J. Zhang and H. Zhang. SEM: a System for Enumerating Models. In Proceedings of the 14th International Joint Conference on Arti cial Intelligence (IJCAI), pages 298-303. Morgan Kaufmann, USA, 1995.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....= x Every Boolean identity of length less than 15, except those excluded by Lemma 1 or 2, is an instance of one of these eight BIs. However, none of these can be a single axiom because each is true in the following non Boolean structure (found by the model searching programs Sem [20] and Mace[8] 0 0 2 0 2 1 0 2 0 2 2 1 3 1 3 3 1 3 1 3 (M 1 ) This completes the proof of Theorem 3, and with it the demonstration that there is no single equational axiom for the She#er stroke shorter than (Sh 1 ) 5. An Exhaustive List of Possible 15 Symbol Single Axioms Using our programs ....

....Number Filename 9 11 13 15 712 She#er mgi 9 11 13 15 356 She#er mgi without mirrors Also, the list of interpretations 1 M 10 is available on line in the file She#er interpretations. 6. Automated Deduction Methods We used special purpose programs to generate candidate axioms, Sem [20] and Mace [8] to search for non Boolean structures that satisfy candidates, Otter to search for proofs that candidates are axioms and to shorten proofs, and Ivy [9] to check Otter s proofs. Disjunction and Negation. The basic approach to searching for a disjunction negation single axiom was to ....

Zhang, J. and H. Zhang: 1995, `SEM: A System for Enumerating Models'. In: Proc. IJCAI-95, Vol. 1. pp. 298--303.

....in a power associative CC loop, we show that all cubes are WIP elements (see Section 8) so that the subloop generated by a square and a cube always is a group. Our investigations were aided by the computer programs OTTER, developed by McCune [14] and SEM, developed by J. Zhang and H. Zhang [19]. 2 Background Let Q be a loop. We shall reformulate the notion of CC loop in terms of autotopisms, the de nition of which we now recall. De nition 2.1 Let Sym(Q) denote the group of all permutations of the set Q, and let I denote the identity element of Sym(Q) A triple ( 2 ....

.... 11 11 8 9 10 14 15 12 13 3 0 1 2 6 7 4 5 12 12 13 14 15 11 8 9 10 6 7 4 5 3 0 1 2 13 13 14 15 12 8 9 10 11 7 4 5 6 0 1 2 3 14 14 15 12 13 9 10 11 8 4 5 6 7 1 2 3 0 15 15 12 13 14 10 11 8 9 5 6 7 4 2 3 0 1 Table 2: A Power Associative WIP CC Loop These examples were found by the program SEM [19]. As usual, once one is given such an example, it is easy to write a very short program (in, e.g. C or java or python) to verify the claimed properties for it. ....

J. Zhang and H. Zhang, SEM: a system for enumerating models, Proc. 14th Int.. Joint Conf. on AI (IJCAI-95), Montreal, 1995, pp. 298 - 303; available at URL: http://www.cs.uiowa.edu/~hzhang/

....been extended by J. Belinfante [1999a, 1999b] and the Mizar group (see Section 10) is also active in the study of set theory. Strikingly different from the preceding problem is that concerning the automated generation of models and counterexamples. Two programs, MACE [McCune, 2001] and SEM [Zhang and Zhang, 1995], have proved useful in finding appropriate counterexamples when a researcher suspects or conjectures that a purported theorem is in fact not provable. Recently, for example, the programs were used to answer several questions concering ortholattices and orthomodular lattices [McCune, 1998; Rose ....

J. Zhang and H. Zhang. SEM: A system for enumerating models. In Proceedings of the international joint conference on artificial intelligence, pages 298--303. Morgan Kaufmann, San Francisco, 1995. 35

.... for a proof, or in parallel with a classical theorem prover) Among these approaches, we can mention the work by Caferra et al. that de ned constrainedbased calculi for simultaneous search for refutations and models [5] the (very successful) works on ecient nite enumeration based model building [24, 26] or Tiomkin s calculus for proving unprovability [25] Use existing proof procedures, with strategies allowing to detect saturation, and extract a model from the formula, in the case in which the procedure terminates without proving unsatis ability. The extraction of the model relies on the ....

