M. Kohlhase. A mechanization of sorted higher-order logic based on the resolution principle. PhD Thesis. Universitat des Saarlandes. Saarbrucken, Germany, 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....[And71] and Huet [Hue72] Whereas the former avoids unification the latter generally delays the computation of unifiers and instead adds unification constraints to the clauses in order to tackle the undecidability problem of HO unification. More recent papers concentrate on the adaption of sorts [Koh94] or theory unification [Wol93] to HO logic. Common to all these approaches is that they do not sufficiently solve the extensionality problem in HO automated theorem proving, i.e. all these approaches require the extensionality axioms to be added into the search space in order to reach Henkin ....

.... alternative completeness proof for a slightly extended version of ER (this version, e.g. employs the instantiation guessing FlexFlex rule) The new proof is motivated as follows: i) it eases the proof of the lifting lemma and avoids the quite complicated notion of clause isomorphisms as used in [BK98a,Koh94], ii) it can be reused to show the completeness for calculi EP and ERUE as well, iii) it prepares the analysis of non normal form resolution calculi, and (iv) it emphasises interesting aspects on rule FlexFlex, unification, and clause normalisation wrt. ER, EP, and ERUE. One such interesting ....

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M. Kohlhase. A Mechanization of Sorted Higher-Order Logic Based on the Resolution Principle. PhD thesis, Universitat des Saarlandes, 1994.

.... are oriented on the ideas of the corresponding completeness proof already presented for extensional higherorder resolution calculus ER in [BK98a] On the other hand we here want to get rid of the rather unintuitive clause isomorphism within the complicate lifting lemma employed in [BK98a] or [Koh94] which are susceptible to errors. This can be achieved by adding the well known FlexFlex unification rule to the calculus. Thus, the Henkin completeness proofs presented in this paper take this additional rule into account and thereby gains clarity and simplicity especially within the lifting ....

....of employing it as a filter. As a resolution approach that generally delays the unification filter until an empty clause is derived can certainly not form the basis of an efficient higher order theorem prover, Kohlhase allows within in his sorted variant of Huet s resolution calculus HORES (see [Koh94]) for eager unification (note that in practice many unification constraints have none or only finitely many solutions and that this information can positively influence the search space) The unsorted variant of HORES also provides the basis for the calculi ER, EP and ERUE discussed in this ....

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Michael Kohlhase. A Mechanization of Sorted Higher-Order Logic Based on the Resolution Principle. PhD thesis, Universitat des Saarlandes, 1994.

....in a literal by a general binding that imitates a logical connective or quantifier. Concerning the adaption of techniques from first order resolution to higherorder resolution some important cornerstones have been established by the sorted higher order resolution calculus described in [Koh94a, Koh94b] or the higher order indexing techniques examined in [Kle97] Indeed, the treatment of equality and extensionality is still a challenging problem for all higher order theorem provers. In fact there is no automated higher order theorem prover known to the author which can effectively and without ....

....systems. In the first part of this paper we introduce an extension of the Huet style cal 1 Logical Engine for Omega. The Leo project is strongly connected to the Omega project [BCF 97] and Leo s main intention is to become a powerful subsystem of Omega. culus introduced in [Koh94a, Koh94b] 2 in order to establish full extensionality. Although a formal proof for this conjecture is still lacking, the different examples discussed in this paper provide strong evidence for the Henkin completeness of the extended resolution calculus. Leo wants to reach a considerable power on its ....

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Michael Kohlhase. A Mechanization of Sorted Higher-Order Logic Based on the Resolution Principle. PhD thesis, Universitat des Saarlandes, 1994.

....above. We make the simplifying assumptions that contrast and parallelism are one and the same notion (we speak of contrastive or cparallelism) and that the properties p used in determining them are restricted to sorts from a given, domainspecific sort hierarchy. Thus we can use sorted type theory [ Kohlhase, 1994 ] to model similarity and contrastive parallelism. 2.1 Sorted Logic Sorts correspond to the basic cognitive concepts. Logically they can either be seen as unary predicates or as refinements of the types. The intuition behind this is that the universe of objects of a type ff is subdivided in ....

....introduced above. The problem at hand is to make colored sorted formulae similar or c parallel. For an algorithm ARP we build up on a sorted version of HOCU (which can be obtained by a straightforward combination of color techniques from [ Hutter and Kohlhase, 1995 ] with sorted methods from [ Kohlhase, 1994 ] but instead of simply having equations for sorted fij equality, we also add the equations for c parallelism and similarity to the unification problem as special equations = p and = s . The ARP calculates sufficient conditions for a given set of input equations by transforming systems of ....

