Istituto per la Ricerca Scientifica e Tecnologica I 38100 Trento ? Loc . Pant?e di Povo ? tel. 0461 ? 314444 Telex 400874 ITCRST ? Telefax 0461 ? 302040 Abstracting into Inconsistent Spaces -LRB- or : the \ False Proof '' Problem -RRB- F. Giunchiglia , T. Walsh April 1989 Technical Report # 8904-02 In Atti del Primo Congresso dell ` Associazione Italiana per l'Intelligenza Artificiale -LRB- AI * IA 89 -RRB- , Trento , Italy , 1989 , pp . 233-239 . Also DAI Research Paper no. 514 , University of Edinburgh . Istituto Trentino di Cultura Abstracting into inconsistent spaces -LRB- Or , the \ false proof '' problem -RRB- Fausto Giunchiglia Mechanized Reasoning Group IRST , Povo , 38100 Trento , Italy Toby Walsh ? Department of Artificial Intelligence University of Edinburgh , 80 South Bridge EH1 1HN Edinburgh , Scotland Abstract Abstraction has been extensively used in many areas of Artificial Intelligence . We present a general framework for analyzing abstraction and using it to explore a hidden danger , the \ false proof problem '' . The existence of this problem is proved , a solution is proposed and an implementation of it that interfaces to a state of the art theorem prover -LRB- Otter -LSB- McC88 -RSB- -RRB- is presented and discussed . 1 Introduction Abstraction can be characterized as the process of mapping a representation of a problem -LRB- also called the \ ground '' representation -RRB- into a new representation -LRB- also called the \ abstract '' representation -RRB- which preserves certain desirable properties and is simpler to handle . The \ desirable properties '' amount to requiring that the abstract solution be of help in solving the problem in the original search space . The notion of \ simplicity '' depends on the application , it may mean decidability or lower complexity . Abstraction functions are usually not bijective . In a previous paper , a theory of abstraction has been outlined which allows us to capture and describe most of the -LRB- informally described -RRB- work on abstraction done in many different areas -LRB- eg . logic programming and theorem proving , planning and common sense ? Supported by a SERC studentship . reasoning -RRB- -LSB- GW89a -RSB- -LRB- an extended version is forthcoming -LSB- GW90 -RSB- -RRB- . The goal of this paper is to make effective use of the theory outlined in -LSB- GW89a -RSB- and to propose a solution to the \ false proof problem '' . Intuitively stated , the false proof problem is as follows . In order to build an abstract space \ simpler '' than the ground space , the trick is to \ forget '' some \ irrelevant '' details and to keep around what is judged important -LRB- according to some theorem proving strategy -RRB- . This , it had been noticed by Plaisted -LSB- Pla80 , Pla86 -RSB- , may cause some problems . In particular it may happen that the abstract space is inconsistent even if the ground space is consistent . In this paper we prove that this problem can not be avoided , in the sense that , once we commit ourselves to certain forms of abstraction -LRB- which , we claim , are just the right ones -RRB- , it is always possible to find a set of axioms such that the abstract space is inconsistent even if the ground space is not . The problem is of particular relevance in applications where the axioms depend on the application . In these cases it is in general not possible to establish a priori whether the abstract space is going to be consistent . The structure of the paper is as follows1 . In section 2 we briefly describe the relevant parts of our theory of abstraction and give some examples . In section 3 the false proof problem is presented and our solution is discussed and compared with other approaches . In section 4 some more implementational issues are discussed . Finally in section 5 the application of our algorithm to some examples is presented and analyzed . 2 A Theory of Abstraction Definition 1 -LRB- Formal system -RRB- : A formal system ? is a triple h ? ; ? ; ? i , where ? is the Language , ? is a set of axioms -LRB- ? ? ? -RRB- and ? is the set of Inference Rules of ? . Intuitively , an abstraction is a mapping from one formal system , ? 