J. Zhang and H. Zhang. SEM: a system for enumerating models. In Proc. IJCAI-95, volume 1, pages 298-303. Morgan Kaufmann, 1995. 169

....where the constraints are highly symmetrical. Symmetries arise because all constraints are universally quanti ed. Known approaches to nite model search have explored ways to tackle those symmetries. MGTP G [3] uses ad hoc axioms to lter out some symmetries statically. FALCON [10] and SEM [11] use a dynamic cut and heuristic procedure (LNH: Least Number Heuristic) to avoid exploring symmetrical subspaces during search. On the other hand, many constraint programs of practical or industrial interest exhibit a subproblem having functional semantics. Our approach generalizes the LNH ....

....structures satisfying axioms in L. Hence L is at the same time an excellent experimentation basis and a eld of application. The concepts introduced in this paper can be extended to richer languages like the many sorted rst order language used as input to the rst order nite model generator SEM [11]. 2.2 Semantics We use traditional naming conventions in the eld of CSP based nite model generation (FALCON [10] SEM [11] Without loss of generality, individuals are taken from the set N = f0; 1; 2; g of natural numbers. Since we are only interested in nite models, we interpret a ....

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Jian Zhang and Hantao Zhang. SEM: a system for enumerating models. In Chris S. Mellish, editor, Proceedings of the Fourteenth International Joint Conference on Articial Intelligence, pages 298-303, 1995.

....present a problem, which we could not solve yet: since functions are not generated data types, we do not have a simple mechanism available, that can find instances of function variables. 7 Related Work and Examples A large number of tools has been developed which try to construct finite models [12, 15, 14], even for first order specifications [4] Since our specifications usually do not have finite models at all, these tools are not directly applicable. Nevertheless an interesting alternative to our approach would be to use them by defining suitable abstractions (e.g. by collapsing all data ....

J. Zhang and H. Zhang. SEM: A system for enumerating models. In IJCAI-95, 1995.

....file can usually be used for both programs. MACE has been used for many applications including quasigroup existence problems [3] ortholattice problems [6] and lattice and Boolean algebra problems [7, 8] Other successful programs that look for finite models of first order statements are SEM [11] and FINDER [10] A related class of programs can produce finite models when the search for a refutation fails. Examples of these are SATCHMO [2] and MGTP [1] At its core, MACE has a Davis Putnam Loveland Logeman propositional decision procedure named ANLDP. ANLDP can be used directly to decide ....

.... (v0,v6,v5) v0,v1,v2) v0,v3,v4) v4,v2,v5) v3,v1,v6) v0,v6,v5) For a domain of 6, each of these (7 variable) clauses produces 9 : 9 propositional clauses. MACE can usually handle this many clauses, but it s hard to fight exponential behavior. The program SEM [11] is usually better than MACE for large clauses or large domains. 6 Differences from Previous Versions Major changes from earlier versions of MACE are listed here. 1. Previous versions of MACE called Otter to parse the input and to produce an intermediate form that was given to a program named ....

J. Zhang and H. Zhang. SEM: A system for enumerating models. In Proceedings of the International Joint Conference on Artificial Intelligence. Morgan Kaufmann, 1995. 10

.... proven by showing that Q is not a logical consequence of the program, in other words by constructing an interpretation in which the program clauses are true (i.e. which is a model of the program) and in which Q is false (or alternatively, the clause false Q is true) First order model generators [27, 28, 35, 24] can be used for this task. They search for an interpretation over a finite domain such that all clauses of a given set evaluate to true. This paper develops an alternative approach which exploits the fact that definite programs have least models. If Q is false in some model based on a ....

