Reuben Hersh confided to us that, about forty years ago, the late Paul Cohen predicted to him that at some unspecified point in the future, mathematicians would be replaced by computers. Rather than focus on computers replacing mathematicians, however, our aim is to consider the (im)possibility of human mathematicians being joined by “artificial mathematicians” in the proving practice—not just as a method of inquiry but as a fellow inquirer.

Theorems in automated theorem proving are usually proved by formal logical proofs. However, there is a subset of problems which humans can prove by the use of geometric operations on diagrams, so called diagrammatic proofs. Insight is often more clearly perceived in these proofs than in the corresponding algebraic proofs; they capture an intuitive notion of truthfulness that humans find easy to see and understand. We are investigating and automating such diagrammatic reasoning about mathematical theorems. Concrete, rather than general (...) diagrams are used to prove particular concrete instances of the universally quantified theorem. The diagrammatic proof is captured by the use of geometric operations on the diagram. These operations are the inference steps of the proof. An abstracted schematic proof of the universally quantified theorem is induced from these proof instances. The constructive -rule provides the mathematical basis for this step from schematic proofs to theoremhood. In this way we avoid the difficulty of treating a general case in a diagram. One method of confirming that the abstraction of the schematic proof from the proof instances is sound is proving the correctness of schematic proofs in the meta-theory of diagrams. These ideas have been implemented in the system, called Diamond, which is presented here. (shrink)

This paper described a formal theory of type judgments for propositional logic notations of PM; I felt the need of my own automated type checker to check their examples. The type checker I wrote did indeed serve to help me referee the paper, but also took a rather different approach to notation and typing for propositional functions of PM, which proved worth writing up independently in our own paper: Holmes, M. Randall, “Polymorphic type– checking for the ramified theory of (...) types of Principia Mathematica”, in Fairouz Kamareddine, ed., Thirty–five Years of Automating Mathematics, Kluwer, 2003, pp. 173-215. (shrink)

We present a set of algorithms for automated simplification of symbolic constants of the form ∑iαixi with αi rational and xi complex. The included algorithms, called SimplifySum2 and implemented in Mathematica, remove redundant terms, attempt to make terms and the full expression real, and remove terms using repeated application of the multipair PSLQ integer relation detection algorithm. Also included are facilities for making substitutions according to user-specified identities. We illustrate this toolset by giving some real-world examples of its usage, (...) including one, for instance, where the tool reduced a symbolic expression of approximately 100 000 characters in size enough to enable manual manipulation to one with just four simple terms.2Available from https://github.com/alexkaiser/SimplifySum. (shrink)

Although resolution-based inference is perhaps the industry standard in automated theorem proving, there have always been systems that employed a different format. For example, the Logic Theorist of 1957 produced proofs by using an axiomatic system, and the proofs it generated would be considered legitimate axiomatic proofs; Wang’s systems of the late 1950’s employed a Gentzen-sequent proof strategy; Beth’s systems written about the same time employed his semantic tableaux method; and Prawitz’s systems of again about the same time are (...) often said to employ a natural deduction format. [See Newell, et al (1957), Beth (1958), Wang (1960), and Prawitz et al (1960)]. Like sequent proof systems and tableaux proof systems, natural deduction systems retain.. (shrink)

nature of modern data collection and storage techniques, and the increases in the speed and storage capacities of computers. Statistics books from 30 years ago often presented examples with fewer than 10 variables, in domains where some background knowledge was plausible. In contrast, in new domains, such as climate research where satellite data now provide daily quantities of data unthinkable a few decades ago, fMRI brain imaging, and microarray measurements of gene expression, the number of variables can range into the (...) tens of thousands, and there is often limited background knowledge to reduce the space of alternative causal hypotheses. In such domains, non-automated causal discovery techniques appear to be hopeless, while the availability of faster computers with larger memories and disc space allow for the practical implementation of computationally intensive automated search algorithms over large search spaces. Contemporary science is not your grandfather’s science, or Karl Popper’s. Causal inference without experimental controls has long seemed as if it must somehow be capable of being cast as a kind of statistical inference involving estimators with some kind of convergence and accuracy properties under some kind of assumptions. Until recently, the statistical literature said not. While parameter estimation and experimental design for the effective use of data developed throughout the 20th century, as recently as 20 years ago the methodology of causal inference without experimental controls remained relatively primitive. Besides a cessation of hostilities from the majority of the statistical and philosophical communities (which has still only partially happened), several things were needed for theories of causal estimation to appear and to flower: well defined mathematical objects to represent causal relations; well defined connections between aspects of these objects and sample data; and a way to compute those connections. A sequence of studies beginning with Dempster’s work on the factorization of probability distributions [Dempster 1972] and culminating with Kiiveri and Speed’s [Kiiveri & Speed 1982] study of linear structural equation models, provided the first, in the form of directed acyclic graphs, and the second, in the form of the “local” Markov condition.. (shrink)