1 into another , ? 2 . Definition 2 -LRB- Abstraction -RRB- : If ? 1 = h ?? 1 ; ?? 1 ; ?? 1i and ? 2 = h ?? 2 ; ?? 2 ; ?? 2i are two formal systems , an abstraction mapping f , written also f : ? 1 7 ! ? 2 , is a triple of functions hf ? ; f ? ; f?i such that : f ? : ?? 1 7 ! ?? 2 f ? : ?? 1 7 ! ?? 2 f ? : ?? 1 7 ! ?? 2 where f ? , f ? and f ? are total and surjective . We require that f ? be total since , computationally , we want to be able to \ translate '' any wff in ? 1 , surjective since we want the abstract language to consist only of what is generated by f ? . Analogous arguments apply for the totality and surjectivity of f ? and f ? . 1In order to keep the paper under a reasonable size some theorems are stated but not proved . All the missing proofs can be found in -LSB- GW90 -RSB- and -LSB- GW89a -RSB- . Example 1 -LSB- Abstraction -RSB- : Let 's consider ? 1 to be first order ND and ? 2 to be propositional ND . In this case : ffl f ? maps a first order language onto a propositional language by discarding quantifiers , keeping the connectives and mapping all predicates with the same predicate symbol onto the same propositional constant ; ffl f ? maps the axioms in the same way as f ? maps the wffs ; ffl f ? maps any of the propositional ND rules onto themselves and all other rules onto IIR , the identity inference rule that any wff is derivable from itself . ? Note that sometimes we may want to capture a mapping between formal systems in which we do not map the individual axioms or inference rules but the whole set of axioms or inference rules . In this case , we could generalise the notion of abstraction to map from the power set of axioms , and the power set of inference rules . This topic is much more extensively covered in -LSB- GW90 -RSB- . Often -LRB- eg . in logic programming -RRB- we want to define abstractions with language and inference rules fixed a-priori but parameterized on the axioms which vary according to the application . In other words the axioms should not be fixed when defining the abstraction mapping but only when constructing the application . Lemma 1 : If ? 1 = h ? 1 ; ? 1 ; ? 1i is a formal system and f : ? 1 7 ! ? 2 , f = hf ? ; g ; f?i is an abstraction , then f ? = -LRB- f ? ; g '' ? ; f ? -RRB- is an abstraction , where ? ? ? 1 and by \ g '' ? '' we mean g restricted to apply to ? . f is called a ? - abstraction . f ? is called a ? - restriction of f . We write ?? to mean the formal system obtained from ? by substituting ? 's set of axioms with ? . Where no confusion arises , we will use the notation f to mean f ? , write f -LRB- ' -RRB- to mean the mapping of the wff ' of ? 1 under the abstraction f . Our notion of abstraction is very general ; we therefore characterise abstractions further by how they preserve provability . Definition 3 -LRB- Truthful -RRB- : An abstraction f : ? 1 7 ! ? 2 is said to be truthful iff , for any wff ' , if ` ? 1 ' then ` ? 2 f -LRB- ' -RRB- . Strictly truthful abstractions are -LRB- the majority of truthful -RRB- abstractions where there is at least one wff , for which ` ? 2 f -LRB- -RRB- but 6 ` ? 1 . A class of abstractions can be defined where if f is -LRB- strictly -RRB- truthful so are any of its ? - restrictions . For example , ? - abstractions which keep the same deductive machinery in the ground and abstract spaces , and for any wff ff which is an axiom , f ? maps the ff onto f ? -LRB- ff -RRB- satisfy this property -LRB- such abstractions are said to be ? / ? - invariant -RRB- . A complete characterization of the class of abstractions which maintain strict truthfulness in their ? - restrictions is outwith the goals of this paper . On the other hand most of the abstractions used in the past and all those used in this paper are ? / ? - invariant . Some of the abstractions defined in the past are not strictly truthful -LRB- see -LSB- GW90 -RSB- for a complete discussion -RRB- . The authors claim that , if abstract theorem proving is to tell you anything , provability preserving properties like truthfulness are what you want your abstraction mapping to have2 . Note that one could also require the opposite property , namely that if f -LRB- ' -RRB- is a theorem in ? 