Mark E. Stickel. Schubert's steamroller problem: Formulations and solutions. Journal of Automated Reasoning, 2:89--101, 1986.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....functions and predicates we can perform complex deductions on sentences with plurals and numbers. Being based on the powerful theorem prover Otter and the efficient model generator Satchmo, RACE is very fast and solves all examples of this paper in a few milliseconds. Schubert s Steamroller [Stickel 1986] a standard theorem proving example takes 70 milliseconds on a 500 MHz Macintosh running Mac OS X. As a first check of RACE s scaling up capability we contrived a contradictory example consisting of 10 complex ACE sentences and altogether 56 auxiliary first order axioms. RACE found all 5 ....

....(short PRQ formulas) that has the expressive power of full first order logic. Torge implemented EP Tableaux as the Prolog program Finfimo. In an earlier publication [Fuchs et al. 2000] we used ACE as a front end to EP Tableaux and successfully solved a data base example and Schubert s steamroller [Stickel 1986]. However, as we found out, EP Tableaux as implemented in Finfimo does not scale up. Here is why. EP Tableaux processes existentially quantified variables at run time by the expansion rule xE(x) E(x c 1 ) E(x c k ....

M. E. Stickel, Schubert's Steamroller Problem: Formulations and Solutions, Journal of Automated Reasoning, vol. 2, pp. 89-101, 1986

....We see an example of its use in section 5.2. A formula not containing a lambda expression is called quantifier free formula. http: cogsci.ed. ac.uk fracas 3 Two applications Schubert s Steamroller was presented in 1978 by Lenhart Schubert as a challenge to automated deduction systems [7]. Its statement in CPE is as follows: Every wolf is an animal. Every fox is an animal. Every bird is an animal. Every caterpillar is an animal. Every snail is an animal. Some wolf exists. Some fox exists. Some bird exists. Some caterpillar exists. Some snail exists. Every grain is a plant. ....

Mark E. Stickel. Schubert's steamroller problem: Formulations and solutions. Journal of Automated lrieasoning, 2:89 101, 1986. 15

....plants. Every animal likes to eat all plants or all animals much smaller than itself that like to eat some plants. Birds do not like to eat snails. Snails are much smaller than birds and like to eat some plants . This is part of the well known, in theorem proving, Schubert s steamroller problem [26]) The object hierarchy of the domain is the following. animal plant bird snail b1 c1 p1 where b1, c1 , and p1 represent arbitrary instances of bird, snail, and plant respectively. The object internal knowledge is as follows (presented in non clausal form, for the sake of readability) V ....

M. E. Stickel, Schubert's Steamroller Problem: Formulations and Solutions, Journal of Automated Reasoning 2 (1986) 89-101.

....to retain fairness. As the simp c2u rule without renaming obviously enjoys the niteness property, one might alternatively interleave renaming and instantiation, but that would amount to ignoring universality for most of the time. 6 There are problems, like e.g. Schubert s Steamroller [13], or the Natural Language problems submitted to this FTP workshop, 5 in which simpli cation with universal variables actually always terminates. To handle these cases eciently, it is advisable to equip a proof procedure with some sort of cycle detection, that only interleaves simpli er ....

Mark E. Stickel. Schubert's steamroller problem: Formulations and solutions. Journal of Automated Reasoning, 2:89-101, 1986. 8

....of the database example can indeed be replaced by a more natural formulation without losing precision. 6 Schubert s Steamroller in ACE As a further test for the productive interplay between ACE and EP Tableaux we chose Schubert s Steamroller a well known problem for automated reasoning systems [15]. The problem is interesting for our approach as it raises the issues of how to express problems unambiguously and of which theorem proving technique is best applied to prove the desired facts. Each sentence of the original natural language version of the Steamroller problem is given below along ....

.... = false) the EP Tableaux method proves as desired the unsatis ability of the formulae (1) 6) and (8) By rephrasing the original natural language version of the Steamroller problem unambiguously in ACE and then successfully proving its conclusions with EP Tableaux we challenge Stickel s [15] warning of the danger of using natural language to try to convey problem statements unambiguously . 7 Conclusion and Outlook We have shown that Attempto Controlled English (ACE) can serve as a natural language front end to the model generator EP Tableaux. Domain specialists who A Natural ....