Kohlhase, Michael 1994. A Mechanization of Sorted Higher-Order Logic Based on the Resolution Principle. Ph.D. Dissertation, Universitat des Saarlandes.

....the erasures of the instantiations of head variables must be general bindings, it is sucient to employ the uncolored rules for general paper.tex; 5 04 2001; 17:42; p.22 23 higher order uni cation. This will allow us to use most of the metatheory directly from the un colored case (see for instance [32, 24]) To keep this paper self contained, let us restate the de nitions. DEFINITION 10. General Binding) We call the formula GB h = Z 1 : Z n : H 1 Z) H m Z) a general binder i = n and has type m one of the following holds: Z j and j = m ....

....cation procedure, i.e. for any given 2 U(E) there is a CUT derivation E CUT E 0 such that E 0 is a C uni cation problem in C solved form, and 0 E is more general than . For this we only need a subset CUT of inference rules that approximate the solution (for details and proofs see[32, 24]) Even though CUT must be nonterminating in general (otherwise HOU would be decidable) we have the following semi termination result. THEOREM 3. If E is a uni cation problem with uni er , then CUT is terminating. CUT conserves the subset of uni ers that are compatible with . paper.tex; ....

Michael Kohlhase. A Mechanization of Sorted Higher-Order Logic Based on the Resolution Principle. PhD thesis, Universitat des Saarlandes, 1994.

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Michael Kohlhase. A Mechanization of Sorted Higher-Order Logic Based on the Resolution Principle. PhD thesis, Universitat des Saarlandes, 1994.

....erasures of the instantiations of head variables must be general bindings, it is sufficient to employ the uncolored rules paper.tex; 5 03 1999; 9:13; p.22 23 for general higher order unification. This will allow us to use most of the meta theory directly from the un colored case (see for instance [32, 24]) To keep this paper self contained, let us restate the definitions. DEFINITION 10. General Binding) We call the formula GB h ffi = Z ff 1 : Z ff n : H 1 Z) H m Z) a general binder iff Gamma ffi = fi n ff and has type fl m ff Gamma one of the following holds: ....

....procedure, i.e. for any given 2 U(E) there is a CUT derivation E CUT E 0 such that E 0 is a C unification problem in C solved form, and oe 0 E is more general than . For this we only need a subset CUT of inference rules that approximate the solution (for details and proofs see[32, 24]) Even though CUT must be nonterminating in general (otherwise HOU would be decidable) we have the following semi termination result. THEOREM 3. If E is a unification problem with unifier , then Gamma CUT is terminating. Gamma CUT conserves the subset of unifiers that are compatible ....

Michael Kohlhase. A Mechanization of Sorted Higher-Order Logic Based on the Resolution Principle. PhD thesis, Universitat des Saarlandes, 1994.

....rule Func eases the implementation and the integration of heuristics. See [Ben97] for a more detailed discussion. 4 Soundness and Completeness Theorem 16 (Soundness of ERES) The calculus ERES is sound with respect to Henkin semantics. Proof. The soundness of HORES is discussed in detail in [Koh94b] the only major difference to the first order case is the treatment of Skolemization, which has been discussed in [Mil83] The soundness of the three new extensionality rules are obvious, as they do only apply the two extensionality principles and the Leibniz definition, which are valid in ....

Michael Kohlhase. A Mechanization of Sorted Higher-Order Logic Based on the Resolution Principle. PhD thesis, Universitat des Saarlandes, 1994.

....notion of completeness, since it does not admit complete calculi [God31] But there is a more general notion of semantics due to Henkin [Hen50] that allows complete calculi and therefore sets the standard for the deductive power of calculi. The core of higher order resolution (HORES , see [Hue73,Koh94a] for details) is a simple extension of the first order resolution method to the higher order language: the only significant difference is that fij equality has to be build in by keeping formulae in normal form and that firstorder unification has to be replaced by higher order unification (i.e. ....

Michael Kohlhase. A Mechanization of Sorted Higher-Order Logic Based on the Resolution Principle. PhD thesis, Universitat des Saarlandes, 1994.

....T is not necessarily complete with respect to Henkin models, the notion of completeness that can be established by this method is a strictly weaker notion than Henkin completeness. As a consequence, the calculi developed for higher order automated theorem proving [And71, Hue72, Hue73, JP72, Mil83, Koh94b, Koh95] and the corresponding theorem proving systems such as Tps [ABI 96] or earlier versions of the authors Leo 1 are not or cannot be proven complete with respect to Henkin models. Moreover, they are not even sound with respect to T, since all of them but [And71] conversion, which is ....