2 then ' is a theorem in ? 1 . However , a theorem prover using this type of abstraction will loose completeness as failure to prove f -LRB- ' -RRB- does not guarantee that the ' is not a theorem . One last problem is when ? 1 and ? 2 work by refutation . In such formal systems usually , the system automatically negates the wff to prove , ' , adds this to the axioms and derives ? which , in effect means proving ' . Truthfulness is therefore the provability preserving property we require of refutation systems . See -LSB- GW90 -RSB- for a longer discussion . Example 2 -LSB- Abstraction -RSB- : we hope to demonstrate some of the power of our theory of abstraction by applying it to a particular example of abstraction , predicate abstractions ; this class of abstraction has been proposed for resolution theorem proving by Plaisted -LSB- Pla81 -RSB- and Tenenberg -LSB- Ten87 -RSB- . Plaisted 's work is , perhaps , closest in spirit to this work ; he defines three classes of abstractions that preserve provability . However , this work is less general than ours as he restricts his attention to ? / ? - invariant ? - abstractions between resolution systems . Additionally , his classes of abstraction fail to capture all abstraction mappings that preserve provability . Predicate mappings are abstractions that map predicate symbols from one first order language to those of another defined as follows : ? 1 : A first order calculus , defined as follows : 1 . ? 1 : clause form ; 2 . ? 1 : resolution . ? 2 : A first order calculus , defined as follows : 1 . ? 2 : clause form ; 2 . ? 2 : resolution . The predicate mapping fpred which is applied to the wffs and axioms is generically defined by : a -RRB- fpred -LRB- p -LRB- x ; : : :-RRB- -RRB- = q -LRB- x ; : : :-RRB- for all p 2 Rq , and where SRq is the set of all possible predicate symbols and Rq \ Rs = ; for q < > s ; b -RRB- fpred -LRB- ff _ -RRB- = fpred -LRB- ff -RRB- _ fpred -LRB- -RRB- ; c -RRB- fpred -LRB- ff ^ -RRB- = fpred -LRB- ff -RRB- ^ fpred -LRB- -RRB- ; 2You might also require stronger properties that define a relationship between the abstract and the unabstract proof . These properties will help map an abstract proof back into an unabstract proof . How to map back is very dependent on the particular abstraction and on what information is lost by the abstraction . d -RRB- fpred -LRB- : ff -RRB- = : fpred -LRB- ff -RRB- . An example of such an abstraction for a commonsense reasoning domain is as follows . The domain consists of various types of containers -LRB- represented by bottle -LRB- x -RRB- , glass -LRB- x -RRB- , ... -RRB- that have common properties -LRB- eg . - bottle -LRB- x -RRB- v graspable -LRB- x -RRB- and - glass -LRB- x -RRB- v graspable -LRB- x -RRB- -RRB- . In such a theory , we might reason about the generic properties of containers by using a predicate mapping to collapse bottle -LRB- x -RRB- and box -LRB- x -RRB- onto a single predicate , container -LRB- x -RRB- . Predicate abstractions allow us to carry out such reasoning as they are truthful . Theorem 1 : fpred is truthful irrespective of the axioms , ? 1 of ? 1 . Proof : By showing that given a resolution proof ?? 1 of ` ? 1 ' , we can build a resolution proof of ` ? 2 fpred -LRB- ' -RRB- . The proof proceeds by induction on the depth N of ?? 1 . We just take the proof ?? 1 and apply fpred to every wff in it . 2 Note that this proof demonstrates why theorem proving in the abstract theory , ? 2 will help us to prove theorems in ? 1 . Since there exists an abstract proof , ?? 2 which has an identical shape to the proof ?? 1 , we have a way of mapping a proof from the abstract theory back into the unabstract theory . ? 3 Solutions to the False Proof Problem A major problem with the use of TI-abstractions is that it may happen that , even if ? 