On the Classical Decision Problem y Yuri Gurevich Author: Hello, my friend. What is on your mind to day? Quisani: Decidable and undecidable fragments of rst-order logic. People refer to this eld as Entscheidungsproblem or the classical decision problem. I wanted to see a global picture, without going into to o many details, and failed. You worked in the eld, didn't you? Can you shed some light? A: I am surprised. Decidability,you joked the other day, is a red herring. Q: I did? Well, there is no doubt in my mind that feasibility is the real issue, butIkeep bumping into the classical decision problem. Most recently, this happ ened when I lo oked up a pap er of Kolaitis and Vardi [KV] on the 0{1 law. (This lawis,by the way, another issue I would like to discuss with you sometime.) Besides, the two issues { decidability and feasibility{ areobviously related. Undecidability implies nonfeasibility, and nonfeasibility pro ofs (e.g. pro ofs of completeness for NP or exp onential time) are often fashioned after undecidability pro ofs. Sp eaking ab out surprises, I was surprised to o. Apparently, the classical decision problem was tremendously p opular among logicians. Even Godel worked on it. This puzzles me. Logicians are so philosophically minded. Why all that interest in what seems to b e a rather technical question? A: Let me start from the b eginning. The original Entscheidungsproblem was p osed, I guess, by Hilb ert. It may b e stated as a satis abilityorvalidity prob- lem: Given a rst-order formula , decide whether is satis able (resp ectively, valid). Pro of theorists usually prefer the validityversion whereas mo del theo- rists prefer the satis abilityversion. I am more used to the satis abilityversion; let me cho ose it to b e the default. Q: What rst-order formulas are you talking ab out? Do you allow equality, function symb ols? They make a big deal out of such details in that eld. Logic in Computer Science Column, The Bulletin of EATCS, Octob er 1990 y Partially supp orted by NSF grant CCR 89-04728. Address: EECS Department, Universityof Michigan, Ann Arb or, MI 48109-2122, USA 1 A: These details do not matter for the original Entscheidungsproblem, but they will matter later, so let us x a version of rst-order logic without equality or individual constants or function symbols. (First-order logic comes with a deduction mechanism, but details of the deduction mechanism will b e irrelevant for our purp oses.) Without loss of generality,wemay restrict attention to sentences, i.e., formulas without free individual variables. Let me also clarify the terminology. Recall that the collection of predicates (i.e. relation symb ols) of a sentence is its signature. A sentence of some signature is satis able if there exists a structure of signature that satis es ,and is valid (or logically true) if every -structure satis es . Itiseasytoseethatthetwoversions of Entscheidungsproblem are easily reducible each to the other. Hilb ert called Entscheidungsproblem \the fundamental problem of mathematical logic" [DG]. Q: Sounds very imp ortant indeed. A: At the time the notion of algorithm was not formalized. Algorithms usually meant feasible algorithms, I think. Just imagine you have a feasible decision algorithm for Entscheidungsproblem. You would b e able to solvenumerous mathematical problems including those most famous. Q: Name one. A: The great \theorem" of Fermat. Its negation is expressed by an existential sentence =(9x; y; z; u)(x; y ; z ; u) in the language of Peano Arithmetic where u u u (x; y; z; u) states that x; y ; z are p ositiveand u>2andx + y = z . Cho ose a nitely axiomatizable fragment PA of Peano Arithmetic suciently richto 0 prove(a; b; c; n)or :(a; b; c; n), whichever is true, for any sp eci c quadruple a; b; c; n of natural numb ers, and let b e the conjunction of axioms of PA . 0 0 It is easy to see that the great \theorem" fails if and only if the implication ! is valid. Now use your algorithm. 0 Q: Sorry, I do not rememb er exactly what Peano Arithmetic is. Is it well known that there exists a nitely axiomatizable fragmentofPeano Arithmetic suciently rich for our purp ose? A: Peano Arithmetic is a standard rst-order formalization of the arithmetic of natural (i.e. nonnegativeinteger) numb ers. It is describ ed in many logic textb o oks; see Kleene's b o ok [Kl] for example. It has a small numb er of sp eci c axioms and one axiom schema that formalizes the induction principle. One well- known nitely axiomatizable fragmentofPeano Arithmetic suciently rich for our purp ose is Robinson's system [Kl] formulated by Raphael Robinson. Q: All right. Allowmetocheck that the great \theorem" fails if and only if the implication ! is valid. First I supp ose that the great \theorem" fails 0 and a quadruple a; b; c; n is a counter-example. Then (a; b; c; n)istrue, PA 0 2 proves (a; b; c; n)and PA proves . Hence the implication ! is valid. 