C. Sch urmann. Automating the Meta-Theory of Deductive Systems. PhD thesis, CarnegieMellon University, 2000. CMU-CS-00-146.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....derivation can then be verified by type checking. Of central importance to this e#ort are the type safety meta theorems (progress, type preservation, and GC safety) These are expressed in Twelf in relational form as logic programs [21, 22] In support of this, Twelf provides a totality checker [26, 24] that ensures that the relations represent total functions, and are therefore valid meta proofs. This is discussed in detail in Section 5. It is important to note that decidable checking of typability is not an aim for TALT. Unlike TAL and TALx86, TALT is a type assignment (or Curry style) ....

....(and input arguments to subcalls) are fully determined. 2. Termination checking: the induction variable(s) decrease in all recursive calls. 3. Totality checking: in every case analysis, all cases are covered. For details, the reader is referred to Pfenning and Schurmann [24] or Schurmann [26]. The specification of TALT in LF consists of 2081 lines of Twelf code, and the complete proof of Theorems 4.1, 4.2 and 4.6 consists of 10137 lines of Twelf code (including comments) A breakdown of the proof code for the interested reader is given in Table 1. The full proof takes approximately ....

Carsten Schurmann. Automating the Meta Theory of Deductive Systems. PhD thesis, Carnegie Mellon University, School of Computer Science, Pittsburgh, Pennsylvania, October 2000.

....remarks appear in Section 5. 2 The Twelf Metalogic We begin with a brief tutorial on the use of Twelf to express logics and meta reasoning. The ideas underlying this methodology originated with Pfenning [23] and were developed further in a variety of papers on Twelf and its predecessor Elf [24, 28, 27, 29, 26]. We assume familiarity with logic programming, and with the methodology of encoding logics in LF [11] wherein syntactic classes and judgements become types, and syntactic objects and derivations becomes terms. 2.1 A Simple Logic As a running example, we will use a very simple logic of natural ....

Carsten Schurmann. Automating the Meta Theory of Deductive Systems. PhD thesis, Carnegie Mellon University, School of Computer Science, Pittsburgh, Pennsylvania, October 2000.

....derivation can then be verified by type checking. Of central importance to this e#ort are the type safety meta theorems (progress, type preservation, and GC safety) These are expressed in Twelf in relational form as logic programs [22, 23] In support of this, Twelf provides a totality checker [27, 25] that ensures that the relations represent total functions, and are therefore valid meta proofs. This is discussed in detail in Section 5. It is important to note that decidable checking of typability is not an aim for TALT. Unlike TAL and TALx86, TALT is a type assignment (or Curry style) ....

....(and input arguments to subcalls) are fully determined. 2. Termination checking: the induction variable(s) decrease in all recursive calls. 3. Totality checking: in every case analysis, all cases are covered. For details, the reader is referred to Pfenning and Schurmann [25] or Schurmann [27]. The specification of TALT in LF consists of 2081 lines of Twelf code, and the complete proof of Theorems 4.1, 4.2 and 4.6 consists of 10137 lines of Twelf code (including comments) A breakdown of the proof code for the interested reader is given in Table 1. The full proof takes approximately ....

Carsten Schurmann. Automating the Meta Theory of Deductive Systems. PhD thesis, Carnegie Mellon University, School of Computer Science, Pittsburgh, Pennsylvania, October 2000.

.... reasoning about HOAS speci cations in an intuitionistic higher order logic [27] and Despeyroux, Pfenning and Sch urmann have developed a modal typed calculus that allows primitive recursive functions on HOAS encoded object language syntax without destroying the adequacy of the encoding [8, 36]. Abstractions as functions from names to terms. The Theory of Contexts 24 [23] reconciles the elegance of higher order abstract syntax with the desire to deal with names at the object level and have relatively simple forms of structural recursion induction. It does so by axiomatising a ....

C. Schurmann. Automating the Meta-Theory of Deductive Systems. PhD thesis, Carnegie-Mellon University, 2000.

....and it is well supported by existing systems for machine assisted reasoning based on typed calculus. It does not lend itself to principles of structural recursion and induction for the encoded object language that are particularly straightforward, but such principles have been developed: see [6, 25]. Abstractions as functions from names to terms. The Theory of Contexts [15] reconciles the elegance of higher order abstract syntax with the desire to deal with names at the object level and have relatively simple forms of structural recursion induction. It does so by axiomatizing a suitable ....

