Weyhrauch, R. W., 1980. Prolegomena to a theory of mechanized formal reasoning, Artificial Intelligence 13, 133-170.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....The LCF approach [GMW79] requires inference procedures to be constructed as tactics that generate a fully expanded proof in terms of low level inferences when applied. Proof objects have also been widely used as a way of validating inference procedures and securing mobile code [Nec97] Reflection [Wey80,BM81] is a way of reasoning about the metatheory of a theory within the theory itself. The di#cult tradeo# with reflection is that the theory has to be simple in order to be reasoned about, but rich enough to reason with. The verification of decision procedures is actually well within the realm ....

Richard W. Weyhrauch. Prolegomena to a theory of mechanized formal reasoning. Artificial Intelligence, 13(1 and 2):133--170, April 1980.

....semantics of the behaviour of the whole (meta level) reasoning system can be described by a set of (intended) temporal models. 1 Introduction In the literature on meta level architectures and reflection (e.g. 28] two separate streams can be distinguished: a logical stream (e.g. 4] 19] [37] ) and a procedural stream (e.g. 10] 11] Unfortunately there is a serious gap between the two streams. In the logical stream one restricts oneself often to static reflections; i.e. of facts the truth of which does not change during the reasoning: e.g. provable(A) with A an object level ....

....and dynamic aspects takes place by introducing the notion of an explicit (declarative) control information state in the object level component. Our logical framework has been partly inspired by Weyhrauch s view on the role of partial models (or simulation structures) in meta level architectures ([37], 19] see also [32] What is different in our case is that the partial models may be dynamic. Furthermore, similarities can be found to the approach called dynamic interpretation of natural language (e.g. see [17] 20] 23] In this approach the dynamic interpretation of a sentence in ....

R.W. Weyhrauch, Prolegomena to a theory of mechanized formal reasoning, Artificial Intelligence 13 (1980), pp. 133-170.

.... situation independent knowledge in the knowledge base (described as a logical theory) and situation specific information which can be used as an input, imported from the world situation outside (that is described as a given situation model) This perspective is similar to the perspective in [18] (and [10] where the models representing specific information are called simulation structures. domain description declarative functionality description (input output function) reasoning component specification (knowledge base) Fig. 1 Domain description, functionality description and reasoning ....

R.W. Weyhrauch, Prolegomena to a theory of mechanized formal reasoning, Artificial Intelligence 13

....an essential extension to the object level theory. Therefore it is excluded to model downward reflection according to reflection rules as sometimes can be found in the literature, e.g. If at the meta level it is provable that Provable(to) then at the object level t o is provable (e.g. see [Wey80]) MT I Provable(to) OT to A reflection rule like this can only be used in a correct manner if the meta theory about provability gives a sincere axiomatisation of the object level proof system, and in that case by downward reflection nothing can be added to the object level that was not ....

R.W. Weyhrauch, Prolegomena to a Theory of Mechanized Formal Reasoning, Artificial Intelligence J. 13 (1980), pp. 133-170

....creates an essential modification of the object level theory. Therefore it is excluded to model downward reflection according to reflection rules as sometimes can be found in the literature, e.g. If at the meta level it is provable that Provable(j) then at the object level j is provable (cf. [Wey80]) Provable(j) OT A reflection rule like this can only be used in a correct manner if the meta theory about provability gives a sincere axiomatization of the object level proof system, and in that case by downward reflection nothing can be added to the object level ....

R.W. Weyhrauch, Prolegomena to a Theory of Mechanized Formal Reasoning, Artificial Intelligence J. 13 (1980), pp. 133-170.

....creates an essential extension to the object level theory. Therefore it is excluded to model downward reflection according to reflection rules as sometimes can be found in the literature, e.g. If at the meta level it is provable that Provable(j) then at the object level j is provable (e.g. see [Wey80]) Provable(j) OT A reflection rule like this can only be used in a correct manner if the meta theory about provability gives a sincere axiomatisation of the object level proof system, and in that case by downward reflection nothing can be added to the object ....

R.W. Weyhrauch, Prolegomena to a Theory of Mechanized Formal Reasoning, Artificial Intelligence J. 13 (1980), pp. 133-170

....specified to the arguments specified. The superclass is the first met in the class precedence list (see Section 5.3) after the object. This corresponds to the super facility of class based systems [3, 5] The way procedural attachment is implemented here is different from the standard way (e.g. [16], Prolog) where evaluable computable predicates are used rather than computable terms. Our way of incorporating procedural attachment gives a more natural representation, corresponding to the attribute value model of objectbased systems. Myers universal attachment mechanism [17] also allows for ....