On a traditional view, the primary role of a mathematical proof is to warrant the truth of the resulting theorem. This view fails to explain why it is very often the case that a new proof of a theorem is deemed important. Three case studies from elementary arithmetic show, informally, that there are many criteria by which ordinary proofs are valued. I argue that at least some of these criteria depend on the methods of inference the proofs employ, and that (...) standard models of formal deduction are not well-equipped to support such evaluations. I discuss a model of proof that is used in the automated deduction community, and show that this model does better in that respect. (shrink)

The refereed proceedings of the 19th International Conference on Automated Deduction, CADE 2003, held in Miami Beach, FL, USA in July 2003. The 29 revised full papers and 7 system description papers presented together with an invited paper and 3 abstracts of invited talks were carefully reviewed and selected from 83 submissions. All current aspects of automated deduction are discussed, ranging from theoretical and methodological issues to the presentation of new theorem provers and systems.

This article will rework the classical question ‘Can a machine think?’ into a more specific problem: ‘Can a machine think anything new?’ It will consider traditional computational tasks such as prediction and decision-making, so as to investigate whether the instrumentality of these operations can be understood in terms of the creation of novel thought. By addressing philosophical and technoscientific attempts to mechanise thought on the one hand, and the philosophical and cultural critique of these attempts on the other, I will (...) argue that computation’s epistemic productions should be assessed vis-à-vis the logico-mathematical specificity of formal axiomatic systems. Such an assessment requires us to conceive automated modes of thought in such a way as to supersede the hope that machines might replicate human cognitive faculties, and to thereby acknowledge a form of onto-epistemological autonomy in automated ‘thinking’ processes. This involves moving beyond the view that machines might merely simulate humans. Machine thought should be seen as dramatically alien to human thought, and to the dimension of lived experience upon which the latter is predicated. Having stepped outside the simulative paradigm, the question ‘Can a machine think anything new?’ can then be reformulated. One should ask whether novel behaviour in computing might come not from the breaking of mechanical rules, but from following them: from doing what computers do already, and not what we might think they should be doing if we wanted them to imitate us. (shrink)

This article will rework the classical question ‘Can a machine think?’ into a more specific problem: ‘Can a machine think anything new?’ It will consider traditional computational tasks such as prediction and decision-making, so as to investigate whether the instrumentality of these operations can be understood in terms of the creation of novel thought. By addressing philosophical and technoscientific attempts to mechanise thought on the one hand, and the philosophical and cultural critique of these attempts on the other, I will (...) argue that computation’s epistemic productions should be assessed vis-à-vis the logico-mathematical specificity of formal axiomatic systems. Such an assessment requires us to conceive automated modes of thought in such a way as to supersede the hope that machines might replicate human cognitive faculties, and to thereby acknowledge a form of onto-epistemological autonomy in automated ‘thinking’ processes. This involves moving beyond the view that machines might merely simulate humans. Machine thought should be seen as dramatically alien to human thought, and to the dimension of lived experience upon which the latter is predicated. Having stepped outside the simulative paradigm, the question ‘Can a machine think anything new?’ can then be reformulated. One should ask whether novel behaviour in computing might come not from the breaking of mechanical rules, but from following them: from doing what computers do already, and not what we might think they should be doing if we wanted them to imitate us. (shrink)

This book constitutes the refereed proceedings of the 13th International Conference on Automated Deduction, CADE-13, held in July/August 1996 in New Brunswick, NJ, USA, as part of FLoC '96. The volume presents 46 revised regular papers selected from a total of 114 submissions in this category; also included are 15 selected system descriptions and abstracts of two invited talks. The CADE conferences are the major forum for the presentation of new results in all aspects of automated deduction. Therefore, (...) the volume is a timely report on the state-of-the-art in the area. (shrink)