C. Schurmann. Automating the Meta-Theory of Deductive Systems. PhD thesis, Carnegie-Mellon University, 2000.

....should be moved to an additional meta level to ensure consistency. A straightforward two level approach such the one in Section 4 is meta theoretically uncertain. C It is dicult to allow (primitive) recursion on higher order syntax and especially to combine it with induction over open terms [30]. C No support for co induction at this time. The simulation of Full HOAS in Isabelle s IFOL seems successful. Indeed, the proofs of subject reduction, determinism of operational semantics, and the progress lemma resemble informal mathematical practice very closely. However, much of the hard ....

C. Schurmann. Automating the Meta-Theory of Deductive Systems. PhD thesis, Carnegie-Mellon University, 2000. CMU-CS-00-146.

....should be moved to an additional meta level to ensure consistency. A straightforward two level approach such the one in Section 4 is metatheoretically uncertain. C It is dicult to allow (primitive) recursion on higher order syntax and especially to combine it with induction over open terms [32]. C No support for co induction at this time. The simulation of Full HOAS in Isabelle s IFOL seems successful. Indeed, the proofs of subject reduction, determinism of operational semantics, and the progress lemma resemble informal mathematical practice very closely. However, much of the hard ....

C. Schurmann. Automating the Meta-Theory of Deductive Systems. PhD thesis, CarnegieMellon University, 2000. CMU-CS-00-146.

....way. This constitutes a reasonable middle ground between the CWA which allows no dynamic assumption but is amenable to negation and the OWA, where assumptions are totally unpredictable. Not surprisingly, the RWA plays also a central part in the study of induction in (meta) logical frameworks [23]. Technically, this regularity under dynamic extension is calibrated so as to ensure that static and dynamic clauses never overlap, by requiring every (dynamic) assumption to be parametric. This property extends to the negative program; in a sense, we maintain a distinction between static and ....

....the transformations has became extensionally quanti ed. Our nal goal is to achieve negation elimination in LF. Acknowledgments. I would like to thank Frank Pfenning for his continuous help and guidance. The notion of context schema is inspired by Sch urmann s treatment of analogous material in [23]. Elimination of Negation in a Logical Framework 425 ....

C. Schurmann. Automating the Meta-Theory of Deductive Systems. PhD thesis, CarnegieMellon University, 2000.

....the analysis of goals. We may thus consider this work as an initial requirements analysis of the properties of proofs of interest to logic programming. It is envisaged that the results of this analysis can then be implemented and utilized by means of an automated proof assistant such as Twelf [22, 23], possibly in conjunction with constraint logic programming techniques [15] 2 Properties of Proofs Computation, in the context of logic programming, is proof search: for a given program and goal, computation determines whether or not the goal is provable. However, this process involves more than ....

Carsten Schurmann, Automating the Meta-Theory of Deductive Systems, PhD thesis, Carnegie-Mellon University, 2000.

....the type family vs indeed represents some proof of value soundness. Such a notion can be given in the form of schema checking which guarantees that a type family such as vs inductively defines a total function from its first three arguments to its fourth argument. A discussion of schema checking [RP96, Sch00] is beyond the scope of these notes. Some material may also be found in the documentation which accompanies the implementation of Elf in the Twelf system [PS99] 1 1 [update on final revision] 3.8. THE FULL LF TYPE THEORY 71 3.8 The Full LF Type Theory The levels of kinds and types in the ....

.... EV#value V # type Case in Structural Induction Constant Declaration Base Case for Axioms Constant of Atomic Type Induction Step Constant of Functional Type A decidable criterion on when a given type family represents a proof of a theorem about a deductive system is subject of current research [RP96, Sch00]. 2 An alternative to this approach is to work in a stronger type theory with explicit induction principles in which we can directly express induction arguments. This approach is taken, for example, in the Calculus of Inductive Constructions [PM93] which has been implemented in the Coq system ....

Carsten Schurmann. Automating the Meta Theory of Deductive Systems. PhD thesis, Department of Computer Science, Carnegie Mellon University, August 2000. Available as Technical Report CMU-CS-00146.