R. W. Weyhrauch, Prolegomena to a theory of mechanized formal reasoning, AI 13 (1980) 133-170.

....symbol is intended to denote ordinary addition over the integers. Clearly it is possible to directly compute the sum of two numerals and thus one can beliefit froin associating the function symbol with such an addition procedure (such procedural attachment in FOL is described by Weyhrauch [1]) However to apply the addition procedure to a term such as [lix] liy] one must solve for its subtcrms, in ,his case lix] and liy] in terms of numeric constants. Thus the usefulness of procedures attached to function or predicate symbols can depend upon the ability to solve for ....

Weyhrauch, R.W., Prolegomena to a Theory of Mechanized Formal Reasoning. Artificial Intelligence 13 (1,2) (April 1980) 81-133.

....(described in An integrated approach to Arti cial Intelligence by L. Stringa, IRST technical report No 9012 11) 1 1 Introduction and motivations It has been argued that knowledge should be structured into sets of facts or theories (often called contexts ) some of the many examples are [11, 21, 10, 45, 7, 32, 29, 48, 5]. In [10, 13] the authors take a further step and introduce a new general kind of formal systems allowing multiple distinct languages and call them MultiLanguage systems (ML systems) In [11] it is argued, in fact, that providing each theory with its own language allows us to give a natural and ....

....and has also a di erent language. Maybe more important, the ideas described here have been incorporated into a system, called GETFOL [12] which gives the user the ability to de ne arbitrary ML systems with arbitrary bridge rules. GETFOL is a total reimplementation extension of the FOL system [45, 44, 22]. The results presented in this paper amount to showing that some forms of multicontextual reasoning that we have mechanized inside GETFOL are consistent and as expressive as the usual modal logics. The main body of the paper (sections 3, 4, 5) concentrates on the class MR of the ML systems ....

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R.W. Weyhrauch. Prolegomena to a Theory of Mechanized Formal Reasoning. Artif. Intell., 13(1):133-176, 1980.

....check a series of computations done by an external algorithm A that to prove that A itself is correct. However, some applications seem to require the latter, where algorithm A itself is coded, proved and run inside the proof assistant. This approach is called deep reflection, and was pioneered in [16, 3]. In Coq, one first application is Boutin s Ring tactic [2] deciding equalities in the theory of rings, mixing shallow and deep reflection. One of the largest deep reflection endeavours is certainly the work by Verma et al. 15] where binary decision diagrams are integrated in Coq through total ....

R. W. Weyhrauch. Prolegomena to a theory of mechanized formal reasoning. Artifical Intelligence, 13(1, 2):133--170, 1980.

....or agents know about their environment, and the reasoning processes that allow them to derive new knowledge from what they already know. In KRR, a notion very similar to context, called LSpair was rstly introduced by Weyhrauch in his Prolegomena to a Theory of Mechanized Formal Reasoning [52]. His goal was to implement the epistemological part of McCarthy s Advice Taker, a program which should possess abilities that in human beings would be called common sense [41] A fundamental assumption of the Advice Taker s project was that formal logic was an appropriate tool for modeling and ....

R.W. Weyhrauch. Prolegomena to a Theory of Mechanized Formal Reasoning. Articial Intelligence, 13(1):133-176, 1980. 42

....components) and store them in the catalogue for future use. Manager agents maintain these four types of agents; e.g. the number of agents of a specific type can be adjusted, depending on contributions of agents to good and bad designs. McAlinden, Florida James, Chao, Norman, Hills and Smith [50] describe how design agents can be integrated to facilitate information and knowledge sharing. In this approach, a central product model of the STEP standard is used, as well as ACL and knowledge based ontologies. Their aim is to incorporate existing and legacy systems without delay in a design ....

....information and knowledge sharing. In this approach, a central product model of the STEP standard is used, as well as ACL and knowledge based ontologies. Their aim is to incorporate existing and legacy systems without delay in a design project. The aforementioned research on distributed design [48, 49, 50] does not include explicit representations for reflective reasoning. Being able to reason about, or even from, the viewpoint of another agent is a means with which, e.g. conflicts can be prevented. In the literature on reflection such as [50, 51, 52, 53, 54] a restricted number of types of ....

