[PDF] - A Prolog T ec hnology Theorem Pro v er | Mark Stic k el PDF Document

SLIDE 1 A Prolog T ec hnology Theorem Pro v er | Mark Stic k el (JAR 1988) 1 Mark E. Stic k el A Prolog T ec hnology Theorem Pro v er: Implemen tation b y an Extended Prolog Compiler Journal

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Automated Reasoning v

l.

4, pp. 353-380, 1988 Discussion led b y John Harrison Univ ersit y

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Cam bridge

PTTP:

history and its place in A TP

Horn

clauses and Prolog

F

rom Prolog to PTTP

Renemen

ts John Harrison Univ ersit y

f

Cam bridge, 26 June 1997

SLIDE 2 A Prolog T ec hnology Theorem Pro v er | Mark Stic k el (JAR 1988) 2 Mo del elimination The deductiv e pro cedure underlying PTTP is Donald Lo v eland's MESON mo del elimination metho d, whic h w as in v en ted in the sixties. Mo del elimination is describ ed b y Lo v eland in JA CM v

l.

15 (1968), pp. 236-251 and MESON is describ ed in his 1978 b

k:

`Automated Theorem Pro ving: A Logical Basis' (North-Holland). ME w as dev elop ed b efore Lo v eland had heard

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resolution. Lo v eland's later dev elopmen t

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linear resolution w as quite separate. ME is a general pro

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metho d for rst

rder

logic, and do es not (directly) supp

rt

equalit y reasoning, arithmetic etc. John Harrison Univ ersit y

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Cam bridge, 26 June 1997

SLIDE 3 A Prolog T ec hnology Theorem Pro v er | Mark Stic k el (JAR 1988) 3 PTTP The idea underlying Stic k el's PTTP w as to implemen t the MESON pro cedure using `Prolog T ec hnology'. That is, he made just a few small mo dications to a standard Prolog system (details later) and

btained

a system complete for rst

rder

logic. It's probably thanks to PTTP that mo del elimination didn't disapp ear completely against the bac kground

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the in tense in terest in resolution. SETHEO (from Munic h), winner

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the 1996 CADE theorem pro ving comp etition, is basically a w ell-engineered v ersion

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PTTP . The second-placed system, Otter, is the curren t resolution agship. There are implemen tations

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similar algorithms in Isab elle (meson_tac) and in HOL (MESON_TAC), though here clauses are interpr ete d not c

mpile

d. John Harrison Univ ersit y

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Cam bridge, 26 June 1997

SLIDE 4 A Prolog T ec hnology Theorem Pro v er | Mark Stic k el (JAR 1988) 4 Where ME b elongs W e can divide the standard rst

rder

theorem pro ving metho ds in to t w

main

groups:

The

b

ttom-up,

`lo cal' metho ds, e.g. resolution (Robinson, JA CM 1965) and the in v erse metho d (Maslo v, Dok. Ak ad. Nauk 1964).

The

top-do wn, `global' metho ds, e.g. mo del elimination and tableaux. In some sense, al l these can b e seen as searc h for a pro

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in cut-free sequen t calculus, using unication to disco v er instan tiations for quan tiers. The b

ttom-up

metho ds start at the assumptions and deduce an ev er-increasing set

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facts till they reac h the conclusion. T

p-do

wn metho d w

rk

bac kw ards from the conclusion, breaking it do wn to subproblems un til the assumptions are reac hed. John Harrison Univ ersit y

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Cam bridge, 26 June 1997

SLIDE 5 A Prolog T ec hnology Theorem Pro v er | Mark Stic k el (JAR 1988) 5 T

p-do

wn vs. b

ttom-up

The b

ttom-up

metho ds ha v e sev eral adv an tages. Eectiv ely they p erform pro

f

at the meta-lev el: w e can regard free v ariables as implicitly univ ersally quan tied. Therefore it is p

ssible

to apply subsumption to the curren t set

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facts, and a v

id

pro ving the same lemma t wice. By con trast, in top-do wn (`global') metho ds, the free v ariables in dieren t subgoals need to b e correlated. Ho w ev er, top-do wn metho ds are more goal-directed: w e don't just gro w a big set

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facts and hop e w e reac h the conclusion. Moreo v er, they are m uc h more economical to implemen t, since w e

nly

need to store the curren t subgoals. In fact, they are al l v ery Prolog-lik e: apart from the PTTP implemen tation

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MESON, there is a complete tableau pro v er called leanT A P that requires

nly

5 lines

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Prolog. John Harrison Univ ersit y

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Cam bridge, 26 June 1997

SLIDE 6 A Prolog T ec hnology Theorem Pro v er | Mark Stic k el (JAR 1988) 6 leanT A P This is due to Bec k ert and P

segga;

see the Journal

f

Automated Reasoning, v

l.