1 is consistent ? 2 may be inconsistent . This problem was already noticed by Plaisted who called it the \ false proof problem '' -LSB- Pla80 , Pla81 -RSB- . The consequences of this fact are more limitative than it could be thought at a first sight . In fact the following theorem can be proved 3 : Theorem 2 : If f : ? 1 7 ! ? 2 is a strictly truthful ? / ? - invariant ? - abstraction such that f -LRB- : ff -RRB- = : f -LRB- ff -RRB- , then there exists an ? - restriction of f , f ? , such that ??? 1 1 is consistent and ??? 1 2 inconsistent . Note that the condition \ f -LRB- : ff -RRB- = : f -LRB- ff -RRB- '' is satisfied by all the abstractions proposed in the past with negation belonging to the language of ? 2 . Several solutions have been proposed to the false-proof problem . All place strong restrictions on the types of abstraction and axioms allowed . Worse of all , all the solutions either use abstractions that are not truthful , or require a test that is not decidable4 . Tenenberg -LSB- Ten87 , Ten88 -RSB- presents two solutions to the false proof problem for predicate abstractions . The first restricts the axioms kept in ? 2 , eliminating those that distinguish in ? 1 between the predicates which are conflated together ; such an abstraction is called a restricted predicate abstraction -LSB- Ten87 -RSB- . Unfortunately \ distinguishability '' is an undecidable property and deciding it may require an arbitrary amount of theorem proving in 3This theorem can actually be proved in stronger forms -LSB- GW90 , GW89a -RSB- . 4Since testing consistency is not , in general , decidable this hardly seems a satisfactory solution . ? 1 . Additionally , restricted predicate abstractions are not truthful ; since ? 2 has fewer axioms than ? 1 -LRB- since we do n't map all of them -RRB- , it is easy to see that ? 2 will have fewer theorems than ? 1 , and that we will lose truthfulness . Tenenberg 's second solution -LSB- Ten88 -RSB- overcomes the objection to undecidability . In this proposal , axioms are kept in ? 2 provided they can be trivially shown not to distinguish in ? 1 between predicates which are conflated together ; thus , instead of performing an arbitrary amount of theorem proving in ? 1 to determine \ indistinguishability '' , we insist that it is an immediate consequence of the axioms5 . Whilst being decidable , this abstraction is not truthful . We claim that losing truthfulness is undesirable as completeness is also lost -LCB- not every theorem will have an abstracted proof . Hobbs -LSB- Hob85 -RSB- , and Imielinski -LSB- Imi87 -RSB- also suggest solutions to the false proof problem for a class of abstractions called \ domain abstractions '' , where the objects in the language -LRB- and not the predicate symbols -RRB- are conflated together . Hobbs ' proposal , however , requires an arbitrary theorem proving in the original theory , ? 1 , whilst Imielinski 's loses truthfulness . We propose a decidable solution which retains truthfulness and applies to most types of abstractions . Truthful abstractions applied to the same ? 1 can be classified into a weak partial order , which we indicate by \ v '' . Definition 4 -LRB- v -RRB- : If f : ? 1 7 ! ? 2 and g : ? 1 7 ! ? 3 are two truthful abstractions then f v g iff for all wffs ' , if ` ? 2 f -LRB- ' -RRB- then ` ? 3 g -LRB- ' -RRB- . We say that f is weaker than g or , dually , that g is stronger than f . g is stronger than f in the sense that there are fewer wffs which are theorems in ? 2 and not in ? 1 than wffs which are theorems in ? 3 and not in ? 1 . \ v '' is in general a weak partial order -LRB- respecting transitivity , antisymmetry and reflexivity -RRB- but not a total order . If , however , we have a set of totally ordered abstractions then the following result holds -LSB- GW90 -RSB- : Theorem 3 : If f1 : ? 1 7 ! ? 12 , ... , fn : ? 