0 0 Next I supp ose that the implication ! is valid. Then PA proves ,and{ 0 0 if PA is consistent{ is true, and the great \theorem" fails. Fine. You need 0 the consistency of PA , but natural numb ers with usual arithmetical op erations 0 (whichyou probably want to representby relations) form, I understand, a mo del for Peano Arithmetic and therefore for PA , so there is no problem there. 0 I likeyour argument indep endently of the decidability issue. It shows that the great \theorem" fails if and only if its negation is provable in a fragment of Peano Arithmetic.Thus, proving that the great \theorem" is indep endent from, say,Peano Arithmetic would mean that it is true. This is interesting. A: This is not my argument of course. It is folklore. Notice that the argument uses only that the great \theorem" is expressible byuniversal sentences of Peano Arithmetic. The Riemann hyp othesis is expressible byauniversal sentence of Peano Arithmetic to o though this is not obvious at all [DMR]. It follows that the Riemann hyp othesis fails if and only if its negation is provable in Peano Arithmetic. The decision algorithm for rst-order logic would decide the Rieman hyp othesis as well. Of course, the applicability of the decision algorithm would not b e restricted to problems expressible byauniversal sentences of Peano Arithmetic. Q: You havemadeyour p oint. What happ ened after Hilb ert p osed the problem? A: The classical decision problem was indeed very p opular with logicians. There were plenty of p ositive and negative results [Ch2]. Some go o d mathematics was done along the waytoo.For example, Ramsey's Theorem, so p opular in combinatorics, was proved in a pap er related to a case of the classical decision problem. Q: Wait, I thoughtyou were still talking ab out the p erio d b efore the formaliza- tion of algorithm. How could one prove negative results at that time? A: The same way that many negative complexity results are proved to day. The key word is \reduction". A class K of sentences is called a reduction class (for satis ability) if there exists an algorithm that, given an arbitrary sentence , 0 0 pro duces a sentence in K such that is satis able if and only if is. K is called decidable (for satis ability) if the satis ability problem SAT(K ) { given a sentence in K , decide whether it is satis able { is decidable. Q: Let me see. We deal with total recursive reductions. Is it true that the decision problem for any recursively enumerable set reduces to the Entschei- dungsproblem? A: Absolutely. 3 Q: Then the satis ability problem for any reduction class is complete for re- cursiveenumerability with resp ect to total recursive reductions. I realize that these notions were not known at the time, and you don't need these notions to understand that the satis ability problem for a reduction class is as dicult as the whole Entscheidungsproblem. A: Right. When Church and Turing formalized the notion of algorithm and proved the undecidability of the original Entscheidungsproblem [Ch1, Tu], re- duction classes were proven to b e undecidable. (More exactly, the satis ability problem for any reduction class is undecidable.) But the eld did not die, though the fo cus shifted. The classical decision problem b ecame sort of a metaproblem: Which fragments of rst-order logic (more exactly, which classes of sentences) are decidable? Q: Why didn't the eld die? The original Entscheidungsproblem was a sp eci c question. Church and Turing answered the question. I guess, it to ok some time for the Church-Turing thesis to sink in and b ecome accepted. But why didn't the eld die after that? Why did the metaproblem attract attention after that? A: First of all, I doubt that Hilb ert saw the original Entscheidungsproblem as a yes-no question. He might think ab out an op en-ended problem of mechanizing mathematics. The ambitious attempt to mechanize mathematics via a decision algorithm for rst-order logic failed. Do es this mean that the eld should b e abandoned? Of course not. One should try to see what can b e done. It is natural to try to isolate sp ecial cases of interest where mechanization is p ossible. There are manyways to de ne the syntactic complexity of logic formulas. It turns out that, with resp ect to some natural de nitions, sentences of low syntactic complexity suce to express many mathematical problems.