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Weyhrauch RW. Prolegomena to a theory of mechanized formal reasoning. Artificial Intelligence 1980;13:133-170.

....sono tanto potenti quanto un dimostratore. 10 3. il sistema verifica che ogni passo di deduzione sia applicabile ed in caso affermativo lo esegue. I punti precedenti descrivono (le versioni iniziali de) i primi proof checkers [Mil72a, Mil72b] LCF [GMW79, GMW77] AUTOMATH [deB70, deB80] e FOL [Wey80, Wey77] L ovvio problema sorto con questa generazione di proof checkers e che costruire dimostrazioni e un operazione lunga, ripetitiva e noiosa. Una parziale soluzione (applicata, per esempio in FOL [Wey80] e di dotare il verificatore non solo di regole primitive ma anche di regole piu ....

....proof checkers [Mil72a, Mil72b] LCF [GMW79, GMW77] AUTOMATH [deB70, deB80] e FOL [Wey80, Wey77] L ovvio problema sorto con questa generazione di proof checkers e che costruire dimostrazioni e un operazione lunga, ripetitiva e noiosa. Una parziale soluzione (applicata, per esempio in FOL [Wey80] e di dotare il verificatore non solo di regole primitive ma anche di regole piu potenti che corrispondano ad interi pezzi di dimostrazione. Tali regole complesse sollevano l utente dall affrontare e risolvere tutti i dettagli. In piu , se si identifica la classe di formule a cui tali regole ....

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R.W. Weyhrauch. Prolegomena to a Theory of Mechanized Formal Reasoning. Artif. Intell., 13(1):133--176, 1980.

....data representing itself in compliance with its semantics. Causal connection in this case means that the representation is linked in a consistent way to the represented objects. 5 The first reflective system to appear in the literature is (to the best of our knowledge) the FOL system by Weyrauch [73]. In FOL, knowledge and metaknowledge are expressed in different contexts, and the user can access them both for expressing and inferring new facts. A FOL context consists of a language L (which is a first order language with sorts and conditional expressions) and a simulation structure S, which ....

R. W. Weyhrauch. Prolegomena to a theory of mechanized formal reasoning. Artificial Intelligence, pages 133--70, 1980.

....principles) to safely define the s contexts corresponding to non monotonic extension of PSC. We also provide an implementation within the GETFOL system [GT91] GETFOL is an interactive system based on Natural Deduction [Pra65] extending the FOL system developed at Stanford by Richard Weyhrauch [Wey80]. GETFOL turns out to be appropriate for our purposes since, among othe features (e.g. the many sorted language) allows the user to define and handle multiple contexts. 5 Acknowledgements We are indebted to Fausto Giunchiglia for the many invaluable GETFOL FOL suggestions we had. The Mechanized ....

R.W. Weyhrauch. Prolegomena to a Theory of Mechanized Formal Reasoning. Artif. Intell., 13(1):133--176, 1980. This article was processed using the L a T E X macro package with LLNCS style 10

....The paper is organized as follows. Section 2 describes the object and meta theories. Section 3 describes how the metatheory has been built to resemble (parts of) the code implementing the object level logic. It also shows how this feature can be exploited by the semantic attachment mechanism [Wey80, GS89] Section 4 shows how the metatheory can express and prove representations of proofs in the object (mathematical) theory. Section 5 shows how the metatheoretic properties described in sections 3, 4 can be used to assert theorems in the object level mathematical theory. Finally, section 6 ....

....reasoning at various levels; in particular we show how it is possible to prove the same result only by object level theorem proving, only by metalevel theorem proving and by a combination of them. 1 GETFOL is implemented on top of a complete re implementation extension [GW91] of the FOL system [Wey80] 2 2 From the object level theory to the metatheory At the moment, GETFOL uses a first order Natural Deduction calculus, very much like that defined in [Pra65] it is presented as a sequent calculus, where a sequent has form Gamma A, where A is a formula and Gamma is a set of formulas. ....

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R.W. Weyhrauch. Prolegomena to a theory of Mechanized Formal Reasoning. Artificial Intelligence. Special Issue on Non-monotonic Logic, 13(1), 1980. 15

....quite different: meta reasoning is studied as a tool for controlling object level inference in automated deduction systems. The important issue here is how meta reasoning can be used to drive object level deduction in a correct, effective and efficient manner. Some examples in proof checking are [11, 17, 9], some examples in theorem proving are [4, 3, 2] We thank Frank van Harmelen for his comments on an early draft of the paper. 1 This paper is a first step towards providing a foundation for meta reasoning in the fields of AI and CS. From this perspective we understand and formalise ....