2nd edition. Many-valued logics are those logics that have more than the two classical truth values, to wit, true and false; in fact, they can have from three to infinitely many truth values. This property, together with truth-functionality, provides a powerful formalism to reason in settings where classical logic—as well as other non-classical logics—is of no avail. Indeed, originally motivated by philosophical concerns, these logics soon proved relevant for a plethora of applications ranging from switching theory to cognitive modeling, and (...) they are today in more demand than ever, due to the realization that inconsistency and vagueness in knowledge bases and information processes are not only inevitable and acceptable, but also perhaps welcome. The main modern applications of (any) logic are to be found in the digital computer, and we thus require the practical knowledge how to computerize—which also means automate—decisions (i.e. reasoning) in many-valued logics. This, in turn, necessitates a mathematical foundation for these logics. This book provides both these mathematical foundation and practical knowledge in a rigorous, yet accessible, text, while at the same time situating these logics in the context of the satisfiability problem (SAT) and automated deduction. The main text is complemented with a large selection of exercises, a plus for the reader wishing to not only learn about, but also do something with, many-valued logics. (shrink)

This article features long-sought proofs with intriguing properties (such as the absence of double negation and the avoidance of lemmas that appeared to be indispensable), and it features the automated methods for finding them. The theorems of concern are taken from various areas of logic that include two-valued sentential (or propositional) calculus and infinite-valued sentential calculus. Many of the proofs (in effect) answer questions that had remained open for decades, questions focusing on axiomatic proofs. The approaches we take are (...) of added interest in that all rely heavily on the use of a single program that offers logical reasoning, William McCune's automated reasoning program OTTER. The nature of the successes and approaches suggests that this program offers researchers a valuable automated assistant. This article has three main components. First, in view of the interdisciplinary nature of the audience, we discuss the means for using the program in question (OTTER), which flags, parameters, and lists have which effects, and how the proofs it finds are easily read. Second, because of the variety of proofs that we have found and their significance, we discuss them in a manner that permits comparison with the literature. Among those proofs, we offer a proof shorter than that given by Meredith and Prior in their treatment of ukasiewicz's shortest single axiom for the implicational fragment of two-valued sentential calculus, and we offer a proof for the ukasiewicz 23-letter single axiom for the full calculus. Third, with the intent of producing a fruitful dialogue, we pose questions concerning the properties of proofs and, even more pressing, invite questions similar to those this article answers. (shrink)

This paper outlines a logical representation of certain aspects of the process of mathematical proving that are important from the point of view of Artificial Intelligence. Our starting point is the concept of proof-event or proving, introduced by Goguen, instead of the traditional concept of mathematical proof. The reason behind this choice is that in contrast to the traditional static concept of mathematical proof, proof-events are understood as processes, which enables their use in Artificial Intelligence in such contexts in which (...) problem-solving procedures and strategies are studied. We represent proof-events as problem-centred spatio-temporal processes by means of the language of the calculus of events, which adequately captures certain temporal aspects of proof-events (i.e. that they have history and form sequences of proof-events evolving in time). Further, we suggest a “loose” semantics for the proof events by means of Kolmogorov’s calculus of problems. Finally, we expose the intented interpretations for our logical model from the fields of automated theorem-proving and Web-based collective proving. (shrink)

Logic has pride of place in mathematics and its 20th century offshoot, computer science. Modern symbolic logic was developed, in part, as a way to provide a formal framework for mathematics: Frege, Peano, Whitehead and Russell, as well as Hilbert developed systems of logic to formalize mathematics. These systems were meant to serve either as themselves foundational, or at least as formal analogs of mathematical reasoning amenable to mathematical study, e.g., in Hilbert’s consistency program. Similar efforts continue, (...) but have been expanded by the development of sophisticated methods to study the properties of such systems using proof and model theory. In parallel with this evolution of logical formalisms as tools for articulating mathematical theories (broadly speaking), much progress has been made in the quest for a mechanization of logical inference and the investigation of its theoretical limits, culminating recently in the development of new foundational frameworks for mathematics with sophisticated computer-assisted proof systems. In addition, logical formalisms developed by logicians in mathematical and philosophical contexts have proved immensely useful in describing theories and systems of interest to computer scientists, and to some degree, vice versa. Three examples of the influence of logic in computer science are automated reasoning, computer verification, and type systems for programming languages. (shrink)