1 7 ! ? n2 are truthful abstractions and f1 v : : : v fn -LRB- f1 ; : : : ; fn are totally ordered -RRB- , then if ? j2 is consistent so is ? i2 for any 1 < = i < = j. If , however , ? k2 is inconsistent then so is ? i2 for any k > = i > = n . This suggests the following program for constructing an ordered set of truthful abstractions that preserve consistency : consistent-abs -LRB- Sigma , SetOfAbs -RRB- : - generate -LRB- Sigma , SetOfAbs -RRB- , strongest -LRB- SetOfAbs , F -RRB- , abstract -LRB- Sigma , F , AbsSigma -RRB- , consistent -LRB- AbsSigma -RRB- . 5That is , derivable applying only one rule of inference . Where : generate -LRB- Sigma , SetOfAbs -RRB- generates a totally ordered set of abstractions , SetOfAbs of the formal system , Sigma ; the strongest abstraction in SetOfAbs should map Sigma into a theory in which consistency is decidable -LRB- eg . propositional logic -RRB- ; strongest -LRB- SetOfAbs , F -RRB- returns the strongest abstraction , F in the ordered set of abstractions , SetOfAbs ; abstract -LRB- Sigma , F , AbsSigma -RRB- maps the formal system , Sigma onto the abstracted formal system , AbsSigma according to the abstraction mapping , F ; consistent -LRB- Sigma -RRB- tests the consistency of the formal system Sigma . Ideally , we want generate/2 to construct the ordered sets of abstractions in an \ intelligent '' way . For example , by identifying the cause of inconsistency in the abstracted theory , we might be able to suggest ways of subtly altering which wffs to collapse together . 4 Implementation We have implemented our solution in PROLOG . To test consistency of the abstracted theory , we have constructed an interface to Otter -LSB- McC88 -RSB- , a fast , \ state-of-the-art '' resolution theorem prover . Since predicate abstractions map first order theories onto first order theories , in general , they are not strong enough to give decidable abstract theories . We therefore compose -LRB- in the obvious componentwise way -RRB- the predicate abstraction , fpred with a propositional abstraction , fprop which maps all predicates with the same predicate name onto a common propositional symbol . For example , p -LRB- x -RRB- ; p -LRB- a -RRB- and p -LRB- b -RRB- all map onto p. fprop can be proved to be truthful -LSB- GW89b -RSB- . By composing fprop with fpred to give fprop ffi fpred -LRB- where ffi denotes abstraction composition -RRB- , we construct an abstraction that maps a first order theory onto a decidable propositional theory . Additionally , since the composition of two truthful abstractions is itself a truthful abstraction , fprop ffi fpred is truthful . Theorem 4 If f : ? 1 7 ! ? 2 and g : ? 2 7 ! ? 3 are both truthful abstractions then g ffi f : ? 1 7 ! ? 3 is also a truthful abstraction . Proof : If ` ? 1 ' then , as f is truthful , ` ? 2 f -LRB- ' -RRB- . But as g is truthful , if ` ? 2 then ` ? 3 g -LRB- -RRB- . Hence ` ? 3 g -LRB- f -LRB- ' -RRB- -RRB- . That is ` ? 3 g ffi f -LRB- ' -RRB- . 2 Composition is a natural way to build ordered sets of abstractions since , if f and g are two truthful abstractions , f v g ffi f . The remaining problem is to generate predicate abstractions automatically . Our false proof algorithm merely decides which predicate symbols must not be collapsed together ; knowing which predicate symbols are worth collapsing together requires lots of domain and problem dependent knowledge . Representing and reasoning with this knowledge is beyond the scope of this paper . We therefore merely present a method for generating all possible predicate abstractions . Of course , generating all the possible predicate abstractions is not a winning strategy since we have an exponential worst case as it will be shown in the next section . We represent a predicate abstraction as a list of sets of predicate symbols ; all predicate symbols in a set are mapped onto the same symbol in the abstracted theory . Our algorithm for generating predicate abstractions is : generate -LRB- Sigma , Abs -RRB- : - language -LRB- Sigma , Language -RRB- , predicates -LRB- Language , Preds -RRB- , generate -LRB- N -RRB- , divide-into -LRB- N , Preds , Abs -RRB- . generate -LRB- 1 -RRB- . generate -LRB- N -RRB- : - generate -LRB- M -RRB- , N is M +1 . Where : language -LRB- Sigma , Language -RRB- returns the language , Language of the formal system , Sigma ; predicates -LRB- Language , Preds -RRB- returns the set of -LRB- unary -RRB- predicate symbols , Preds found in the language , Language ; divide-into -LRB- N , Set , Subsets -RRB- divides the set , Set into a list of N subsets , Subsets . On backtrack , the set is divided in all possible ways . Note that generate/2 is restricted to conflating unary predicate symbols together ; it would not be difficult to generalise generate/2 to predicates of all arities . It is often worth abstracting the unary predicates of a theory as they can express sort information -LRB- eg . box -LRB- a -RRB- , container -LRB- b -RRB- , ... -RRB- and object properties -LRB- eg . movable -LRB- c -RRB- , breakable -LRB- d -RRB- , ... -RRB- ; the existence of sort hierarchies and the interdependence of object properties allow for these unary predicates to be collapsed together in meaningful ways . generate/2 will span the complete set of -LRB- unary -RRB- predicate abstractions , starting with the -LRB- usually uninformative -RRB- predicate abstraction that maps all unary predicate symbols onto a single predicate symbol , then moving onto all predicate abstractions that map onto two predicate symbols , three predicate symbols , and so on . Fact 1 : generate/2 constructs all possible unary predicate abstractions . An iterative deepening search is used to explore the space of all possible -LRB- unary -RRB- predicate abstractions ; this guarantees that predicate abstractions that preserve consistency are found in order of their strength ; that is , those abstractions that map the language into the fewest predicate symbols are found first . Our goal is to find the strongest abstraction which preserves consistency . Unfortunately , the number of different -LRB- unary -RRB- predicate abstractions of a theory can be quite large . Theorem 5 If the language , ? , of a formal system , ? has n unary predicate symbols , then the number of different -LRB- unary -RRB- predicate abstractions , P -LRB- n -RRB- of ? is given by : P -LRB- n -RRB- = nX i = 1 D -LRB- n ; i -RRB- Where , D -LRB- n ; i -RRB- = i ? D -LRB- n ? 1 ; i -RRB- + D -LRB- n ? 1 ; i ? 1 -RRB- D -LRB- n ; n -RRB- = 1 And , D -LRB- m ; 1 -RRB- = 1 Proof : The problem is equivalent to finding the number of ways of dividing a set , A of n elements into subsets . This is just the sum of ways of dividing the set into i subsets for i = 1 to n. To divide a set of n elements into i subsets , we take the first element of the set and put it in a subset of its own -LRB- leaving D -LRB- n ? 1 ; i ? 1 -RRB- ways of dividing the remaining n ? 1 elements into i ? 1 subsets -RRB- . Or we take the first element , divide the remaining n ? 1 elements into i subsets -LRB- in D -LRB- n ? 1 ; i -RRB- ways -RRB- , and add the first element to one of these i subsets -LRB- giving i ? D -LRB- n ? 1 ; i -RRB- possible combinations -RRB- . D -LRB- n ; i -RRB- is also known as the second number of Stirling -LSB- Mol89 -RSB- . 2 The following table demonstrates how quickly P -LRB- n -RRB- grows . n P -LRB- n -RRB- 1 1 2 2 3 5 4 15 5 52 6 203 7 877 8 4140 9 21147 10 1.16 e05 The table above suggests the need of guiding , in a problem dependent way , the process of generating abstractions . Moreover , just because one of the many possible predicate abstractions maps into a consistent abstracted theory does not make it useful . A good choice of abstraction will not only guarantee consistency of the abstract theory , but will also consider lots of domain dependent information about which predicate symbols are worth collapsing together . Conflating together unrelated symbols will give abstracted proofs which save us nothing in theorem proving in the original theory . 