....for a specific safety policy. In addition to checking whether a given program fulfills a specific safety policy, it is equally important to verify properties of the safety policy, for example its soundness. This is especially important if we change and extend the policies. Pfenning and Schurmann [35, 44] demonstrated that it is feasible to automate inductive reasoning about logical specifications and complemented the higher logic programming interpreter with the meta theorem prover Twelf. A key component of the meta theorem prover is to derive a goal by applying program clauses, lemmas, ....

....not only serves as a foundation for Prolog like first order logic programming, but also for higher order logic programming languages such as Elf or Prolog. To search for a proof of a goal, interpreters Elf or Prolog uses depth first search. The theorem prover Twelf is based on iterative deepening [44]. Based on uniform proofs, Cervesato [4] developed a proof theoretic view of compilation for logic programming languages. However so far there has been no effort to develop a proof theoretic foundation for tabled logic programming. Tamaki and Sato designed tabled resolution as a refinement of the ....

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Carsten Schurmann. Automating the meta theory of deductive systems. PhD thesis, Department of Computer Sciences, Carnegie Mellon University, 2000. available as Technical Report CMU-CS-00-146.

....functions. Those functions are called realizers and can be interpreted as proofs. The overall goal of this work is to use the meta logic M 2 as formal basis for an automated theorem prover. The reader can find a detailed presentation of M 2 and its meta theory in the forthcoming thesis [Sch00]. M 2 is a two level system. LF on the lower level provides the data which the partial functions on the M 2 level manipulate. M 2 inherits not only LF s expressive power, but also the properties which are associated with hypothetical judgments, namely, substitutions, weakening, ....

....we cannot use the universal quantifier to bind them. Universal quantifiers are designed to bind variables whose structure can be analyzed by case distinction. In our example, there exists only one basic building block, in the general case however, Phi can be described by several different blocks [Sch00]. We distinguish each of them by labels, and assign the label L to the one block in our example. Phi : Delta j Phi; u : nd pGq; h : hyp pGq) L Blocks of well typed parameter variables ranging over those blocks are called variable blocks, Variable block: ae : Delta j ae; x : A and they ....

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Carsten Schurmann. Automating the Meta-Theory of Deductive systems. PhD thesis, Carnegie-Mellon University, 2000. forthcoming.

....database in a speci c regular way. This constitutes a reasonable middle ground between the CWA which allows no dynamic assumption but is amenable to negation and the OWA, where assumptions are totally unpredictable. The RWA is also a promising tool in the study of the meta logical frameworks [18]. Technically, this regularity under dynamic extension is calibrated so as to ensure that static and dynamic clauses never overlap. This property extends to the negative program; in a sense, we maintain a distinction between static and dynamic information, but at a much ner level, i.e. inside the ....

....the transformations has became extensionally quanti ed. Our nal goal is to achieve negation elimination in LF. Acknowledgments. I would like to thank Frank Pfenning for his continuous help and guidance. The notion of context schema is inspired by Sch urmann s treatment of analogous material in [18]. 424 Alberto Momigliano ....

C. Schurmann. Automating the Meta-Theory of Deductive Systems. PhD thesis, Carnegie-Mellon University, 2000. forthcoming.

....Published by Elsevier Science B. V. conducted. Those domains include various deductive systems, such as logics and type systems. In addition, recent work on theorem proving about encodings that are represented in LF using higher order representation techniques have led to the meta logic # [Sch00] and a prototype implementation as part of the Twelf implementation. The quintessential di#erence between theorem proving in first order logics where the subjects of reasoning are encoded using relations, and theorem proving in logical frameworks is that in LF the theorem prover can take ....

....equivalent modulo # conversion. The substitution lemma that occurs in an informal proof is expressed by a simple LF level application (E # 1 V 2 ) in the definition of ev app. The meta logical foundation for reasoning about deductive systems that are encoded in LF is discussed in detail in [Sch00] The underlying concept of the meta theorem prover implemented in the Twelf system is that of a proof state, which consists of a set of assumptions and a goal. The type preservation theorem, for example, exp. #V : exp. #T : tp. #D : eval E V . of E T . #Q : of V T . 1) ....

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Carsten Schurmann. Automating the Meta-Theory of Deductive Systems. PhD thesis, Carnegie Mellon University, 2000. CMU-CS-00-146.

