n 1 n be the left hand side of the sequent FPHP n 1 n ....

J. Kraj'icek. Bounded Arithmetic, Propositional Logic and Complexity Theory, Cambridge Univ. Press, 1985.

....the induction schema is restricted to bounded formulas. To enable such theory to formalize polynomial time computations, one has also to add an axiom saying that x log x is a total function. There are various presentations of this theory denoted by I Delta 0 Omega 1 , S 2 , T 2 etc. see [2]. In bounded arithmetic it is possible to formalize not only deterministic polynomial time computations (for this a smaller fragment suffices) but also computations with any finite number of alternations, put otherwise, computations with an oracle in the Polynomial Hierarchy. Finite two player ....

J. Kraj'icek, Bounded Arithmetic, Propositional Logic and Complexity Theory, Cambridge Univ. Press, 1995. 7

.... Gamma be a monotone sequent with n variables. If Sigma Gamma has an LK proof of size S, then Sigma Gamma has a tree like MLK proof with S O(1) lines and size S O(1) Delta n O(logn) Proof : By Theorem 2 and the well known result that tree like LK polynomially simulates LK [Kra95], it will be sufficient to simulate tree like LK De Morgan proofs by tree like MLK proofs. Let P be a tree like LK De Morgan proof of Sigma Gamma of size S. By the previous lemma and Theorem 2, for each k 2 f0; ng we obtain tree like MLK proofs of the sequents th n k (x 1 ; x ....

....size MLK proofs. Theorem 5 FPHP n 1 n and OPHP n 1 n have tree like MLK proofs of size polynomial in n. Proof : Buss proved that PHP n 1 n has a Frege proof of size polynomial in n, and therefore, so do FPHP n 1 n and OPHP n 1 n . Since tree like LK polynomially simulates any Frege system [Kra95], they also have polynomial size tree like LK proofs. We first consider FPHP n 1 n . For every i 2 f1; n 1g and j 2 f1; ng, let ij be the formula W j 0 6=j p i;j 0 where j 0 ranges over f1; ng. Let LFPHP n 1 n be the left hand side of the sequent FPHP n 1 n , ....

J. Kraj'icek. Bounded Arithmetic, Propositional Logic and Complexity Theory, Cambridge Univ. Press, 1985.

....i RFN(P ) is a 8 b i formula. The quanti ed propositional proof systems we will be working with in this paper are formulated over the sequent calculus and will allow the following rules of inference: a) structural rules, propositional rules, and the cut rule for the system LK as de ned in [14]. As we make frequent use of the cut rule and the right rule, we list them as examples: A PROPOSITIONAL PROOF SYSTEM FOR R i 2 11 (Cut rule) A A; right) A ; B ; A B (b) propositional quanti er rules de ned as follows: 8:left) A( B) 8 xA( x) ....

....proof system that uses a nite axiom schemata, and has modus ponens as its only rule of inference. We are requiring to be in language. The usual de nition of Frege proof system is more general in that it does not have this condition and it allows so called 12 CHRIS POLLETT Frege rules (see [11, 14]) as opposed to modus ponens; however, our de nition will suit our present purpose. A Frege proof is a sequence of propositions A 1 ; An where each A i is either a substitution instance of an axiom or follows from earlier propositions by modus ponens. An extended Frege proof system is a ....

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J. Krajcek. Bounded Arithmetic, Propositional Logic and Complexity Theory. Cambridge University Press, 1995.

....one extreme of the trade off we get quasipolynomialsize monotone proofs of PHP n 1 n . At the other extreme of the trade off we get subexponentialsize bounded depth monotone proofs of PHP n 1 n . This comes close to known lower bounds for bounded depth Frege (or LK) of the pigeonhole principle [22], although the dependence on the depth parameter is still substantially different in upper and lower bounds. In general, the trade off may be expressed as follows: for every s n, the PHP n 1 n admits monotone proofs of size n s log s (n) and depth 2 log s (n) The abovementioned extremes occur ....