....higher order models and ensure totality of our evaluation functions by a saturation condition (cf. De nition 4.9) in our abstract consistency classes. This does not restrict the applicability of our model existence theorems, since saturation is relatively simple to prove for a given calculus (cf. Koh94b, Koh98, BK97b] For all the notions of a model we present model existence theorems tying the di erentiating conditions of the models to suitable conditions in the abstract consistency classes (cf. Section 4.4) We can use the classical construction in all cases: abstract consistent sets are ....

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Michael Kohlhase. A Mechanization of Sorted Higher-Order Logic Based on the Resolution Principle. PhD thesis, Fachbereich Informatik, Universitat des Saarlandes, 1994.

....fX Bg is uncolored and there is no applicability condition on either of the reduction relations. Thus we can lift all the known theoretical results from the simply typed calculus to the colored case. In particular, fij reduction is terminating and confluent 4. For a sorted calculus see [KOH 94] HOCU: a linguistic application 11 in the presence of ff conversion (alphabetic renaming of bound variables, which we consider as built into the system) and we can decide fij equality by reducing to fij normal form. Higher order unification computes substitutions oe such that oe(M) fij oe(N) ....

KOHLHASE M., A Mechanization of Sorted Higher-Order Logic Based on the Resolution Principle . PhD thesis, Universitt des Saarlandes, 1994.

....complete resolution calculus by combining results of Huet [Hue72] on higher order resolution with the first order framework of multi valued deduction mentioned above. Our system is further refined in this paper to a logic HOL n and a resolution calculus HR n using techniques and results from [Koh94]. Thus the proofs omitted in this paper can be easily adapted from those in these sources. The resulting framework can be combined with the sort techniques developed in [Koh94] to obtain a higher order formalization of mathematics in the spirit of [KK94] Naturally the results reported here are ....

....is further refined in this paper to a logic HOL n and a resolution calculus HR n using techniques and results from [Koh94] Thus the proofs omitted in this paper can be easily adapted from those in these sources. The resulting framework can be combined with the sort techniques developed in [Koh94] to obtain a higher order formalization of mathematics in the spirit of [KK94] Naturally the results reported here are much more widely applicable, they extend to all logical systems that combine multiple truth values with higher order features, such as binding and fij conversion. Even if the ....

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Michael Kohlhase. A Mechanization of Sorted Higher-Order Logic Based on the Resolution Principle. PhD thesis, Universitat des Saarlandes, 1994.

....of this approach is proven by developing a sorted record calculus with dependent sorts and labeled abstraction that is well suited both for formalizing mathematical practice and supporting efficient inference services. This mathematical vernacular is an extension of the sorted calculus from [Koh94] by records, dependent record sorts and selection sorts. 1 Introduction Around 1995, an anonymous group of authors put forward the QED Manifesto [QED95] which advocates building up a mathematical knowledge base (and supporting software systems) as a kind of human genome project for the ....

....hierarchy is not necessarily fixed: A new logical system can be incorporated by specifying a logic morphism to any of the existing systems. In Figure 1, we have specified some of the logical systems, we will discuss in this paper. 3 Relativizing Type Theory into Set Theory ZF SZF T Set RZF [Koh94] [Oho95] BMV MV iZF iT Set iRZF Fig. 1. Hierarchy The main goal of this paper is to construct a hierarchy of representation languages culminating in a high level logical systems BMV and MV for formalizing mathematics. BMV is a joint generalization of Ohori s record calculus [Oho95] and the ....

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Michael Kohlhase. A Mechanization of Sorted Higher-Order Logic Based on the Resolution Principle. PhD thesis, Universitat des Saarlandes, 1994.

.... Especially example 111, which cannot be solved by any of the above provers, is trivial for Leo (10msec on a Pentium Pro 200) Conclusion and Availability The next logical steps to enhance the deductive power of Leo will be to extend the system to sorted logics [8], to extend the indexing scheme from cosimplification to higher order pattern unification [9] to fine tune the heuristics for extensionality treatment and finally to extend the system by a treatment for primitive equality. The source code and proof examples (including detailed proofs for the ....

M. Kohlhase. A Mechanization of Sorted Higher-Order Logic Based on the Resolution Principle. PhD thesis, Universitat des Saarlandes, 1994.

....notion of completeness, since it does not admit complete calculi [God31] But there is a more general notion of semantics due to Henkin [Hen50] that allows complete calculi and therefore sets the standard for the deductive power of calculi. The core of higher order resolution (HORES , see [Hue73,Koh94] for details) is a simple extension of the first order resolution method to the higher order language: the only significant difference is that fij equality has to be build in by keeping formulae in normal form and that first order unification has to be replaced by higher order unification (i.e. ....