This paper draws attention to the tendency of a range of technologies to reduce pedagogical interactions to a series of datafied transactions of information. This is problematic because such transactions are always by definition reducible to finite possibilities. As the ability to gather and analyse data becomes increasingly fine-grained, the threat that these datafied approaches over-determine the pedagogical space increases. Drawing on the work of Hegel, as interpreted by twentieth century French radical philosopher Alexandre Kojève, this paper develops a model (...) of relational pedagogy which highlights three points of incompatibility with a datafied learning environment reduced to finite measures. Firstly: Kojève’s accont of authority in Hegel posits two aspects to the mimetic relation between teacher and student: recognition and realisation, which belong to the ipseity or about-self-ness of the subject, and are incompatible with a general definition of data. Secondly, the Hegelian approach to human historical time, in particular the assertion that time and desire are begun in the future, not the past, renders it incompatible with mathematical time as used in data processing. Finally, from these it is possible to derive a distinctive notion of the work of pedagogy, grounded in Kojève’s realist reading of Hegel, irreducible to information processing. In consequence of this threefold irreducibility, the paper draws attention to a need for relations of human pedagogical work to be inherent in the design of educational technologies and highlights the dangers of presuming a machine intelligence model in the design of learning environments. (shrink)

Traditional logic as a part of philosophy is one of the oldest scientific disciplines. Mathematical logic, however, is a relatively young discipline and arose from the endeavors of Peano, Frege, Russell and others to create a logistic foundation for mathematics. It steadily developed during the 20th century into a broad discipline with several sub-areas and numerous applications in mathematics, informatics, linguistics and philosophy. While there are already several well-known textbooks on mathematical logic, this book is unique in that (...) it is much more concise than most others, and the material is treated in a streamlined fashion which allows the professor to cover many important topics in a one semester course. Although the book is intended for use as a graduate text, the first three chapters could be understood by undergraduates interested in mathematical logic. These initial chapters cover just the material for an introductory course on mathematical logic combined with the necessary material from set theory. This material is of a descriptive nature, providing a view towards decision problems, automated theorem proving, non-standard models and other subjects. The remaining chapters contain material on logic programming for computer scientists, model theory, recursion theory, Godel’s Incompleteness Theorems, and applications of mathematical logic. Philosophical and foundational problems of mathematics are discussed throughout the text. The author has provided exercises for each chapter, as well as hints to selected exercises. About the German edition: …The book can be useful to the student and lecturer who prepares a mathematical logic course at the university. What a pity that the book is not written in a universal scientific language which mankind has not yet created. - A.Nabebin, Zentralblatt. (shrink)

In practice, mathematical proofs are most often the result of careful planning by the agents who produced them. As a consequence, each mathematical proof inherits a plan in virtue of the way it is produced, a plan which underlies its “architecture” or “unity”. This paper provides an account of plans and planning in the context of mathematical proofs. The approach adopted here consists in looking for these notions not in mathematical proofs themselves, but in the agents who produced them. The (...) starting point is to recognize that to each mathematical proof corresponds a proof activity which consists of a sequence of deductive inferences—i.e., a sequence of epistemic actions—and that any written mathematical proof is only a report of its corresponding proof activity. The main idea to be developed is that the plan of a mathematical proof is to be conceived and analyzed as the plan of the agent(s) who carried out the corresponding proof activity. The core of the paper is thus devoted to the development of an account of plans and planning in the context of proof activities. The account is based on the theory of planning agency developed by Michael Bratman in the philosophy of action. It is fleshed out by providing an analysis of the notions of intention—the elementary components of plans—and practical reasoning—the process by which plans are constructed—in the context of proof activities. These two notions are then used to offer a precise characterization of the desired notion of plan for proof activities. A fruitful connection can then be established between the resulting framework and the recent theme of modularity in mathematics introduced by Jeremy Avigad. This connection is exploited to yield the concept of modular presentations of mathematical proofs which has direct implications for how to write and present mathematical proofs so as to deliver various epistemic benefits. The account is finally compared to the technique of proof planning developed by Alan Bundy and colleagues in the field of automated theorem proving. The paper concludes with some remarks on how the framework can be used to provide an analysis of understanding and explanation in the context of mathematical proofs. (shrink)

