Goldfarb W., Lewis H.R. "The decision problem for formulas with a small number of atomic subformulas" J. Symbolic Logic 38(3), pp.471--480, 1973.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... (that is without function symbol and with an eventually infinite number of constants) 8t9u8v Delta Delta Delta 8w(A 1 A 2 Delta Delta Delta An ) where the A i are atomic positive or negative formulas, the satisfiability of the 5 subformula case has been shown to be undecidable in [23]. It is established that this problem is equivalent to the halting problem of 2 counter machines (which is undecidable) The 3 and 4 subformula problems remain open. 23 7.4. The Proof via Post This proof [28] is not based on the Conway functions but on the better known Post problem [42] ....

Goldfarb W., Lewis H.R. "The decision problem for formulas with a small number of atomic subformulas" J. Symbolic Logic 38(3), pp.471--480, 1973.

....in the class, whether or not it is satisfiable. Considering the quantificational formulas with a small number of subformulas, the satisfiability of the class 8t 1 ; t 2 ; t 3 ; t 4 [P (t 1 ) Q(t 2 ) R(t 3 ) S(t 4 ) where P , Q, R and S are any positive or negative predicates, is open [9]. A particular subclass (the only interesting one) is the following one : 8t 1 ; t 2 ; t 3 ; t 4 [P (t 1 ) P (t 2 ) P (t 3 ) P (t 4 ) for which the problem of the consistency corresponds to the problem ot the existence of solution for the Prolog program : 8 : P (t 1 ) P (t 2 ....

....with four subformulas, decides in a finite number of steps, whether or not the formula is consistent. This result solves the last open problem in mathematical logic concerning the satisfiability of quantificational formulas with a small number of subformulas. The 5 formulas case was solved in [9]. 6 Linear Horn Clause and Other Subcases We have proved that when the Horn clause is right or left linear, the problems were undecidable. Now it is natural to study the behaviour of this small program depending on the linearity of the terms goal, left, right and fact. We prove that the ....

Goldfarb W., Lewis H.R. "The decision problem for formulas with a small number of atomic subformulas" J. Symbolic Logic 38(3), pp.471-- 480, 1973.

....and other program patterns) 1 Introduction This paper is about the computational power of classes of quantificational formulas specified by restrictions on the number of atomic subformulas. Important works have been done about decision problems for such classes. W. Goldfarb and H.R. Lewis in [10] established the undecidability of the class of those formulas containing five atomic formulas as follows 8x9w8z 1 Delta Delta Delta 8zm [ A 1 A 2 A 3 ) A 4 A 5 ) Indeed, the satisfiability of such a class is equivalent to the halting problem for two counter machines which is ....

Goldfarb W. and Lewis H.R. "The decision problem for formulas with a small number of atomic subformulas" J. Symbolic Logic 38(3), pp.471--480, 1973.

....the classification of prenex classes has been completed. Accounts of the classical results in this area can be found in several books [3, 7, 9, 25] More The results included in section 4 have been published as a part of [29] recent results have been obtained by H.R. Lewis and W. Goldfarb [13, 14, 26]. A short survey of the research in this area can be found in [19] see also the introduction to [15] In 1962, in a short note, D. Scott [31] proved that the satisfiability problem for L 2 was decidable. His proof was based on a reduction of this problem to the problem of satisfiability of ....

H. Lewis and W. Goldfarb. The decision problem for formulas with a small number of atomic subformulas. Journal of Symbolic Logic, 38:471--480, 1973.

....W. Ackermann, P. Bernays, K. Godel, L. Kalm ar, M. Schonfinkel, T. Skolem, H. Wang [1, 2, 6, 20, 21, 32, 52, 53, 54, 58] and many others. Accounts of the classical results in this area can be found in several books [3, 8, 19, 37] More recent results have been obtained by H.R. Lewis and W. Goldfarb [22, 23, 38]. A short survey of the research in this area can be found in [28] Questions concerning decidability of clause implication and similar problems were also studied by computer scientists motivated by problems in the area of artificial intelligence and automated deduction. An easy algorithm deciding ....

H. Lewis and W. Goldfarb. The decision problem for formulas with a small number of atomic subformulas. Journal of Symbolic Logic, 38:471--480, 1973.

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