....approach, our motivations lead us to have working hypotheses quite different from those of logicians. In particular, we require the object and meta theory to be two distinct logical theories, each theory with its own language, axioms and deductive machinery. This approach was first suggested in [17] in which the possibility of having a hierarchy of logical meta theories, deductively linked to each other by reflection rules, was first outlined. These ideas have been pushed further and put into place in [9] However a theoretical foundation to this work has never previously been given. The ....

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R.W. Weyhrauch. Prolegomena to a Theory of Mechanized Formal Reasoning. Artif. Intell., 13(1):133--176, 1980.

....Finally, MT expressions are in a one to one mapping with executable programs (all of this being described in section 6) As a consequence, they can be executed. The execution mechanism of MT strategies is based on the simulation structure machinery and the attach mechanism first introduced in FOL [36] (see sections 6.1 and 6.2) This achieves obj 6: executability of reasoning strategies and automatization of reasoning. 6 MC systems in GETFOL MC systems have been implemented inside GETFOL [11] GETFOL has commands which allow the user to define arbitrary MC systems. First of all, it is ....

....in the language of MT there is a corresponding data structure in the code; for any function symbol in the language of MT , there is a corresponding function in the code. We are therefore able to use the semantic attachment and the simulation structure machinery, implemented originally in FOL [36] (but see also [11] to execute declarative specifications of reasoning strategies. Let us consider, as an example, the term rdown( T H( A ; C ) P SC ; C ) rdown corresponds to the function rdown in the code. T H( A ; C ) is the meta theoretic constant which denotes the data ....

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R.W. Weyhrauch. Prolegomena to a Theory of Mechanized Formal Reasoning. Artificial Intelligence, 13(1):133--176, 1980. 46

....whose prefix contains either no quantifiers or only (finitely many) existential quantifiers or only (finitely many) universal quantifiers or (finitely many) universal quantifiers followed by (finitely many) existential quantifiers. 2 GETFOL is a complete reimplementation of the FOL system [Wey80]. 1 The decider is composed of a set of decision procedures for simpler subclasses of FOL. The first step is always the scanning of the goal to be proved. This allows to identify the complexity of the goal and thus which is the right decision procedure to apply 3 . This operation takes ....

R.W. Weyhrauch. Prolegomena to a theory of Mechanized Formal Reasoning. Artificial Intelligence. Special Issue on Non-monotonic Logic, 13(1), 1980. 11

....write control strategies as programs (usually called tactics) in ML to guide the search for a proof of a theorem. In the second paradigm, from now on called declarative, the metalevel is a logical metatheory and metareasoning is performed by deduction on metalevel statements. One example in AI is [Wey80] one in theorem proving is [How88] Both approaches are sometimes incorporated and alternatively used; thus, for instance, in NuPrl [CAB 86] and Isabelle [Pau89] both ML and a declarative logical metatheory can be used to build derived inference rules. In logic programming, metainterpreters ....

....to behave as the procedural metalevel of the system. Section 3 describes MT and how it can be automatically generated from the implementation code. In section 4, it is proved that MT represents all the object level deductions and 1 GETFOL is a reimplementation extension of the FOL system [Wey80] GETFOL has, with minor variations, all the functionalities of FOL plus extensions, some of which described here, to allow metatheoretic theorem proving. 2 The notions of correctness and completeness here involved are sometimes called adequacy and faithfulness, respectively. 2 that deduction ....

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R.W. Weyhrauch. Prolegomena to a Theory of Mechanized Formal Reasoning. Artif. Intell., 13(1):133--176, 1980. 15

....use in efficiently introducing derived rules into a theorem prover. In this case, there may be no extension to the set of formulas that are theorems at the object level, but formulas that are already provable can be justified as theorems more quickly. Reflection is the approach taken by Weyhrauch [Wey80] but he allows metatheorems to be asserted as axioms, so there is no guarantee that the extension is sound. However, the work is motivated more by an interest in formal theories of reasoning than in safely extending theorem provers. The logic considered is first order. Davis and Schwartz [DS79] ....