5 Results We illustrate the results of our implementation with two problems ; the first is taken from -LSB- Ten87 -RSB- , whilst the second is a classic challenge problem from the theorem proving literature , Schubert 's steamroller . Tenenberg demonstrates his solution to the false proof problem with the following example theory : - bottle -LRB- x -RRB- v madeofglass -LRB- x -RRB- - bottle -LRB- x -RRB- v graspable -LRB- x -RRB- - glass -LRB- x -RRB- v madeofglass -LRB- x -RRB- - glass -LRB- x -RRB- v graspable -LRB- x -RRB- - glass -LRB- x -RRB- v open -LRB- x -RRB- - box -LRB- x -RRB- v graspable -LRB- x -RRB- - bottle -LRB- x -RRB- v - glass -LRB- x -RRB- - bottle -LRB- x -RRB- v - box -LRB- x -RRB- - box -LRB- x -RRB- v - glass -LRB- x -RRB- bottle -LRB- x -RRB- v glass -LRB- x -RRB- v box -LRB- x -RRB- - graspable -LRB- x -RRB- v movable -LRB- x -RRB- - madeofglass -LRB- x -RRB- v breakable -LRB- x -RRB- - bottle -LRB- x -RRB- v milkbottle -LRB- x -RRB- v winebottle -LRB- x -RRB- - open -LRB- x -RRB- v - graspable -LRB- x -RRB- v pourable -LRB- x -RRB- open -LRB- a -RRB- We have not bothered to list 3 \ axioms '' from -LSB- Ten87 -RSB- as they are provable from the other axioms . The false proof algorithm generates all possible predicate abstractions of this theory that preserve consistency . We can make several general observations about the results . The main problem is box , which has to be mapped with care as it has different properties than bottles or glasses . For example , one of the strongest predicate abstractions found that preserved consistency maps all the different types of bottles and glasses together , maps all the properties -LRB- like pourable etc -RRB- together but keeps box distinct from everything else . Not surprisingly bottle , winebottle and milkbottle can be safely mapped together . Deciding which of the consistency preserving abstractions will be useful is connected to the issue of mapping back -LRB- see conclusions -RRB- . The second problem we consider is Schubert 's steamroller , well known for the large search space it generates . The axioms , of which all except one can be represented as Horn clauses , are : wolf -LRB- wolfy -RRB- - wolf -LRB- X -RRB- v animal -LRB- X -RRB- fox -LRB- foxy -RRB- - fox -LRB- X -RRB- v animal -LRB- X -RRB- bird -LRB- tweety -RRB- - bird -LRB- X -RRB- v animal -LRB- X -RRB- caterpillar -LRB- maggie -RRB- - caterpillar -LRB- X -RRB- v animal -LRB- X -RRB- snail -LRB- slimey -RRB- - snail -LRB- X -RRB- v animal -LRB- X -RRB- grain -LRB- wheaty -RRB- - grain -LRB- X -RRB- v plant -LRB- X -RRB- - caterpillar -LRB- X -RRB- v - bird -LRB- Y -RRB- v X < Y - snail -LRB- X -RRB- v - bird -LRB- Y -RRB- v X < Y - bird -LRB- X -RRB- v - fox -LRB- X -RRB- v X < Y - fox -LRB- X -RRB- v - wolf -LRB- Y -RRB- v X < Y - bird -LRB- X -RRB- v - caterpillar -LRB- Y -RRB- v eats -LRB- X , Y -RRB- - caterpillar -LRB- X -RRB- v -LRB- plant -LRB- p1 -LRB- X -RRB- & eats -LRB- X , p1 -LRB- X -RRB- -RRB- -RRB- - snail -LRB- X -RRB- v -LRB- plant -LRB- p2 -LRB- X -RRB- & eats -LRB- X , p2 -LRB- X -RRB- -RRB- -RRB- - wolf -LRB- X -RRB- v - fox -LRB- Y -RRB- v - eats -LRB- X , Y -RRB- - wolf -LRB- X -RRB- v - grain -LRB- Y -RRB- v - eats -LRB- X , Y -RRB- - bird -LRB- X -RRB- v - snail -LRB- Y -RRB- v - eats -LRB- X , Y -RRB- - animal -LRB- X -RRB- v - animal -LRB- Y -RRB- v - animal -LRB- Z -RRB- v - Z < X v - plant -LRB- U -RRB- v - eats -LRB- Z , U -RRB- v eats -LRB- X , Y -RRB- v eats -LRB- X , Z -RRB- When applied to this axiom set , our false proof algorithm suggests that NO possible predicate abstraction can be safely used . Of course , the algorithm merely proves that fprop ffi fpred maps onto an inconsistent theory -LRB- which is decidable -RRB- ; it does not tell us whether it is fpred that is responsible for introducing the inconsistency -LRB- as this is not decidable -RRB- , or whether it is the interaction of fprop and fpred that is to blame . Indeed , it is easy to see that one simple predicate abstraction , the one that maps plant and grain onto to the same predicate symbol , preserves consistency . This result suggests one reason why Schubert 's steamroller is so difficult to prove and so hard on unification . That is , there is little redundancy in the terms and you can not totally forget about them and still maintain consistency6 . 