15, pp. 339-358, 1995. prove((E,F),A,B,C,D) :- !,prove(E,[F|A],B,C,D). prove((E;F),A,B,C,D) :- !,prove(E,A,B,C,D), prove(F,A,B,C,D). prove(all(I,J),A,B,C,D) :- !, \+length(C,D),copy_term((I,J,C),(G,F, C)), append(A,[all(I,J)],E),prove(F,E,B,[G |C],D) . prove(A,_,[C|D],_,_) :- ((A=

(B);-(A)=B)
>

(unify(B,C);prove(A,[],D,_,_))). prove(A,[E|F],B,C,D) :- prove(E,F,[A|B],C,D). This sort

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naiv e tableau pro v er is the core

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Isab elle's fast_tac and HOL's TAB_TAC. John Harrison Univ ersit y

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Cam bridge, 26 June 1997

SLIDE 7 A Prolog T ec hnology Theorem Pro v er | Mark Stic k el (JAR 1988) 7 Horn clauses and Prolog A clause is a disjunction

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literals, where a literal is either an atomic form ula

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its negation: L 1 _

_

L n W e sa y it is a Horn clause if it has at most

ne

unnegated literal. In this case w e can write it as L 1 ^

^

L k 1 ^ L k +1 ^

^

L n = ) L k

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simply `L 1 ' if n = 1. These are the clauses that are allo w ed in a Prolog database. The Prolog syn tax for the protot ypical Horn clause is: L k :-

L

1 ; : : : ; L k 1 ; L k +1 ; : : : ; L n Prolog allo ws us to deduce an atomic form ula from suc h a database b y bac k c haining through the rules, using unication to instan tiate v ariables (written in upp er case in Prolog). John Harrison Univ ersit y

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Cam bridge, 26 June 1997

SLIDE 8 A Prolog T ec hnology Theorem Pro v er | Mark Stic k el (JAR 1988) 8 Wh y is Prolog inadequate? Prolog certainly has a limited abilit y to pro v e theorems. Ho w ev er it is inadequate as a general rst

rder

pro v er for three reasons:

Most

Prolog implemen tations ha v e unsound unication

Prolog

is limited to Horn clauses

Prolog's

depth-rst searc h strategy is incomplete. W e arriv e at PTTP b y xing eac h

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these problems. W e will consider them in turn. John Harrison Univ ersit y

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Cam bridge, 26 June 1997

SLIDE 9 A Prolog T ec hnology Theorem Pro v er | Mark Stic k el (JAR 1988) 9 Unsound unication It has long b een usual for Prolog implemen tations to

mit

the so-called `o ccurs c hec k', e.g. allo wing X and f (X ) to b e unied. This is either for (probably b

gus)

eciency reasons,

r

b ecause circular data structures are sometimes considered useful. Ho w ev er it's disastrous for theorem pro ving, e.g. it w

uld

allo w us to deduce SUC(Y) < Y from X < SUC(X). The x is easy: just do unication prop erly . John Harrison Univ ersit y

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Cam bridge, 26 June 1997

SLIDE 10 A Prolog T ec hnology Theorem Pro v er | Mark Stic k el (JAR 1988) 10 Limitation to Horn clauses It is not alw a ys p

ssible

to reduce theorem pro ving problems to Horn clause sets acceptable to Prolog. F

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example, w e migh t w an t to use the facts A _ B and A = ) B to deduce B . Ho w ev er there is no equiv alen t in terms

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Horn clauses. The solution adopted in PTTP is to extend the notion

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`Horn clause': A 1 ^

A

n = ) B to allo w an y

r

all

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the literals in v

lv

ed to b e negated. No w w e can tak e an y problem and reduce it to something based

n

these pseudo-Horn clauses. John Harrison Univ ersit y

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Cam bridge, 26 June 1997

SLIDE 11 A Prolog T ec hnology Theorem Pro v er | Mark Stic k el (JAR 1988) 11 Con trap

sitiv

es W e tak e the fact w e w an t to pro v e (ma yb e an implication under a set

f

assumptions), negate it, Sk

lemize

it and reduce it to clausal form. W e w an t to deriv e ?. F

r

eac h clause: P 1 _ : : : _ P n w e form n c

ntr

ap

sitives
f

the form: P 1 ^

^

P i1 ^ P i+1 ^

^

P n = ) P i and

ne

more

f

the form: P 1 ^ : : : ^ P n = ) ? No w w e try to solv e the goal ?

a

la Prolog. John Harrison Univ ersit y

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Cam bridge, 26 June 1997

SLIDE 12 A Prolog T ec hnology Theorem Pro v er | Mark Stic k el (JAR 1988) 12 Incompleteness Unfortunately , while Prolog-st yle bac k c haining is complete for true Horn clauses, this is not so for pseudo-Horn clauses. Consider the in tended example