....[Sch01] which allows quanti ers to range over open objects whose free variables are declared in a regularly formed (and hence subject to reasoning) context. Those regular contexts are characterized by so called worlds whose structure may be freely chosen, but must be xed ahead of time [Sch00] The main contribution of this paper is an induction principle for higher order encodings that extends rst order intuitionistic logic. Its quanti ers range over objects which live in the dependent type theory LF [HHP93] for which canonical forms are known to exist. A prototype theorem prover ....

.... this setting may be generalized; we let x : A stand for the assumption that there exists an open object that is valid in some context 2 L( But how can we express assumptions about the form of , such as that contains certain parameters The answer to this question has already been proposed in [Sch00,Sch01] A new concept of block variables L is required, that ranges over valid parameter blocks in accordance with the world (written as L : Block variables are of the form : j ; x : A. Each block variable follows the structure 2 in L : some 1 block 2 and is labeled with ....

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Carsten Schurmann. Automating the Meta-Theory of Deductive Systems. PhD thesis, Carnegie Mellon University, 2000. CMU-CS-00-146.

....4 Type System T We begin now with the formal presentation of the type system T which supports function de nition by recursion as used above for the de nition of Gentzen s algorithm ndseq. A detailed presentation of T and its meta theory can be found in the author s thesis [Sch00] T s recursive functions range over higherorder LF objects and can therefore take advantage of all properties which are associated with hypothetical judgments, such as, substitutions, weakening, contraction, and exchange. T s type system is simple. It provides a dependent function ....

....in , and binding is de ned componentwise. Although our informal syntax of T functions used in Figure 3 hides variable blocks, they are important in the formal exposition. In our example, is de ned in terms of one block, but in the general case it can be de ned in terms of several [Sch00] We distinguish each of them by labels, and assign the label L to the one block in our example. j ; u : nd G; h : hyp G) L Variable blocks L are indexed by the appropriate label L. is a context that denotes not only the set of free LF variables, but also what is know about the ....

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Carsten Schurmann. Automating the Meta-Theory of Deductive Systems. PhD thesis, Carnegie Mellon University, 2000. CMU-CS-00-146.

....realizes the proof that CPS conversion preserves well typedness must be de ned with respect to the higher order nature of the encoding of typing derivations. However, only few languages support this feature, and hence it is best implemented as a total function (realizer) in the meta logic M 2 [12] of type: 8E : exp: 8T : tp: 8D : of E T : 9C : exp: 9T 0 : tp: 9R : E exp = C : 9R : T tp = T 0 : 9Q : of C T 0 : where the participating judgments are represented in LF as types. pis expression q = exp exp : type pis index expression q = iexp iexp : type pis type q = tp ....

Carsten Schurmann. Automating the Meta-Theory of Deductive Systems. PhD thesis, Carnegie Mellon University, 2000. CMU-CS-00-146.

....total functions. Those functions are called realizers and can be interpreted as proofs. The overall goal of this work is to use the meta logic M 2 as formal basis for an automated theorem prover. The reader can nd a detailed presentation of M 2 and its meta theory in the author s thesis [Sch00] 7 4. TYPE SYSTEM M 2 M 2 is a two level system. LF on the lower level provides the data which the partial functions on the M 2 level manipulate. M 2 inherits not only LF s expressive power, but also the properties which are associated with hypothetical judgments, namely, ....

....there are parameter variables x that can only bind parameters, and lists of parameter variables , or so called variable blocks that range over entire parameter blocks. In our example, there exists only one basic block, in the general case, however, can be described by several di erent blocks [Sch00] We distinguish each of them by labels, and assign the label L to the one block in our example. j ; u : nd pGq; h : hyp pGq) L Variable blocks L are indexed by labels L corresponding to the label of the block they are an instantiation of. Third, M 2 assigns names x to ....

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Carsten Schurmann. Automating the Meta-Theory of Deductive Systems. PhD thesis, Carnegie Mellon University, 2000. CMU-CS-00-146.

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C. Sch urmann. Automating the Meta-Theory of Deductive Systems. PhD thesis, CarnegieMellon University, 2000. CMU-CS-00-146.

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Carsten Schurmann. Automating the Meta Theory of Deductive Systems. PhD thesis, Department of Computer Science, Carnegie Mellon University, August 2000. Available as Technical Report CMU-CS-00146.

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