Alasdair Urquhart. Complexity of proofs in classical propositional logic. In Yiannis N. Moschovakis, editor, Logic from Computer Science, pages 597-608. Springer, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....is established or all these possibilities failed. Of course we always choose as small as possible. In this article we use only as an abbreviation, what produces sometimes a super uous blow up of the formula (consider e.g. the formulas p 0 p 1 : pn p 0 p 1 : pn of [13]) Using a two sided sequent calculus with special rules for solves this problem. A serious problem is the super uous backward application of ( Consider the following example of backward proof search in (K fP Qg) S;2 (we omit the history in the sequents) P; Q; R; P P; Q; R; Q ( ....

Alasdair Urquhart. Complexity of proofs in classical propositional logic. In Yiannis N. Moschovakis, editor, Logic from Computer Science, pages 597-608. Springer, 1992.

....that the Haj os calculus cannot be polynomially simulated by the tree like Haj os calculus. This loss of efficiency when we restrict to tree like Haj os derivations can be compared to similar losses in efficiency in weak propositional proof systems, such as Resolution and cut free Gentzen systems [Urq]. 2 The Haj os calculus We define a graph G to be a pair (V; E) where V is a finite set of positive integers, and E a set of unordered pairs of elements of V . The size of a graph G, jGj, is the number of its edges. A 3 coloring of a graph is an assignment of one of 3 distinct colors to each of ....

Urquhart, A., "Complexity of proofs in classical propositional logic," Manuscript 1992.

....analysis (which is still feasible on today s computers) but over 2 Theta 10 18 branches in a minimal closed tableau. Technically speaking, these examples serve the purpose of filling a gap in the classification of conventional proof systems in terms of the p simulation relationship (see [CR74, CR79, Urq90]) truth tables and Smullyan s analytic tableaux are incomparable proof systems 5 . We relate this phenomenon to a basic inadequacy of the tableau branching rules which affects the complexity of tableau refutations in general and not only in the case of the hard examples described. These are ....

A. Urquhart. Complexity of proofs in classical propositional logic. In Y. Moschovakis, editor, Logic from Computer Science. Springer-Verlag, 1990. Forthcoming.

.... methods [2, 6] or canonical proofs [4, 5] Even if the later one allows to reduce some non determinism sources, through particular proof forms, it does not propose a good management of the additive connectives [4] Thus, to increase the pruning of the proof search tree in MALL, continuing works of [1, 2, 3, 6, 11], we propose a tableaux like proof search method, based on a speci c deducibility relation dealing with additive and multiplicative contexts. To improve proof search, we can propose strategies in this new proof system. 2 Additives management in proof search Let us start to present some ....

A. Urquhart, Complexity of proofs in classical propositional logic, In Logic from Computer Science, ed. Y. Moschovakis, Springer-Verlag, 596608, 1992.

....set H of formulae with S 1 proofs of size O(f(n) and one can prove that the length of their shortest S 2 proofs cannot be bounded above by any polynomial function of f(n) then one can conclude that S 2 cannot p simulate S 1 . For the formal definitions and the resulting classification see [10, 11, 36]. The examples used in [13] to show the speed up of the truth tables over Smullyan s tableaux are expressions in conjunctive normal form, defined as follows: given a sequence of k atomic variables P 1 ; P k , consider all the possible clauses containing as members, for each i = 1; 2; ....

A. Urquhart. Complexity of proofs in classical propositional logic. In Y. Moschovakis, editor, Logic from Computer Science, pages 596--608. SpringerVerlag, 1992.

....we obtain Theorem 5. The decision problem for ALL can be solved in polynomial time in the numbers of literals. Compared with the last approach, we do not prove twice the same statement. The proof search is represented by an acyclic graph which allows to share the previous result, see also [6, 17]. Roughly, we can say that a brute recursive method is memoryless whereas here we keep in mind previous established lemma. Let us remark that we can propose some optimizations as to deduce that (U; V ) could be provable only by analyzing atoms or to recognize proof templates (modulo instantiation) ....

A. Urquhart, Complexity of proofs in classical propositional logic, In Logic from Computer Science, ed. Y. Moschovakis, Springer-Verlag, 596608, 1992. This article was processed using the L A T E X macro package with LLNCS style

....can be presented anyhow. Therefore, it is of course impossible to give a general treatment of efficiency. In relative sense it seems from our own investigations that BDDs for propositional logic are polynomially incomparable to standard techniques such as semantical tableaux and resolution (see [16] for the details of polynomial simulations to compare the different methods) Such results carry over to the setting with predicate logic. But we have not investigated this any further. One might obtain some insight by implementing the proposed algorithm. As this is a far from trivial job, we ....

A. Urquhart. Complexity of proofs in classical propositional logic. In Y. Moschovakis, editor, Logic from Computer Science, Springer-Verlag, pp. 596-608, 1992.

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