M. E. Stickel. Schubert's steamroller problem: Formulations and solutions. Journal of Automated Reasoning, 2, 1986.

....have shown that Attempto Controlled English (ACE) can replace first order logic as input language to the modelgenerator EP Tableaux. As a further test for the productive interplay between ACE and EP Tableaux we chose Schubert s Steamroller a well known problem for automated reasoning systems [7]. We rephrased the original natural language version unambiguously in ACE and then successfully proved the conclusions with EP Tableaux thus challenging Stickel s [7] warning of the danger of using natural language to try to convey problem statements unambiguously . Both examples support our ....

.... interplay between ACE and EP Tableaux we chose Schubert s Steamroller a well known problem for automated reasoning systems [7] We rephrased the original natural language version unambiguously in ACE and then successfully proved the conclusions with EP Tableaux thus challenging Stickel s [7] warning of the danger of using natural language to try to convey problem statements unambiguously . Both examples support our claim that domain specialists who may not be familiar with formal methods can formulate their problems in controlled natural language, and can then verify and validate ....

M. E. Stickel. Schubert's steamroller problem: Formulations and solutions. Journal of Automated Reasoning, 2, 1986.

....imprecise and unclear statements mixing relevant and irrelevant information, di erent levels of abstraction, and di erent degrees of granularity. This has been clearly exhibited when problem descriptions in natural language e.g. the Library Data Base Problem [14] or Schubert s Steamroller [6] were manually translated into their formal equivalents. Di erent researchers achieved widely di ering results depending on their understanding, ad hoc interpretation and shallow parsing of the natural language texts. To minimize the semantic distance between problem domains and formal speci ....

....section 4. Thus, we have shown that the dicult and unfamiliar formal statement of the database example can indeed be replaced by a more natural formulation without losing precision. 11 8 Steamroller Example We chose Schubert s Steamroller a well known problem for automated reasoning systems [6, 9] as a further test for the amalgamation of Attempto with EP Tableaux. The problem is interesting for our approach in that on the one hand it raises issues of how to express problems unambiguously and on the other hand of which theorem proving technique is best applied. We formulated the problem ....

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Stickel M. E. Schubert's steamroller problem: Formulations and solutions. Jour. of Automated Reasoning, 2, 1986.

....to retain fairness. As the simp c2u rule without renaming obviously enjoys the finiteness property, one might alternatively interleave renaming and # instantiation, but that would amount to ignoring universality for most of the time. There are problems, like e.g. Schubert s Steamroller [14], or the Natural Language problems submitted to this FTP workshop, 7 in which simplification with universal variables actually always terminates. To handle these cases e#ciently, it is advisable to equip a proof procedure with some sort of cycle detection, that only interleaves simplifier ....

Mark E. Stickel. Schubert's steamroller problem: Formulations and solutions. Journal of Automated Reasoning, 2:89--101, 1986.

....FC clause becomes relevant only after p is asserted. r. g p q q m : p, q. m. f Next, we will focus on the pruning power of SATCHMORE s total relevancy requirement. First, we will consider Schubert s Steamroller, a problem taken from the theorem proving literature (see [Sti86] for a discussion of this problem) As we shall see, SATCHMO performs effectively on this problem, but fails miserably on a slight modification unless relevancy testing is incorporated. 19 Example 18 Consider the following set of clauses (Schubert s Steamroller) wolf(X) fox(Y) ....

M.E. Stickel. Schubert's steamroller problem: Formulations and solutions. J. of Automated Reasoning, 2:89--101, 1986.

....imprecise and unclear statements mixing relevant and irrelevant information, different levels of abstraction, and different degrees of granularity. This has been clearly exhibited when problem descriptions in natural language e.g. the Library Data Base Problem [14] or Schubert s Steamroller [6] were manually translated into their formal equivalents. Different researchers achieved widely differing results depending on their understanding, ad hoc interpretation and shallow parsing of the natural language texts. To minimize the semantic distance between problem domains and formal ....