R. W. Weyhrauch. Prolegomena to a theory of mechanized formal reasoning. Artificial Intelligence, 13:133--170, 1980.

....(more generally, propositional attitudes) and provability, two areas where modal logics have been extensively used. 1 Introduction and motivations It has been argued that knowledge should be structured into sets of facts or theories (often called contexts ) some of the many examples are [GW88, Giu93, Wey80, McC87, KK90]. In [Giu91, GS91] the authors take a further step and introduce a new general kind of formal systems allowing multiple distinct languages and call them Multi Language systems (ML systems) In [Giu91] it is argued, in fact, that providing each theory with its own language allows us to give a ....

....of the interaction with the implemented system plays a central role in our research. The ideas described here have been incorporated into a system, called GETFOL, which gives the user the ability to define arbitrary ML systems (GETFOL is a total re implementation extension of the FOL system [Wey80, GW91]) The paper is structured as follows. Section 2 gives a short description of some basic notions concerning ML systems (but see [Giu91] for a much longer presentation) Section 3 introduces the class MR, the MR system MBK for the representation of propositional attitudes and the MR system MK for ....

R.W. Weyhrauch. Prolegomena to a Theory of Mechanized Formal Reasoning. Artif. Intell., 13(1):133--176, 1980. 15

....The most important major examples in theorem proving are LCF and its descendants ( 12, 7, 21, 20] In LCF forward reasoning is performed by using formal inference rules and backward reasoning is performed at the metalevel by using tactics. Another example in AI is GOAL [5] that provides FOL [28] with a goal oriented language for interactive proof construction. At the third level, which we call the logical level, bidirectional reasoning is totally performed by bidirectional deductions inside a well de ned formal system. As far as we know, this approach has never been explored before. In ....

....a nite set of formulae. Conclusions, or more generally goals, are represented as B , where B; have the same meaning as in sequents. Brie y, the language L can be de ned as the set of sequents and goals over a propositional language. 1 GETFOL ( 11] is a reimplementation of the FOL system ([28]) GETFOL has, with minor variations, all the functionalities of FOL plus extension, some of which described here, to perform bidirectional reasoning. 2 THE FORMAL SYSTEM FB 3 2.1 Forward and backward systems The inference rules in R but one are either sequent or goal versions of the natural ....

R.W. Weyhrauch. Prolegomena to a Theory of Mechanized Formal Reasoning. Articial Intelligence, 13(1):133-176, 1980.

....principle is stated externally to them, by means of a set of inference rules. In this case we have an informal representation of (1) which is not a formula in a formal system. Such examples are the linking rules de ned in [8] in Metalogic Programming) and the re ection rule of the FOL system [36]. Its clear that, if the object theory and the meta theory are completely distinct, then (1) cannot be stated in any of the two theories. Examples of such approaches are [36, 17] in Arti cial Intelligence) and [21, 22] in Metalogic Programming) A second observation about (1) concerns its role ....

....Such examples are the linking rules de ned in [8] in Metalogic Programming) and the re ection rule of the FOL system [36] Its clear that, if the object theory and the meta theory are completely distinct, then (1) cannot be stated in any of the two theories. Examples of such approaches are [36, 17] (in Arti cial Intelligence) and [21, 22] in Metalogic Programming) A second observation about (1) concerns its role namely, how and to which purpose the re ection principle is used. A rst possibility is that (1) has a descriptive role. This means that (1) is a statement that is true for a ....

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R.W. Weyhrauch. Prolegomena to a Theory of Mechanized Formal Reasoning. Articial Intelligence, 13(1):133-176, 1980. 45

....reasoning is meant to hold no matter what solutions one adopts to these fundamental issues. 3 Forms of contextual reasoning Mechanisms for contextual reasoning have been studied in di erent disciplines, though with di erent goals. A very partial list includes:re ection and metareasoning [22, 14], entering and exiting context, lifting, transcending context [16, 20, 5] local reasoning, switch context [12, 4] parochial reasoning and context climbing [7] changing viewpoint [1] focused reasoning [17] As a matter of fact, it is very dicult to see the relationship between these di erent ....

R.W. Weyhrauch. Prolegomena to a Theory of Mechanized Formal Reasoning. Articial Intelligence, 13(1):133-176, 1980.

