Automated Reasoning 1 Automated Reasoning John Harrison Univ - PDF document

Automated Reasoning 1 Automated Reasoning John Harrison Univ ersit y of Cam bridge � What is automated reasoning? Automatic vs. in teractiv e. � Successes of the AI and logic approac hes � Dev elopmen t of formal logic � History of automated reasoning � V eri�cation � Curren t researc h topics John Harrison Univ ersit y of Cam bridge, 22 Jan uary 1997

Automated Reasoning 2 What is automated reasoning? In one sense w e'll in terpret our title narro wly: w e are in terested in reasoning in logic and mathematics, rather than ev eryda y life. The �eld is also called automate d the or em pr oving . In another sense w e in terpret it broadly: w e don't just consider making computers pro v e theorems automatically , but also w a ys in whic h they can supp ort h umans. The correct title migh t b e me chanize d the or em pr oving . W e'll divide the discussion in to (fully) automatic systems, and inter active systems. John Harrison Univ ersit y of Cam bridge, 22 Jan uary 1997

Automated Reasoning 3 The limits of automated reasoning It's almost certainly imp ossible, ev en in principle, that a computer can pro v e automatically all the mathematical theorems w e are in terested in. This follo ws from T arski's the or em on the unde�nability of truth (1936), whic h implies that the set of true facts of arithmetic is not ev en semic omputable . Ho w ev er w e can set up logical systems that are capable of deducing man y , p erhaps most, in teresting theorems, suc h that the set of logically v alid form ulas is at least semic omputable . F or example, the system of Zermelo-F r aenkel set the ory based on �rst or der lo gic has this prop ert y . But this do esn't include al l true facts (G� odel 1930, also a corollary to T arski's theorem). And it is still not c omputable . John Harrison Univ ersit y of Cam bridge, 22 Jan uary 1997

Automated Reasoning 4 Decidable systems In fact, for some limited areas of mathematics or logic, there are systems for whic h v alidit y is actually c omputable . A simple example is prop ositional logic. W e can decide if : ( p _ q ) ) : p ^ : q is v alid simply b y considering cases, e.g. writing a truth table. F or a more in teresting example, the �rst order theory of reals with m ultiplication is decidable (T arski 1948). This theory includes man y non trivial problems. In general, note that a system that is c omplete and semic omputable is also c omputable . (This follo ws from a classic theorem in computabilit y theory that if a set and its complemen t are b oth RE, the set is recursiv e.) John Harrison Univ ersit y of Cam bridge, 22 Jan uary 1997

Automated Reasoning 5 Do wn to earth Despite these promising facts, t w o similar problems remain. 1. Ev en if a theory is decidable in principle, the time or space usage of the decision pro cedure ma y mak e it ine�ectiv e in practice. This applies to the �rst order theory of reals, for example. 2. In systems where v alidit y is semic omputable , w e just ha v e to k eep searc hing un til w e �nd the theorem. This is also impractical in man y cases; t ypically , w e use ingenious tric ks to cut do wn the searc h space. The tric ks are usually dra wn either from lo oking at human b ehaviour or considering the or ems fr om lo gicians . There w as (is?) still a con tro v ersy o v er whether the h uman-orien ted `AI' approac h or the `logic' approac h is b etter. John Harrison Univ ersit y of Cam bridge, 22 Jan uary 1997

Automated Reasoning 6 A theorem in geometry One of the early successes in automated theorem pro ving (the AI side) w as the pro of of the follo wing theorem: A � A � A � A � A � A � A � A � A B C If the sides AB and AC are equal (i.e. the triangle is isoseles), then the angles AB C and AC B are equal. John Harrison Univ ersit y of Cam bridge, 22 Jan uary 1997

Automated Reasoning 7 The usual pro of The usual pro of pro ceeds b y dropping a p erp endicular do wn from the p oin t A to the side B C , meeting it at a p oin t D : A � A � A � A � A � A � A � A � A B C D and then using the fact that the triangles AB D and AC D are congruen t. John Harrison Univ ersit y of Cam bridge, 22 Jan uary 1997

Automated Reasoning 8 The computer's pro of The computer found an ingenious pro of whic h had b een missed b y most writers on geometry (though it had already b een used b y P appus). A � A � A � A � A � A � A � A � A B C Simply , the triangles AB C and AC B are congruen t. Q.E.D. John Harrison Univ ersit y of Cam bridge, 22 Jan uary 1997

Automated Reasoning 9 The Robbins Conjecture (1) A v ery recen t success in automated reasoning, this time on the logic side, w as the pro of b y McCune's program EQP of the Robbins Conjecture. Hun tington (1933) presen ted the follo wing basis for Bo olean algebra: x + y = y + x ( x + y ) + z = x + ( y + z ) n ( n ( x ) + y ) + n ( n ( x ) + n ( y )) = x Shortly thereafter, Herb ert Robbins conjectured that the Hun tington equation can b e replaced with a simpler one: n ( n ( x + y ) + n ( x + n ( y ))) = x John Harrison Univ ersit y of Cam bridge, 22 Jan uary 1997

Automated Reasoning 10 The Robbins Conjecture (2) This conjecture w en t unpro v ed for more that 50 y ears, despite b eing studied b y man y mathematicians, ev en including T arski. It b ecause a p opular target for researc hers in automated reasoning. In Ma y 1996, it w as claimed that a pro of had b een found automatically using the REVEAL pro v er. Ho w ev er this w as traced to a bug in REVEAL. The in Octob er 1996 a correct pro of w as found b y McCune's program EQP . The successful searc h to ok ab out 8 da ys on an RS/6000 pro cessor and used ab out 30 megab ytes of memory . John Harrison Univ ersit y of Cam bridge, 22 Jan uary 1997

Automated Reasoning 11 Origins of mec hanization The idea of mec hanizing reasoning in a manner similar to arithmetic calculation is an old one, going bac k at least to Hobb es. Reason [. . . ] is nothing but Rec k oning. F or as Arithmeticians teac h to adde and subtract in numb ers [...] The Logicians teac h the same in consequences of w ords [...] And as in Arithmetique, unpractised men m ust, and Professors themselv es ma y often erre, and cast up false; so also in an y other sub ject of Reasoning the ablest, most atten tiv e, and most practised men, ma y deceiv e themselv es, and inferre false conclusions. Leibniz en visaged a c alculus r atio cinator . First ho w ev er w e need a char acteristic a universalis. John Harrison Univ ersit y of Cam bridge, 22 Jan uary 1997

Automated Reasoning 12 Dev elopmen t of formal logic W e can highligh t sev eral imp ortan t phases in the dev elopmen t of formal logic. � The So cratic metho d � Aristotle's syllogisms � Leibniz's attempts at a char acteristic a � Bo ole's algebra of logic � F rege's Be gri�sschrift � P eano's F ormulair e � Russell and Whitehead's Principia Mathematic a . � Hilb ert's programme � Metamathematical studies (G� odel, T arski, Ch urc h, T uring, . . . ) John Harrison Univ ersit y of Cam bridge, 22 Jan uary 1997

Automated Reasoning 13 Early computer exp erimen ts The earliest uses of computers in theorem pro ving w ere in the late 50s and early 60s. Among the pioneers w ere: � New ell and Simon (AI) � Gelen tner's geometry mac hine (AI) � Gilmore (logical) � W ang (logical) � Pra witz (logical) The logical approac h pro v ed successful, but so on reac hed its limits. Pra witz's metho d is quite close to mo dern table aux pro v ers. But more p o w erful metho ds w ere needed. John Harrison Univ ersit y of Cam bridge, 22 Jan uary 1997