....4. Thus, we have shown that the difficult and unfamiliar formal statement of the database example can indeed be replaced by a more natural formulation without losing precision. 11 8 Steamroller Example We chose Schubert s Steamroller a well known problem for automated reasoning systems [6, 9] as a further test for the amalgamation of Attempto with EP Tableaux. The problem is interesting for our approach in that on the one hand it raises issues of how to express problems unambiguously and on the other hand of which theorem proving technique is best applied. We formulated the ....

[Article contains additional citation context not shown here]

Stickel M. E. Schubert's steamroller problem: Formulations and solutions. Jour. of Automated Reasoning, 2, 1986.

....is the translation of an automatically generated ALC [12] database to first order logic. ALC is a decidable concept language used for knowledge representation. The next two examples are formulations of two well known puzzles: the Lion and the Unicorn puzzle [6] and Schubert s Steamroller [13]. The Steamroller problem is extended such that its sort theory gets more complicated: We added a father function for all animals and referred to it in the main clause of the problem. Both examples as well as the following examples are contained in the SPASS distribution (see Section 4) The ....

Mark Stickel. Schubert's steamroller problem: Formulations and solutions. Journal of Automated Reasoning, 2(1):89--101, 1986.

....information in a wide range of domains. A number of demonstrations have shown that deduction systems based on sorted logic can be much more efficient than their counterparts based on unsorted logic [Cohn, 1985; Ohlbach and Schmidt Schau , 1985; Walther, 1985; Frisch, 1986b; Walther, 1987; Stickel, 1986; Cohn, 1989] Though evidence has mounted, little has been done to examine closely how and why this improvement occurs nor have detailed measurements been made of how much speedup is obtained. This paper addresses these issues by comparing and analyzing the performance of a sorted deduction ....

....in an efficiently accessible data structure. 5 A number of studies have reported that a deduction system based on such a unification procedure can be more efficient than the corresponding unsorted system [Cohn, 1985; Ohlbach and Schmidt Schau , 1985; Walther, 1985; Frisch, 1986b; Walther, 1987; Stickel, 1986; Cohn, 1989] This should come as no surprise; after all, what we have essentially done is to relocate part of the deduction problem into the unification procedure and solve that part with a special purpose method table lookup rather than with a general purpose one. We claim that the use of ....

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Mark E. Stickel. Schubert's Steamroller problem: Formulations and solutions. Journal of Automated Reasoning, 2(1):89--101, 1986.

....indication of the difficulty or significance of the problems. The significance and difficulty of a problem, with respect to the current state ofthe art in ATP systems, is hard to assess by newcomers to the field. Existing test problems are often not adequate anymore (e.g. Schubert s Steamroller [Sti86] while others may be solvable only with specialized techniques (e.g. LIM [Ble90] and therefore are much too hard to start with. ffl are inconsistent in their presentation of equally named problems. Many copies and variants of the same original problem may exist in different libraries. ....

M.E. Stickel. Schubert's Steamroller Problem: Formulations and Solutions. Journal of Automated Reasoning, 2(1):89--101, 1986.

....and tuned specifically for a particular ATP system. Fourthly, the significance and difficulty of a problem, with respect to the current state of the art in ATP systems, is hard to assess by newcomers to the field. Existing test problems are often not adequate anymore (e.g. Schubert s Steamroller [Sti86] while others may be solvable only with specialized techniques (e.g. LIM [Ble90] and therefore are much too hard to start with. Finally, many copies and variants of the same original problem may exist in different libraries. Coupled with a lack of documentation, this means that ....

M.E. Stickel. Schubert's Steamroller Problem: Formulations and Solutions. Journal of Automated Reasoning, 2:89 -- 101, 1986.

....indication of the significance or difficulty of the problems. The significance and difficulty of a problem, with respect to the current state ofthe art in ATP systems, is hard to assess by newcomers to the field. Existing test problems are often not adequate anymore (e.g. Schubert s Steamroller [Sti86] while others may be solvable only with specialized techniques (e.g. LIM [Ble90] and therefore are much too hard to start with. ffl are inconsistent in their presentation of equally named problems. Many copies and variants of the same original problem may exist in different collections. ....

M.E. Stickel. Schubert's Steamroller Problem: Formulations and Solutions. Journal of Automated Reasoning, 2(1):89--101, 1986.