....in context Gamma if some x i occurs free in Q. 8 view, meta level architectures have been used extensively in the realm of mechanical theorem proving [3, 2, 18, 20] since in many cases it is quite straightforward to construct a proof by means of syntactic analysis of the problem at hand [34, 1]. Here, the important issue is how meta programming and meta reasoning can be used to represent software development steps together with expressing a certain semantics of these steps. In a first step one encodes syntactic categories and the proof theory of QED within itself following the approach ....

R. W. Weyhrauch. Prolegomena to a Theory of Mechanized Formal Reasoning. Artificial Intelligence, 13(1):133--170, 1980. 14

....D, by QExts(S) fE 2 Exts(S) j E 2 d2E J (d; S)g: That is, E is a component admissible extension of S just in case each element of E approves (via J ) of the way it occurs in E relative to S. The admissible extensions AExts(S) are stipulated as a subset of 12 [Minsky 1965] Doyle 1980] [Weyhrauch 1980], Frisch and Allen 1982] Smith 1982] 8 the component admissible extensions, or formally, AExts(S) QExts(S) Exts(S) If E 2 AExts(S) we also write S Delta E. Putting all these definitions together, we say that each choice of (D; I; 6 S ; J ; Delta ) or alternatively, each choice of ....

....the definition of something like inevitability in terms of circumscription. Here one has axioms stating the presence or absence of some state components, and other axioms giving the interpretations of state components and general restrictions (in other words, embedding the 39 See, for example, [Weyhrauch 1980], Konolige and Nilsson 1980] Doyle 1980] 40 There are sometimes computationally tractable ways of doing this, see for example [Reiter 1982] 64 theory of reasoned assumptions in the agent s own language) Circumscribing these with respect to the present predicate on state components ....

Weyhrauch, R. W., 1980. Prolegomena to a theory of mechanized formal reasoning, Artificial Intelligence 13, 133-170.

....lifts the search space from the object theory to the meta theory, which is often better behaved, i.e. has a smaller search space. One example of meta reasoning is reflection, in which theorems in a meta theory are related, via reflection rules, to theorems in an object theory, and vice versa, Weyhrauch, 1980 ] Another example is proof planning,inwhich meta reasoning is used to build a global outline of a proof, which is then used to guide the detailed proof, Bundy, 1991 ] A third example is analogy, in which an old proof is used as a plan to guide the proof of a new theorem, Owen, 1990 ] ....

R. W. Weyhrauch. Prolegomena to a theory of mechanized formal reasoning. Artificial Intelligence, 13:133--170, 1980.

....state spaces of astronomical sizes, and mimicking this enumeration at least naively, see related work below with proof rules inside the proof assistant would be foolish. One remedy to this model checker (automatic, fast) vs proof assistant (expressive, safe) antinomy is to use reflection [32, 7, 1, 4, 6], which in this context can roughly be thought of as replacing proofs in the logic by computations (see Section 3) Reflection is particularly applicable in Coq, since Coq is based on the calculus of inductive constructions, a logic which is essentially a typed lambda calculus, a quintessential ....

R. W. Weyhrauch. Prolegomena to a theory of mechanized formal reasoning. Artifical Intelligence, 13(1, 2), 1980.

....enumerate state spaces of astronomical sizes, and mimicking this enumeration at least naively, see Section 1 with proof rules inside the proof assistant would be foolish. One remedy to this model checker (automatic, fast) vs. proof assistant (expressive, safe) antinomy is to use reflection [Wey80, BM81, ACHA90, BC93, Bou97] The general idea to prove a given property P applied to some term t is as follows. Assume we have a proof assistant in which you can also describe and prove programs: NQTHM [BM79] Coq and Lego are three examples. Then write a program that takes t as input and ....

....decision procedures with proof assistants is concerned, using reflection is both not new and not the only possibility. We have already mentioned the work of Samuel Boutin [Bou97] but the idea of reflection in theorem proving predates it, and was already used in Richard Weyhrauch s FOL system [Wey80] The idea is the same: replace deductions by computations at the meta level, and reflect this by allowing the system to accept meta level computations as actual proofs, thereby taking shortcuts in proofs. This idea goes a bit further in Coq, in that there is no real separation in levels, as the ....

Richard W. Weyhrauch. Prolegomena to a theory of mechanized formal reasoning. Artifical Intelligence, 13(1, 2):133--170, 1980.