....uses abduction and failure analysis in the constraint checking. The second one 2 translates the constraints into finite domain constraints and uses a finite domain solver for the constraint checking. Section 5 discusses alternative approaches: the use of model generators for first order logic [27, 28, 35, 24], of type analysis [19, 10] a query fails if its inferred type is empty) and of program specialisation [22, 14] the query fails if the program for the given query can be specialised into the empty program) In section 6, the different approaches are compared. Finally, in section 7, we draw ....

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J. Zhang and H. Zhang. SEM: a system for enumerating models. In Proc. IJCAI95, Vol. 1, pages 298--303. Morgan Kaufmann, 1995. 26

.... n, a binary functor f h : D D D h , and a binary functor f b : D h D D. Then we replace every occurrence f (t 1 ; t 2 ; t 3 ) by f b ( f h (t 1 ; t 2 ) t 3 ) In a recent paper, Peltier [Pel98] presents a new system FMC AT INF which claims to do a better failure analysis than FINDER and SEM [ZZ95], and exploits symmetries to further improve the pruning of the search space. As other finite model builders for first order logic, it enumerates the full interpretation, not only the pre interpretation. Our understanding is that the concept of covering refutation used to prune the search is very ....

J. Zhang and H. Zhang. Sem: a system for enumerating models. In Proc. IJAI-95, vol 1, pages 298--303, Cambridge MA, 1995. Morgan Kaufmann.

....than enforcing AC. It would be interesting to identify the features of QG5 that gives an algorithm that maintains GAC such an advantage over MAC, and those of QG6 and QG7 that lessen this advantage. We now compare our results with those of FINDER [23] MACE [16] MGTP [11] SATO [26] and SEM [27]. Table 4 shows that our simple Solver code, by maintaining GAC, outperforms MGTP and FINDER by orders of magnitude, and explores less branches than SEM. SEM, MACE, and SATO have very sophisticated branching heuristics and complex rules for the elimination of symmetric models that are far more ....

....GAC, outperforms MGTP and FINDER by orders of magnitude, and explores less branches than SEM. SEM, MACE, and SATO have very sophisticated branching heuristics and complex rules for the elimination of symmetric models that are far more powerful than the symmetry breaking constraint we use [27]. It is therefore impressive that our simple Solver program is competitive with well developed systems like SEM and SATO. To conclude, despite the addition of non binary constraints that disturb the structure of the constraint graph, maintaining GAC significantly reduces search and run times on ....

J. Zhang, H. Zhang, SEM: A system for enumerating models, in: Proc. IJCAI-95, Montreal, Quebec, Vol. 1, 1995, pp. 298--303.

....proofs of diassociativity could always be replaced by the veri cation of a nite number of instances of diassociativity, which could result in a simpli cation. Our investigations were aided by the automated reasoning tools OTTER, developed by McCune [10] and SEM developed by J. Zhang and H. Zhang [14]. SEM nds nite models of systems of axioms, and was used to produce the three examples in Section 4. OTTER derives statements from axioms, and was used to derive enough instances of diassociativity from WRIF for the pattern to become clear. 2 Basics Following Bruck [1] see IV.1) the inner ....

....0 Table 1: A Flexible C non RIF Loop Proof. Consider the loop in Table 2. The nucleus is N = f0; 1; 2g, and all squares are in N , so that L=N is the 4 element boolean group. This is not exible because 3 (6 3) 6= 3 6) 3. The three examples in this section were found using the program SEM [14]. We do not see a really simple way of checking that Examples 4.1 and 4.2 are both C loops, with the rst one also exible. However, the reader can easily write the obvious computer program (entering each loop as an array) to check these facts; it is not necessary to verify that the code for SEM ....

J. Zhang and H. Zhang, SEM: a system for enumerating models, Proc. 14th Int.. Joint Conf. on AI (IJCAI-95), Montreal, 1995, pp. 298 - 303; available at URL: http://www.cs.uiowa.edu/~hzhang/

....The other approach uses a dynamic lookahead strategy to both lter out inconsistent domain values and select the most appropriate branching variable according to a rst fail heuristic. Because of its e ciency and source code availability, we chose to use the nite model generator SEM ([10]) as a software framework. We have obtained order of magnitude improvements on some problems, and successfully solved nite groups of order 30 and more. Keywords : nite model generation, constraint propagation, boolean propagation, lookahead heuristics, symmetries, nite group 1 ....