Depending on what one means by the main connective of logic, the -if..., then... -, several systems of logic result: classic and modal logics, intuitionistic logic or relevance logic. This book presents the underlying ideas, the syntax and the semantics of these logics. Soundness and completeness are shown constructively and in a uniform way. Attention is paid to the interdisciplinary role of logic: its embedding in the foundations of mathematics and its intimate connection with philosophy, in particular the philosophy (...) of language. Set theory is presented both as a conditio sine qua non for logic and as a interesting exact ontology. The study of infinite sets yields perplexing results. Formalization of informal number theory results in formal number theory; Godel's incompleteness is treated. At appropriate places attention is paid to paradoxes, intuitionism, conditionals, the historical development of logic, to logic programming and automated theorem proving for classical logic.". (shrink)

Although there are a few books available that give brief surveys of a variety of nonstandard logics, there is a growing need for a critical presentation providing both a greater depth and breadth of insight into these logics. This book assembles a wider and deeper view of the many potentially applicable logics. Three appendixes provide short tutorials on classical logic and modal logics, and give a brief introduction to the existing literature on the logical aspects of probability theory. These tutorials (...) and the bibliography included in the appendixes provide useful reference material for the reader. (shrink)

We argue that the mathematization of science should be understood as a normative activity of advocating for a particular methodology with its own criteria for evaluating good research. As a case study, we examine the mathematization of taxonomic classification in systematic biology. We show how mathematization is a normative activity by contrasting its distinctive features in numerical taxonomy in the 1960s with an earlier reform advocated by Ernst Mayr starting in the 1940s. Both Mayr and the numerical taxonomists sought to (...) formalize the work of classification, but Mayr introduced a qualitative formalism based on human judgment for determining the taxonomic rank of populations, while the numerical taxonomists introduced a quantitative formalism based on automated procedures for computing classifications. The key contrast between Mayr and the numerical taxonomists is how they conceptualized the temporal structure of the workflow of classification, specifically where they allowed meta-level discourse about difficulties in producing the classification. (shrink)

The ongoing explosion of interest in artificial intelligence is fueled in part by recently developed techniques in machine learning. Those techniques allow automated systems to process huge amounts of data, utilizing mathematical methods that depart from traditional statistical approaches, and resulting in impressive advancements in our ability to make predictions and uncover correlations across a host of interesting domains. But as is now widely discussed, the way that those systems arrive at their outputs is often opaque, even to the (...) experts who design and deploy them. Is it morally problematic to make use of opaque automated methods when making high-stakes decisions, like whether to issue a loan to an applicant, or whether to approve a parole request? Many scholars answer in the affirmative. However, there is no widely accepted explanation for why transparent systems are morally preferable to opaque systems. We argue that the use of automated decision-making systems sometimes violates duties of consideration that are owed by decision-makers to decision-subjects, duties that are both epistemic and practical in character. Violations of that kind generate a weighty consideration against the use of opaque decision systems. In the course of defending our approach, we show that it is able to address three major challenges sometimes leveled against attempts to defend the moral import of transparency in automated decision-making. (shrink)

Mathematical proofs generally allow for various levels of detail and conciseness, such that they can be adapted for a particular audience or purpose. Using automated reasoning approaches for teaching proof construction in mathematics presupposes that the step size of proofs in such a system is appropriate within the teaching context. This work proposes a framework that supports the granularity analysis of mathematical proofs, to be used in the automated assessment of students' proof attempts and for the presentation (...) of hints and solutions at a suitable pace. Models for granularity are represented by classifiers, which can be generated by hand or inferred from a corpus of sample judgments via machine-learning techniques. This latter procedure is studied by modeling granularity judgments from four experts. The results provide support for the granularity of assertion-level proofs but also illustrate a degree of subjectivity in assessing step size. (shrink)

In patent law most of the crucial legal questions such as patentability and infringement are linked to the patent claims. The European Patent Office regards patent claims as a set of independent features which are examined separately in a more or less formal way. The author has found that this approach allows for developing a simple mathematical model which treats patent claim features as logical statements and patent claims as compound statements wherein the individual statements are connected by logical connectives. (...) The proposed mathematical model provides a uniform system for examining various legal questions that are dealt with separately under current case law, moreover, it allows for developing an expert system for resolving complex legal situations and for automating the evaluation of a large number of patent claim variants that is currently not possible. (shrink)