.... of the project This project builds on and extends Richard Weyhrauch s work on the FOL system, in particular his work on meta, reflection principles, and simulation structures (where, using Weyhrauch s terminology, simulation structures are the mechanizable analogue of the notion of model) [10]. It can be described as an attempt to push the idea of linking computation in the code of a mechanized system and deduction in the system itself. From an implementational point of view, GETFOL has been developed on top of a reimplementation of the FOL system [10] described in [9] GETFOL has, ....

....analogue of the notion of model) 10] It can be described as an attempt to push the idea of linking computation in the code of a mechanized system and deduction in the system itself. From an implementational point of view, GETFOL has been developed on top of a reimplementation of the FOL system [10], described in [9] GETFOL has, with minor variations, all the functionalities of FOL plus extensions, some of which described here, to allow for metatheoretic theorem proving. From a conceptual point of view, the close connection with Weyhrauch s work can be seen by analyzing the relation between ....

R.W. Weyhrauch. Prolegomena to a Theory of Mechanized Formal Reasoning. Artif. Intell., 13(1):133--176, 1980. 6

....its metatheory MT. We call the process described in figure 1, introspective metatheoretic reasoning (IMR) Introspective because the system is able to formalize and reason This work has been done at IRST as part of the MAIA project. 3 GETFOL is a reimplementation extension of the FOL system [Wey80, GW91] GETFOL has, with minor variations, all the functionalities of FOL plus extensions, some of which described here, to allow for metatheoretic theorem proving. Basic Inference Procedures Derived Optimized Rules 6 Lifted Axioms Automated Reasoning Inference Theorems ....

R.W. Weyhrauch. Prolegomena to a Theory of Mechanized Formal Reasoning. Artificial Intelligence, 13(1):133--176, 1980. This article was processed using the L a T E X macro package with LLNCS style

....and the synthesis of a logic tactic implementing a normalizer in negative normal form is presented as a case study. 1 Introduction As pointed out in [GMMW77] interactive theorem proving [GMW79, CAB 86, Pau89] has been growing up in the continuum existing between proof checking [deB70, Wey80] on one side and automated theorem proving [Rob65, And81, Bib81] on the other. Interactive theorem provers were built with the goal in mind to overtake the deficiencies of the extreme solutions: proof checkers force the user to lengthy and laborious interactions, while automated theorem provers ....

....and proof strategies. 2 Although this work is largely independent of the particular logic chosen, we shall concentrate on a classical Natural Deduction style [Pra65] first order object theory OT, in which inference rules apply to pairs 1 GETFOL is a reimplementation extension of the FOL system [Wey80] GETFOL has, with minor variations, all the functionalities of FOL plus extensions, some of which described here, to allow metatheoretic theorem proving. 2 The formal metatheory MT presented in this paper is, with minor variations, the MT originally defined in [GT91a, GT92, GT93] 2 ( Gamma; ....

R.W. Weyhrauch. Prolegomena to a Theory of Mechanized Formal Reasoning. Artificial Intelligence, 13(1):133--176, 1980. 14

....work by Goedel [28] metareasoning has been one of the most studied research topics in formal reasoning. Work has been done in mathematical logic (e.g. 15, 51, 40] in philosophical logic (e.g. 42] in logic programming (e.g. 5] in many subfields of AI, such as mathematical reasoning (e.g. [54, 11]) planning (e.g. 49] programming languages (e.g. 48] and so on. These citations are by no means exhaustive. Our interests are in theorem proving with metatheories. Similar to previous work in automated deduction, we have mechanized an object theory OT and its metatheory MT. The mechanization ....

....way. 3 The project This project builds on and extends Richard Weyhrauch s work on the FOL system, in particular his work on Meta, reflection principles, and simulation structures (where, using Weyhrauch s terminology, simulation structures are the mechanizable analogue of the notion of model) [54]. It can be described as an attempt to push the idea of linking computation in the code of a mechanized system and deduction in the system itself. From an implementational point of view, GETFOL has been developed on top of a reimplementation of the FOL system, described in [27] GETFOL has, with ....

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R.W. Weyhrauch. Prolegomena to a Theory of Mechanized Formal Reasoning. Artificial Intelligence, 13(1):133--176, 1980.