....of some expected conjecture. Model generation can also be used to produce a model instance of technical or industrial interest (like a schedule or a con guration instance) Many methods have been implemented to deal with many sorted or uni sorted theories: FINDER [6] FMSET [2] SATO [7] SEM [10], FMC [4] are known systems which solved some open problems. The method SEM (System for Enumerating Models) introduced by J. Zhang and H. Zhang in [10] is one of the most powerful known methods for solving problems expressed as many sorted theories. Many sorted theories are de ned by sets of ....

[Article contains additional citation context not shown here]

J. Zhang and H. Zhang. SEM: a system for enumerating models. In Chris S. Mellish, editor, Proceedings of the Fourteenth International Joint Conference on Articial Intelligence, pages 298303, 1995. 14

....2 elements model (in 0. 01 seconds) 8x; y:ln(x; y) 0 8x; y:pt(x; y) 0 8x; y:dipt(x; y) false 8x; y:diln(x; x) false 8x; y:diln(x; y) false 8x; y: x 6= 1 y 6= 1) con(x; y) false con(1; 1) true 8x; y:apt(x; y) false We note that other model builders, such as Sem, [9], Finder, 7] and Mace [5] could be used also. 4 Conclusion Theorem provers are, in general, unable to detect the satisfiability (resp. nonvalidity) of a formula, even if the formula admits a very simple model (resp. counterexample) The examples treated in this paper though rather simple ....

J. Zhang and H. Zhang. SEM: A system for enumerating models. In Proc. IJCAI-95, volume 1, pages 298--303. Morgan Kaufmann, 1995. 7

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J. Zhang and H. Zhang. SEM: a System for Enumerating Models. In Proceedings of the 14th International Joint Conference on Arti cial Intelligence (IJCAI), pages 298-303. Morgan Kaufmann, USA, 1995.

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J. Zhang and H. Zang. SEM: a system for enumerating models. In Proceedings of the International Joint Conference on Arti cial Intelligence, volume 1, pages 298-303, 1995. 116

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J. Zhang, H. Zhang, SEM: A system for enumerating models, in: Proc. IJCAI-95, Montreal, Quebec, Vol. 1, 1995, pp. 298--303.

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H. Zhang J. Zhang. SEM: a system for enumerating models. In Proc. of International Joint Conference on Artificial Intelligence (IJCAI'95), 1995.

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J. Zhang and H. Zhang. SEM: a System for Enumerating Models. In Proceedings of IJCAI-95, volume 1, pages 298-303, 1995. 172

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Zhang J., Zhang, H. SEM: A System for Enumerating Models. IJCAI-95, pp.298-303, Morgan Kaufmann.

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H. Zhang J. Zhang. SEM: a system for enumerating models. In Proc. of International Joint Conference on Arti cial Intelligence (IJCAI'95), 1995.

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J. Zhang and H. Zhang. SEM: a System for Enumerating Models. In Proceedings of the 14th International Joint Conference on Arti cial Intelligence (IJCAI), pages 298-303. Morgan Kaufmann, USA, 1995.

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J. Zhang and H. Zhang. SEM: A system for enumerating models. In Proceedings of the International Joint Conference on Artificial Intelligence. Morgan Kaufmann, 1995.

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J. Zhang and H. Zhang. SEM: A system for enumerating models. In Proceedings of the International Joint Conference on Artificial Intelligence. Morgan Kaufmann, 1995. 10

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J. Zhang and H. Zhang. SEM: A system for enumerating models. Proc. of International Joint Conference on Artificial Intelligence, 1995.

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J. Zhang and H. Zhang. SEM: a System for Enumerating Models. In Proceedings of the 14th International Joint Conference on Arti cial Intelligence (IJCAI), pages 298-303. Morgan Kaufmann, USA, 1995.

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