....of size n O(logn) Corollary 2 The sequent PHP n has depth d MLK proofs of size 2 O(n 3=d ) for every constant d 2. Corollaries 1 and 2 obviously hold also for bounded depth LK. The lower bound for the size of d depth LK proofs of the pigeonhole principle is Omega Gamma n (1=6) d ) [22]. Thus, the dependence on d is an exponential higher than in Corollary 2. This makes a noticeable difference: Corollary 2 implies that there are proofs of quasipolynomial size and depth O(log n= log log n) the lower bound implies only that proofs of quasipolynomial size must have depth ....

J. Kraj'icek. Bounded Arithmetic, Propositional Logic and Complexity Theory. Cambridge University Press, 1995. 14

....S i k to be BASIC k Sigma b i LIND and T i k to be BASIC k Sigma b i IND. Given a sequence of superscripted theories T i we write T for [ i T i . So S k : i S i k . The theories S k are conservative extensions of the Wilkie Paris [15] theories I Delta 0 Omega k [9]. Sequent calculus systems for the former theories are often more convenient than for the latter, although the latter theories do have the virtue of all being define over the same language. The above axiomatization of S i 2 was given in Pollett [11] and shown to be equivalent to the original one ....

J. Kraj'icek. Bounded Arithmetic, Propositional Logic and Complexity Theory. Cambridge University Press, 1995.

....TC 0 is the smallest class of functions that contains 0, i n k , s 0 , s 1 , multiplication Delta, #, jxj, Bit and which is closed under composition and CRN. 3 Theories of Bounded Arithmetic We briefly review the necessary background on Bounded Arithmetic, for more information see [4] or [10]. The language L 2 of Bounded Arithmetic comprises the usual signature of arithmetic 0; S; Delta; together with function symbols for b 1 2 xc, MSP (x; i) bx=2 i c, jxj and #. A quantifier of the form 8x t , 9x t with x not occurring in t is called a bounded quantifier. ....

J. Kraj'icek. Bounded Arithmetic, Propositional Logic and Complexity Theory. Cambridge University Press, 1995.

....in this paper is to generalize the natural notion of space complexity to other propositional proof systems and complexity measures, and research its properties. The rst arising question is how to measure the memory content at any given moment of time for a speci ed proof system. Recall (see e.g. [Kra95]) that the most customary measures for size complexity of propositional proofs are the bit size and the number of lines. Of these two, the bit size is by far more important, and can be de ned analogously, and naturally, in the context of space complexity. The only simpli cation we allowed ....

J. Krajcek. Bounded arithmetic, propositional logic and complexity theory. Cambridge University Press, 1995. 34

....proofs is based on valuations in boolean algebras. This has actually been used implicitly in [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 Kraj icek [8] and a forthcoming survey by the first author [ 2 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 is a blatant lie for you, but the judge, and ....

Jan Kraj'icek. Bounded Arithmetic, Propositional Logic and Complexity Theory. Cambridge University Press, 1995.

....bounded quanti cation and bounded existential (resp. universal) quanti cation. BASIC denotes a set of quanti er free axioms specifying the interpretations of the function symbols of the language, see Buss [3] and Takeuti [16] For more background concerning Bounded Arithmetic see Kraj cek [10] or [3] We de ne some terms that will be used frequently below: 2 jxj : 1#x mod2(x) x : 2 b 1 2 xc Bit(x; i) mod2(MSP (x; i) 2 min(x;jyj) MSP (2 jyj ; jyj : x) LSP (x; i) x : 2 min(i;jxj) MSP (x; i) a (w; i) MSP (LSP (w; Si jaj) i jaj) so that LSP ....

J. Krajcek. Bounded Arithmetic, Propositional Logic and Complexity Theory. Cambridge University Press, 1995.

....pigeons, m, is also a parameter. See also [Urq87, CS88, BP96] for other bounds on the complexity of resolutions, and [Juk97] for a generalization of the Haken Buss Tur an bound to the case of semantic resolutions. All these lower bounds trivialize when m n 2 . As mentioned in [BT88] also see [Kra94, page 31]) it is an open question whether PHP n 2 n has a poly size resolution proof, and this is open even for regular resolutions. More generally, it is open whether there is any m (as a function of n) for which PHP m n has a resolution proof of size polynomial in n. The only non trivial upper bound ....