....nested groups of representational items and Wilks [WB79] advocates the use of distinct sets of beliefs. Moreover, in the field of artificial intelligence, John McCarthy has a notion of contexts as objects [McC89] while Giunchiglia and Weyhrauch [GW88] use multiple contexts inside the FOL system [Wey80] to formalise a form of non monotonic reasoning. The advantages seem obvious also from an implementational point of view: facts can be clustered in subtheories which capture the structure of the problem (e.g. multiple meta and or object theories, each on a different topic) thus allowing a more ....

....The ML system obtained by extending MB to allow arbitrary nesting and by adding the bridge rules for common belief and circumscriptive ignorance has been used to machineprove the three wise man problem. The proof has been built in GETFOL [GT91] an extension re implementation of the FOL system [Wey80]. GETFOL allows the definition of multiple theories with their own languages, axioms and inference rules and of arbitrary bridge rules among them. 4 Conclusions We have presented a new notion of formal system which allows the use of multiple distinct languages. Some motivations for this choice ....

R.W. Weyhrauch. Prolegomena to a theory of Mechanized Formal Reasoning. Artificial Intelligence. Special Issue on Non-monotonic Logic, 13(1), 1980. 10

....some phenomenon. This thesis is articulated discussing the example about non monotonic reasoning reported in [MD80] No formal issues are faced, the focus is only on the representational issues. The system used to implement the proof is not described (FOL , First Order Logic) an overview is in [Wey80]. All the necessary information is given when needed. The paper follows this path: the next section begins with the description of the example. Here contexts are described and it is briefly discussed why they are the right structure to think when solving this kind of problems. Section three is ....

....this kind of reasoning can be monotonically described inside first order logic by axiomatizing its metatheory (metatheories) So far contexts have been described appealing to the intuitive meaning that the word context carries. More precisely, contexts are the natural development of what in [Wey80] Weyhrauch called LS pairs. An FOL context is a finite data structure which acts like a (partial) theory of the world. This data structure is a triple: hL; SS; F i where L is a language which defines the words which can be used and for what (i.e. is a function) SS is a simulation ....

R.W. Weyhrauch. Prolegomena to a Theory of Mechanized Formal Reasoning. Artif. Intell., 13(1):133--176, 1980.

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Weyhrauch, R. W., 1980. Prolegomena to a theory of mechanized formal reasoning, Artificial Intelligence 13, 133-170.

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Weyhrauch, R. W., 1980. Prolegomena to a theory of mechanized formal reasoning, Artificial Intelligence 13, 133-170.

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Weyhrauch, R. W. 1979. Prolegomena to a theory of mechanized formal reasoning. Artificial Intelligence 13, 1--2, 133--170.

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R.W. Weyhrauch. Prolegomena to a Theory of Mechanized Formal Reasoning. Artificial Intelligence, 13(1):133--176, 1980.

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R.W. Weyhrauch, Prolegomena to a Theory of Mechanized Formal Reasoning, A.I. 13 (1980), pp. 133-170.

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R.W. Weyhrauch. Prolegomena to a Theory of Mechanized Formal Reasoning. Artificial Intelligence, 13(1):133--176, 1980. 14

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R.W. Weyhrauch. Prolegomena to a Theory of Mechanized Formal Reasoning. Artificial Intelligence, 13(1):133--176, 1980. 41

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R.W. Weyhrauch. Prolegomena to a Theory of Mechanized Formal Reasoning. Artif. Intell., 13(1):133--176, 1980.

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R.W. Weyhrauch. Prolegomena to a theory of mechanized formal reasoning. Artificial Intelligence, 13, 133-- 176, 1980.

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Weyhrauch, R.W.: Prolegomena to a theory of mechanized formal reasoning. Artificial Intelligence (1980) 133--70

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Richard W. Weyhrauch. Prolegomena to a theory of mechanized formal reasoning. Artificial Intelligence, 13:133--170, 1980. 23

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R. W. Weyhrauch. Prolegomena to a theory of mechanized formal reasoning. Artificial Intelligence, 13:133--170, 1980. Reprinted in [15]. 44

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R. Weyhrauch. `Prolegomena to a theory of mechanized formal reasoning'. Artificial Intelligence Journal, 13. (1), 133--170 (1980). 17, D. Friedman and M. Wand. `Reification: reflection without meta-phy~ic'. ACM Conference on LISP and Functional Programming. ACM. Austin. August 1984. pp. 348--355.

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R.W. Weyhrauch. Prolegomena to a Theory of Mechanized Formal Reasoning. Artificial Intelligence, 13(1):133--176, 1980.

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