This research aims at establishing the impact of information and communication technologies (ICT) on e-commerce development of industrial enterprises by means of economic and mathematical modelling. The goal was achieved using the following methods: theoretical generalization, analysis and synthesis (to critically analyse the scientific approaches of scientists regarding the expediency of using mathematical models in the context of enterprises’ e-commerce development), target, comparison and grouping (to reveal innovative methodological approach to assessing ICT impact on e-commerce development of industrial enterprises), tabular, (...) analytical and integral method (for summarizing the analysis results of enterprises readiness to implement ICT, ICT use in the activities of industrial enterprises of Ukraine and the analysis of e-commerce development), mathematical modelling (to build a regression model determining impact of changes in ICT use on the market share occupied by industrial enterprises), generalization (to determine promising directions of e-commerce developing of industrial enterprises). The implementation of a comprehensive approach to understanding the importance of ICT influence on e-commerce development of industrial enterprises will ensure acceleration of the digitalization of business processes, will contribute to the speed increase of enterprises response to customer requests, and increase the market share occupied by enterprises. A new vision of directions for developing e-commerce of industrial enterprises is suggested, which are determined by the need for enterprise rebranding, the development of e-commerce tools and technologies, the importance of outsourcing service automation and promotion of subscription trade. ICT is considered as integration factor that determines prospects for e-commerce development of industrial enterprises and contributes to increasing efficiency of online business management. Research results demonstrate that the use of economic and mathematical modelling is an important tool for assessing ICT impact, and its absence can negatively affect the accuracy and validity of online business management. (shrink)

This paper describes model building and manipulation in the solution of problems in mechanics. An automatic problem solver, MECHO, solving problems in several areas of mechanics, employs (1) a knowledge base representing the semantic content of the particular problem area, (2) a means-ends search strategy similar to GPS to produce sets of simultaneous equations and (3) a “focusing” technique, based on the data within the knowledge base, to guide the GSP-like search through possible equation instantiations. Sets of predicate logic statements (...) are employed to describe this model building activity. These clauses are used to give information content to the knowledge base and provide both basis and guidance for the goal driven search.It is hypothesized that humon subjects solving mechanics problems employ similar model building techniques. Protocols of several subjects are presented and comparisons are drawn with the traces of the automated problem solver. Adjustments are made to the program to provide a better fit of traces to protocols. Some implications are presented for using a rule based model for describing human problem solving performance in solving mechanics problems. (shrink)

In this contribution we propose an agent architecture for theorem proving which we intend to investigate in depth in the future. The work reported in this paper is in an early state, and by no means finished. We present and discuss our proposal in order to get feedback from the Calculemus community.

Examples in the history of Automated Theorem Proving are given, in order to show that even a seemingly ‘mechanical’ activity, such as deductive inference drawing, involves special cultural features and tacit knowledge. Mechanisation of reasoning is thus regarded as a complex undertaking in ‘cultural pruning’ of human-oriented reasoning. Sociological counterparts of this passage from human- to machine-oriented reasoning are discussed, by focusing on problems of man-machine interaction in the area of computer-assisted proof processing.

Instantiation Theory presents a new, general unification algorithm that is of immediate use in building theorem provers and logic programming systems. Instantiation theory is the study of instantiation in an abstract context that is applicable to most commonly studied logical formalisms. The volume begins with a survey of general approaches to the study of instantiation, as found in tree systems, order-sorted algebras, algebraic theories, composita, and instantiation systems. A classification of instantiation systems is given, based on properties of substitutions, degree (...) of type strictness, and well-foundedness of terms. Equational theories and the use of typed variables are studied in terms of quotient homomorphisms and embeddings, respectively. Every instantiation system is a quotient system of a subsystem of first-order term instantiation. The general unification algorithm is developed as an application of the basic theory. Its soundness is rigorously proved, and its completeness and efficiency are verfied for certain classes of instantiation systems. Appropriate applications of the algorithm include unification of first-order terms, order-sorted terms, and first-order formulas modulo alpha-conversion, as well as equational unification using simple congruences. (shrink)

We present a search engine for mathematical formulae. The MathWebSearch system harvests the web for content representations (currently MathML and OpenMath) of formulae and indexes them with substitution tree indexing, a technique originally developed for accessing intermediate results in automated theorem provers. For querying, we present a generic language extension approach that allows constructing queries by minimally annotating existing representations. First experiments show that this architecture results in a scalable application.