....is, that rule Func eases the implementation and the integration of heuristics. See [Ben97] for a more detailed discussion. 4 Soundness and Completeness Theorem 3 (Soundness of ER) The calculus ER is sound with respect to Henkin semantics. Proof. The soundness of HORES is discussed in detail in [Koh94] the only major difference to the first order case is the treatment of Skolemization, which has been discussed in [Mil83] The soundness of the three new extensionality rules are obvious, as they do only apply the two extensionality principles and the Leibniz definition, which are valid in ....

Michael Kohlhase. A Mechanization of Sorted Higher-Order Logic Based on the Resolution Principle. PhD thesis, Universitat des Saarlandes, 1994.

....for HOL: is possible to prove an instance of the axiom of choice that is known to be independent of HOL [And73] using naive Skolemization. Dale Miller has investigated this problem in depth [Mil83] We will use a variant of his technique that the author has developed for higher order resolution [Koh94]. Skolemization in first order logic is a syntactic trick that allows to use the occurs in check in unification to keep track of the dependencies that the sequencing of quantifiers induces on variables. Variables Y that were existentially quantified in the scope of a universal quantifier 8X may ....

....we have Gamma 0 = Gamma [ C and R 0 = R. All of these rules are used with the understanding that all formulae are reduced w normal form after each rule application. Note that this last set of rules directly corresponds to the rules of higherorder pre unification as they can be found in [Koh94], which generalize Huet s pre unification transformations (see for instance [Sny91] for variable conditions. With these rules we use the tableau mechanism to construct Huet s unification tree [Hue76] We call a branch Theta in a higher order tableau T closed, iff Theta ends in a flex flex pair ....

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Michael Kohlhase. A Mechanization of Sorted Higher-Order Logic Based on the Resolution Principle. PhD thesis, Universitat des Saarlandes, 1994.

....E, and ffl the new variables in the domain of 0 do not occur in Phi. Proof sketch: The derivation D 0 is constructed along the line of D. In order to do this, it is essential to maintain a close correspondence between the clause sets involved (see the notion of a clause set isomorphism in [Koh94]) Note that the clause normal form transformations from D can also be applied to the corresponding clauses in Phi with the exception of the case, where the clause in Phi contains a flexible literal, whose head instantiates with a formula 5 RESOLUTION (HR N ) 17 whose head is a logical ....

....be a starting point for the development of a higher order logic with partial functions. In order for an adequate treatment of quantification (which must exclude the undefined element for a higher order account of partial functions) it will be necessary to combine it with the sort techniques of [Koh94] in the spirit of [KK94] This will yield a suitable basis for formalizing and mechanizing informal mathematical vernacular. Similarly, given a more general treatment of generalized quantifiers we will obtain a higher order mechanization of presuppositions, as a basis for an adequate integration ....

Michael Kohlhase. A Mechanization of Sorted Higher-Order Logic Based on the Resolution Principle. PhD thesis, Universitat des Saarlandes, 1994.

.... 1997) particularly, since the generalization of the colored unification algorithms needed to manipulate the search restrictions has already worked out (Hutter and Kohlhase, 1997) HIGHER ORDER AUTOMATED THEOREM PROVING 461 Acknowledgments While most of the work reported here is based on (Kohlhase, 1994; Kohlhase, 1995) important parts of the treatment of extensionality (Benzm ller and Kohlhase, 1997a; Benzm ller and Kohlhase, 1997b) have been developed in collaboration with Christoph Benzm ller and was funded by the DFG in Project HOTEL. Karsten Konrad has implemented of the HOT theorem prover ....

Kohlhase, M.: 1994, `A Mechanization of Sorted Higher-Order Logic Based on the Resolution Principle'. Ph.D. thesis, Universität des Saarlandes.

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Michael Kohlhase. A Mechanization of Sorted Higher-Order Logic Based on the Resolution Principle. PhD thesis, Universitat des Saarlandes, 1994.

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M. Kohlhase. A mechanization of sorted higher-order logic based on the resolution principle. PhD Thesis. Universitat des Saarlandes. Saarbrucken, Germany, 1994.

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M. Kohlhase. A mechanization of sorted higher-order logic based on the resolution principle. PhD Thesis. Universitat des Saarlandes. Saarbrucken, Germany, 1994.

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M. Kohlhase. A mechanization of sorted higher-order logic based on the resolution principle. PhD Thesis. Universitat des Saarlandes. Saarbrucken, Germany, 1994.

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M. Kohlhase. A mechanization of sorted higher-order logic based on the resolution principle. PhD Thesis. Universitat des Saarlandes. Saarbrucken, Germany, 1994.

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Michael Kohlhase. A Mechanization of Sorted Higher-Order Logic Based on the Resolution Principle. PhD thesis, Universitat des Saarlandes, 1994.

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