....but not snails. Caterpillars and snails like to eat some plants. Therefore, there is an animal that likes to eat a grain eating animal. 10 This problem has been attempted by a wide variety of theorem provers, and the fastest time reported is 6 seconds. That time is reported by Stickel [23], running a connection graph resolution theorem proving program on a Symbolics 3600 and applying it to the result of already transforming the problem into clausal form. In comparison, OSCAR does this problem in 13 seconds on a Symbolics 3600. OSCAR was not constructed with speed in mind, and is ....

....important difference is that it has nothing to do with Horn clauses. The disjuncts stored in the members of all schemes can be of arbitrary logical complexity. Furthermore, on the steamroller problem, OSCAR outperforms resolution based theorem provers using unit resolution. For instance, Stickel [23] lists twelve variations of his theorem prover, all giving strong preference to unit resolution, and OSCAR beats all but the six second version. It is a mistake to put too much emphasis on all detachment. It is just one part of the theorem prover, albeit an important part. The rules for ....

Stickel, M. E., Schubert's steamroller problem: formulations and solutions. Journal of Automated Reasoning 2 (1986) 89-101.

....indication of the difficulty or significance of the problems. The significance and difficulty of a problem, with respect to the current state of the art in ATP systems, is hard to assess by newcomers to the field. Existing test problems are often not adequate anymore (e.g. Schubert s Steamroller [106]) while others may be solvable only with specialized techniques (e.g. LIM [13] and therefore are much too hard to start with. ffl are inconsistent in their presentation of equally named problems. Many copies and variants of the same original problem may exist in different collections. This ....

M.E. Stickel. Schubert's Steamroller Problem: Formulations and Solutions. Journal of Automated Reasoning, 2(1):89--101, 1986. THE TPTP PROBLEM LIBRARY 31

....indication of the significance or difficulty of the problems. The significance and difficulty of a problem, with respect to the current state ofthe art in ATP systems, is hard to assess by newcomers to the field. Existing test problems are often not adequate anymore (e.g. Schubert s Steamroller [Sti86] while others may be solvable only with specialized techniques (e.g. LIM [Ble90] and therefore are much too hard to start with. ffl are inconsistent in their presentation of equally named problems. Many copies and variants of the same original problem may exist in different collections. ....

M.E. Stickel. Schubert's Steamroller Problem: Formulations and Solutions. Journal of Automated Reasoning, 2(1):89--101, 1986.

....a unique sort, in an order sorted logic a term can belong to a number of different sorts. For example, Figure 1. 1 shows a simple (tree structured) sort hierarchy similar to that used for solving the Schubert steamroller deduction problem using an order sorted logic [ Cohn, 1985; Walther, 1985a; Stickel, 1986 ] Under this hierarchy, a variable of sort animal can be unified with any term of sort bird, fox or snail, but not with a term of sort plant. Animal Bird Snail Plant Grain Fox Top Figure 1.1: A simple sort hierarchy Order sorted logics can be roughly classified as monomorphic or polymorphic, ....

....Frisch, 1986b ] and natural language processing [ Veronis, 1987 ] among other areas. Sorts have been shown to improve the efficiency of deductive systems (given the appropriate formulation of the problem at hand) in [ Cohn, 1985 ] Walther, 1985a ] Ohlbach and Schmidt Schau , 1985 ] Stickel, 1986 ] Cohn, 1989a ] and [ Abrams and Frisch, 1989 ] Walther [ 1985b ] examines sorted unification in the monomorphic case, allowing an infinite number of sorts, and shows how the existence and cardinality of a set of most general unifiers depend on properties of the lattice of sorts. ....

Mark E. Stickel. Schubert's Steamroller problem: Formulations and solutions. Journal of Automated Reasoning, 2(1):89--101, 1986.

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Mark E. Stickel. Schubert's steamroller problem: Formulations and solutions. Journal of Automated Reasoning, 2:89--101, 1986.

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Stickel M. E.: Schubert's Steamroller Problem: Formulations and Solutions. Journal of Automated Reasoning. 2 (1986) 89--101

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Mark E. Stickel. Schubert's steamroller problem: Formulations and solutions. Journal of Automated Reasoning, 2:89--101, 1986.

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