J. Kraj'icek. Bounded arithmetic, propositional logic and complexity theory. Cambridge University Press, 1994.

....on the complexity of resolutions, and [Juk96] for a generalization of the Haken Buss Tur an bound to the case of semantic resolutions. All these bounds trivialize when m n 2 , and all lower bounds techniques discovered so far become void with that many pigeons. As mentioned in [BT88] see also [Kra94, page 31]) it is an open question whether PHP n 2 n has a poly size resolution proof, and this is open even for regular resolutions. More generally, it is open whether there is any m (as a function of n) for which PHP m n has a resolution proof of size polynomial in n. The only non trivial upper ....

J. Kraj'icek. Bounded arithmetic, propositional logic and complexity theory. Cambridge University Press, 1994.

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J. Kraj ' icek. Bounded arithmetic, propositional logic and complexity theory. Cambridge University Press, 1994.

....that contain non trivial bounded quantifiers, non triviality meaning that they quantify over the full domain of objects whose length is comparable to the length of initial parameters. This technique is not considered in this paper, and the interested reader is referred e.g. to the monographs [10, 17, 22]. The only formulae we are dealing with in this paper are Sigma b 0 formulae. These are essentially the formulae in which all quantifiers are sharply bounded i.e. quantify over some domain whose size is comparable to the length of initial parameters. The bad news about these formulae is that ....

....news is that the provability of Sigma b 0 formulae in theories of Bounded Arithmetic is closely related to the existence of short propositional proofs for certain propositional tautologies associated with the formula. We do not give here exact definitions or details, they can be found e.g. in [22]. The best way for a complexity oriented reader to imagine this correspondence is to invoke the familiar analogy with uniform vs. non uniform computational models. A Sigma b 0 formula corresponds to a language, provability in Bounded Arithmetic is analogous to computability within specified ....

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J. Kraj ' icek. Bounded arithmetic, propositional logic and complexity theory. Cambridge University Press, 1994.

....the perspective of methods available rather than surveying known partial results (i.e. lower bounds for weaker proof systems) We discuss neither motivations for proving lower bounds for propositional logic nor relations to other problems in logic or complexity theory. The reader is referred to [20] for the background information (as well as for all details missing in this paper) The paper is aimed at curious non specialists. The style of our exposition is accordingly informal at places and we do not burden the text (especially in the introduction) with exhausting references not directly ....

....The style of our exposition is accordingly informal at places and we do not burden the text (especially in the introduction) with exhausting references not directly related to our main objective. The reader starving for details can find them, together with all original references, in [20] (see also expository articles [25, 32] Introduction The language of propositional calculus contains constants 0 and 1 (false and true) connectives : and with their usual meaning, and atoms p 0 ; p 1 ; A propositional calculus is given by a finite number of axiom schemes (like A:A) ....

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J. Kraj'icek : Bounded arithmetic, propositional logic and complexity theory, Cambridge University Press, in print.

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J. Krajcek. Bounded arithmetic, propositional logic and complexity theory. Cambridge University Press, 1995.

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J. Kraj cek. Bounded Arithmetic, Propositional Logic and Complexity Theory. Cambridge University Press, 1995.

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J. Krajcek. Bounded arithmetic, propositional logic and complexity theory. Cambridge University Press, 1995.

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J. Kraj cek, Bounded Arithmetic, Propositional Logic and Complexity Theory, Cambridge University Press, Cambridge, UK, 1995.

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J. Krajcek. Bounded arithmetic, propositional logic and complexity theory. Cambridge University Press, 1995.

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Krajcek, J.: Bounded arithmetic, propositional logic and complexity theory. Cambridge University Press (1995)

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