Rigorous formal proof is one of the key techniques in the natural sciences, engineering, and of course also in the formal sciences. Progress in automated reasoning increasingly enables computer systems to support, and even teach, users to conduct formal a.

Computer assisted theorem proving is an increasingly important part of mathematical methodology, as well as a long-standing topic in artificial intelligence (AI) research. However, the current generation of theorem proving software have limited functioning in terms of providing new proofs. Importantly, they are not able to discriminate interesting theorems and proofs from trivial ones. In order for computers to develop further in theorem proving, there would need to be a radical change in how the software functions. Recently, machine learning results (...) in solving mathematical tasks have shown early promise that deep artificial neural networks could learn symbolic mathematical processing. In this paper, I analyze the theoretical prospects of such neural networks in proving mathematical theorems. In particular, I focus on the question how such AI systems could be incorporated in practice to theorem proving and what consequences that could have. In the most optimistic scenario, this includes the possibility of autonomous automated theorem provers (AATP). Here I discuss whether such AI systems could, or should, become accepted as active agents in mathematical communities. (shrink)

During the second half of the 20th century, attempts were made to operationally redefine various social activities, including those related to science, the military, administration and industry. These attempts were aided by scientific and technical innovations developed in the Second World War, and subsequently by the increase in use of automation in various domains. This Ph.D. thesis addresses these attempts from a sociohistorical perspective, focusing on the specific case of archaeology. During this period, the domain of archaeology underwent a process (...) of disciplinarisation and professionalisation. The same occurred in applied mathematics and then computer science: this thesis focuses on the relationships between these three domains. In France, during the 1950's and 1960's, there were significant methodological and conceptual innovations. Their subsequent scientific recognition, was, however, relatively minor. In archaeology, innovations related to applied mathematics and automatics did not lead to the emergence of an archaeological speciality based on computation. This situation was in striking contrast to what happened in other scientific domains and in archaeology in other countries, where new theoretical and methodological Anglophone definitions in ‘New Archaeology’ were spreading worldwide. This thesis explores three collective attempts to redefine the conceptual and methodological basis of archaeology, led by Georges Laplace, Jean-Claude Gardin and Jean Lesage, across France, Spain and Italy. These cases are completed by other people who had significant careers in both engineering and archaeology. In general, this thesis studies a scientific activity by investigating the cognitive and social aspects of peoples’ methodological contributions. Three models of the relationships between experts in a scientific domain and experts in an applied science (here mathematics and computing) are empirically identified and described. The effects of introducing mathematical and automation procedures on the division of labour and the distribution of recognition are analysed. The success or failure of the methodological propositions are discussed with reference to several factors and models of scientific innovation. This thesis generates new information on the development of rescue and preventive archaeology and on the use of digital technologies in human sciences. The analysis draws on 82 interviews, 23 archives and several bibliometric datasets (extracted from pre-existing databases or constructed for the purpose of this research). Mirroring the archaeological propositions under study, this research also intends to illustrate the possible use of computing and formalised procedures in social sciences. The documentation and demonstrative principles underlying this work, implemented by using Wiki, the methods of literate programming and reproducible research, are themselves analysed. (shrink)

The implication relationship between subsystems in Reverse Mathematics has an underlying logic, which can be used to deduce certain new Reverse Mathematics results from existing ones in a routine way. We use techniques of modal logic to formalize the logic of Reverse Mathematics into a system that we name s-logic. We argue that s-logic captures precisely the “logical” content of the implication and nonimplication relations between subsystems in Reverse Mathematics. We present a sound, complete, decidable, and (...) compact tableau-style deductive system for s-logic, and explore in detail two fragments that are particularly relevant to Reverse Mathematics practice and automated theorem proving of Reverse Mathematics results. (shrink)

ABSTRACT A computational logic, PLPR (Predicate Logic using Polymorphism and Recursion) is presented. Actually this logic is the object language of an automated deduction system designed as a tool for proving mathematical theorems as well as specify and verify properties of functional programs. A useful denotationl semantics and two general deduction methods for PLPR are defined. The first one is a tableau algorithm proved to be complete and also used as a guideline for building complete calculi. The second is (...) a sound and complete natural deduction system. Moreover a fixed point induction rule is introduced for formulas called continuous. The strategies for mechanizing the proofs of the final automated system are based on the previous deduction methods. As the examples of the use of the system show, the implemented theorem prover outperforms humans to a certain extent, retaining logic and calculi generality. (shrink)