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deducing B from A _ B and A = ) B . The con trap

sitiv

es are: B = ) ? :A = ) B :B = ) A :A ^ :B = ) ? A = ) B :B = ) :A A ^ :B = ) ? It is immediate that no Prolog-st yle searc h can terminate in success b ecause there are no unit clauses. John Harrison Univ ersit y

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Cam bridge, 26 June 1997

SLIDE 13 A Prolog T ec hnology Theorem Pro v er | Mark Stic k el (JAR 1988) 13 Ancestor unication W e can restore completeness b y an extra rule: as w ell as unication with the conclusion

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a rule, w e allo w unication with the ne gation

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an anc estor. This is treated as a unit clause and can solv e a goal; note that the v ariables, if an y , are correlated. F

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example ? B A :B No w w e can unify :B and the negation

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B . The logical justication is simple: if w e are trying to pro v e a goal, here B , w e ma y assume its negation :B , since if that is false w e are immediately nished. John Harrison Univ ersit y

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Cam bridge, 26 June 1997

SLIDE 14 A Prolog T ec hnology Theorem Pro v er | Mark Stic k el (JAR 1988) 14 Searc h strategy Although this is no w complete as a c alculus, the usual Prolog depth-rst searc h with rules tried in

rder

is trivially incomplete. F

r

example, the rules P (f (X )) = ) P (X ) and P (f (a)) cannot solv e the goal P (a) b ecause Prolog will k eep applying the rst rule ad innitum: P (a) P (f (a)) P (f (f (a))) P (f (f (f (a))))

W

e need to use a searc h strategy that will allo w all p

ssible

pro

fs

to b e found. The most

b

vious is breadth-rst searc h. But this blo ws up the storage requiremen t: the minimal storage usage is

ne
f

PTTP's strong p

in

ts. Instead, most implemen tations use depth rst iter ative de ep ening: searc h for pro

fs
f

depth 1, then if that fails, depth 2, then if that fails, depth 3, and so

n.

John Harrison Univ ersit y

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Cam bridge, 26 June 1997

SLIDE 15 A Prolog T ec hnology Theorem Pro v er | Mark Stic k el (JAR 1988) 15 Searc h alternativ es V arious searc h mo des ha v e b een exp erimen ted with

Lo

v eland

riginally

used iterativ e deep ening where `depth' is maxim um heigh t

f

pro

f

tree.

Stic

k el then used depth = size

f

pro

f

tree (n um b er

f

no des).

P

aulson uses b est-rst searc h (not iterativ e deep ening).

SETHEO

allo ws heigh t

r

size b

unds;

the former is b etter

n

a v erage.

Harrison

uses an

ptimized

v ersion

f

size b

unds,

whic h seems b etter still. John Harrison Univ ersit y

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Cam bridge, 26 June 1997

SLIDE 16 A Prolog T ec hnology Theorem Pro v er | Mark Stic k el (JAR 1988) 16 Implemen tation renemen ts

Making

larger DFID incremen ts automatically (e.g. if all the clauses are units

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ha v e 3 assumptions, w e can incremen t b y 3 eac h time).

Av
iding

rep etitions do wn a branc h

f

the pro

f

tree. (Though these ma y

nly

app ear after later instan tiation, so there are trade-os to b e made.)

P

erforming `in telligen t bac ktrac king', e.g. if a goal is solv ed using a unit clause

r

ancestor without instan tiation

f

the goal, no

ther

solutions need b e tried. More adv anced

ptimizations

can in teract badly with iterativ e deep ening.

Extending

the system to nonclausal assertions,

a

la Prolog, e.g. P ^ (Q _ R ) = ) S instead

f

t w

separate

clauses. This a v

ids

rep eating the solution

f

P . John Harrison Univ ersit y

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Cam bridge, 26 June 1997

SLIDE 17 A Prolog T ec hnology Theorem Pro v er | Mark Stic k el (JAR 1988) 17 Renemen ts to the calculus It suces to generate rules with conclusion ?

nly

if all the literals in the clause are negated. Often this is just the

riginal

goal. Also, there is Plaisted's `p

sitiv

e renemen t'. There are alternativ e v ersions

f

ME that

nly

use `natural' con trap

sitiv

es, e.g. Lo v eland's Ne ar-Horn Pr

lo

g, Plaisted's Mo die d Pr

blem

R e duction F

rmat

and Baumgartner & F urbac h's R estart Mo del Elimination. There are also v arious tec hniques for cac hing and lemmatizing. These w ere

riginally

used b y Lo v eland, and fell

ut
f

fa v

ur,

but are no w attracting atten tion again. F

r

example, SETHEO uses sev eral quite sophisticated tec hniques. John Harrison Univ ersit y

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Cam bridge, 26 June 1997