6This consideration must be taken with further care since , when building abstraction , we have not 6 Final remarks and conclusions In this paper we have proposed strictly truthful abstractions as the right ones for using abstraction - based planning techniques in the implementation of inference engines . We have proved that the false proof problem can not be escaped , namely it is always true that the abstract space can be inconsistent even if the ground space is consistent . We have then proposed an algorithm which solves the problem and given some examples with results . The algorithm proposed is part of a general purpose \ abstract theorem prover '' under development . We have not faced the problem of \ mapping back '' , namely the problem of how to use the proof of the abstract theorem in order to help prove the goal in the ground space . A possible way of doing this is to use the shape of the proof in the abstract space , unabstracting all the theorems in it and then \ bridging '' the holes by theorem proving -LRB- even if a wff can be derived from another in the abstract space , this may not be the case in the ground space -RRB- . This and more complicated techniques are under evaluation . Acknowledgements The research described in this paper owes a lot to the openess and sharing of ideas which exists in the Mathematical Reasoning group in Edinburgh and Mechanized Reasoning Group in Trento . The authors thank Alan Bundy , Jane Hesketh , Luciano Serafini , Alex Simpson and Paolo Traverso for the many discussions on the topic . References -LSB- GW89a -RSB- F. Giunchiglia and T. Walsh . Abstract Theorem Proving . In Proceedings of the 11th IJCAI . International Joint Conference on Artificial Intelligence , 1989 . Also available as DAI Research Paper No 430 , Dept. of Artificial Intelligence , Edinburgh . -LSB- GW89b -RSB- F. Giunchiglia and T. Walsh . Theorem Proving with Definitions . In Proceedings of AISB 89 . Society for the Study of Artificial Intelligence and Simulation of Behaviour , 1989 . Also available as DAI Research Paper No 429 , Dept. of Artificial Intelligence , Edinburgh . -LSB- GW90 -RSB- F. Giunchiglia and T. Walsh . A Theory of Abstraction . Research paper 516 , Dept. of Artificial Intelligence , University of Edinburgh , 1990 . Accepted to Journal of Artificial Intelligence . -LSB- Hob85 -RSB- J.R. Hobbs . Granularity . In Proceedings of the 9th IJCAI , pages 432 -LCB- 435 . International Joint Conference on Artificial Intelligence , 1985 . considered binary predicates . -LSB- Imi87 -RSB- T. Imielinski . Domain abstraction and limited reasoning . In Proceedings of the 10th IJCAI , pages 997 -LCB- 1003 . International Joint Conference on Artificial Intelligence , 1987 . -LSB- McC88 -RSB- W. W. McCune . Otter users ' guide 0.91 , 1988 . Maths and CS . Division , Argonne National Laboratory , Argonne , Illinois . -LSB- Mol89 -RSB- M. Mollard , 1989 . Private email communication . Laboratoire Structures Discretes , Institut IMAG , Grenoble , France . -LSB- Pla80 -RSB- D.A. Plaisted . Abstraction mappings in mechanical theorem proving . In 5th Conference on Automated Deduction , pages 264 -LCB- 280 . 5th Conference on Automated Deduction , 1980 . -LSB- Pla81 -RSB- D.A. Plaisted . Theorem proving with abstraction . Artificial Intelligence , 16:47 -LCB- 108 , 1981 . -LSB- Pla86 -RSB- D.A. Plaisted . Abstraction using generalization functions . In 8th Conference on Automated Deduction , pages 365 -LCB- 376 . 8th Conference on Automated Deduction , 1986 . -LSB- Ten87 -RSB- J.D. Tenenberg . Preserving Consistency across Abstraction Mappings . In Proceedings of the 10th IJCAI , pages 1011 -LCB- 1014 . International Joint Conference on Artificial Intelligence , 1987 . -LSB- Ten88 -RSB- J.D. Tenenberg . Abstraction in Planning . PhD thesis , Computer Science Department , University of Rochster , 1988 . Also TR 250 .

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O B O O O O O O O O O O O O O O O O O O O O B O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O