This paper is concerned with decision proceedures for the 0-valued ukasiewicz logics,. It is shown how linear algebra can be used to construct an automated theorem checker. Two decision proceedures are described which depend on a linear programming package. An algorithm is given for the verification of consequence relations in, and a connection is made between theorem checking in two-valued logic and theorem checking in which implies that determing of a -free formula whether it takes the (...) value one is NP-complete problem. (shrink)

The history of building automated theorem provers for higher-order logic is almost as old as the field of deduction systems itself. The first successful attempts to mechanize and implement higher-order logic were those of Huet [13] and Jensen and Pietrzykowski [17]. They combine the resolution principle for higher-order logic (first studied in [1]) with higher-order unification. The unification problem in typed λ-calculi is much more complex than that for first-order terms, since it has to take the theory of (...) αβη-equality into account. As a consequence, the higher-order unification problem is undecidable and sets of solutions need not even always have most general elements that represent them. Thus the mentioned calculi for higher-order logic have take special measures to circumvent the problems posed by the theoretical complexity of higher-order unification. In this paper, we will exemplify the methods and proof- and model-theoretic tools needed for extending first-order automated theorem proving to higherorder logic. For the sake of simplicity take the tableau method as a basis (for a general introduction to first-order tableaux see part I.1) and discuss the higherorder tableau calculi HT and HTE first presented in [19]. The methods in this paper also apply to higher-order resolution calculi [1, 13, 6] or the higher-order matings method of Peter [3], which extend their first-order counterparts in much the same way. Since higher-order calculi cannot be complete for the standard semantics by Gödel’s incompleteness theorem [11], only the weaker notion of Henkin models [12] leads to a meaningful notion of completeness in higher-order logic. It turns out that the calculi in [1, 13, 3, 19] are not Henkin-complete, since they fail to capture the extensionality principles of higher-order logic. We will characterize the deductive power of our calculus HT (which is roughly equivalent to these calculi) by the semantics of functional Σ-models. To arrive at a calculus that is complete with respect to Henkin models, we build on ideas from [6] and augment HT with tableau construction rules that use the extensionality principles in a goal-oriented way.. (shrink)

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)

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.

Formal verification involves the use of logical and computational methods to establish claims that are expressed in precise mathematical terms. These can include ordinary mathematical theorems, as well as claims that pieces of hardware or software, network protocols, and mechanical and hybrid systems meet their specifications. In practice, there is not a sharp distinction between verifying a piece of mathematics and verifying the correctness of a system: formal verification requires describing hardware and software systems in mathematical terms, at which point (...) establishing claims as to their correctness becomes a form of theorem proving. Conversely, the proof of a mathematical theorem may require a lengthy computation, in which case verifying the truth of the theorem requires verifying that the computation does what it is supposed to do. The gold standard for supporting a mathematical claim is to provide a proof, and twentieth-century developments in logic show most if not all conventional proof methods can be reduced to a small set of axioms and rules in any of a number of foundational systems. With this reduction, there are two ways that a computer can help establish a claim: it can help find a proof in the first place, and it can help verify that a purported proof is correct. Automated theorem proving focuses on the “finding” aspect. Resolution theorem provers, tableau theorem provers, fast satisfiability solvers, and so on provide means of establishing the validity of formulas in propositional and first-order logic. Other systems provide search procedures and decision procedures for specific languages and domains, such as linear or nonlinear expressions over the integers or the real numbers. Architectures like SMT combine domain-general search methods with domain-specific procedures. Computer algebra systems and specialized mathematical software packages provide means of carrying out mathematical computations, establishing mathematical bounds, or finding mathematical objects. A calculation can be viewed as a proof as well, and these systems, too, help establish mathematical claims. Automated reasoning systems strive for power and efficiency, often at the expense of guaranteed soundness. Such systems can have bugs, and it can be difficult to ensure that the results they deliver are correct. In contrast, interactive theorem proving focuses on the “verification” aspect of theorem proving, requiring that every claim is supported by a proof in a suitable axiomatic foundation. This sets a very high standard: every rule of inference and every step of a calculation has to be justified by appealing to prior definitions and theorems, all the way down to basic axioms and rules. In fact, most such systems provide fully elaborated “proof objects” that can be communicated to other systems and checked independently. Constructing such proofs typically requires much more input and interaction from users, but it allows us to obtain deeper and more complex proofs. The Lean Theorem Prover aims to bridge the gap between interactive and automated theorem proving, by situating automated tools and methods in a framework that supports user interaction and the construction of fully specified axiomatic proofs. The goal is to support both mathematical reasoning and reasoning about complex systems, and to verify claims in both domains. (shrink)

A growing number of applications of automated reasoning exhibitsthe necessity of flexible deduction systems. A deduction system should beable to execute inference rules which are appropriate to the given problem.One way to achieve this behavior is the integration of different calculi. Thisled to so called hybrid reasoning [22, 1, 10, 20] which means the integrationof a general purpose foreground reasoner with a specialized background reasoner. A typical task of a background reasoner is to perform special purposeinference rules according to (...) a built-in theory. The aim of this paper is togo a step further, i.e. to treat the background reasoner as a hybrid systemitself. The paper formulates sufficient criteria for the construction of complete calculi which enable reasoning under hybrid theories combined fromsub-theories. For this purpose we use a generic approach described in [20].This more detailed view on built-in theories is not covered by the knowngeneral approaches [1, 3, 6, 20] for building in theories into theorem provers.The approach is demonstrated by its application to the target calculi of thealgebraic translation [9] of multi-modal and extended multi-modal [7] logicto first-order logic. (shrink)

This books presents the refereed proceedings of the Fifth International Workshop on Analytic Tableaux and Related Methods, TABLEAUX '96, held in Terrasini near Palermo, Italy, in May 1996. The 18 full revised papers included together with two invited papers present state-of-the-art results in this dynamic area of research. Besides more traditional aspects of tableaux reasoning, the collection also contains several papers dealing with other approaches to automated reasoning. The spectrum of logics dealt with covers several nonclassical logics, including modal, (...) intuitionistic, many-valued, temporal and linear logic. (shrink)

THINKER is an automated natural deduction first-order theorem proving program. This paper reports on how it was adapted so as to prove theorems in modal logic. The method employed is an indirect semantic method, obtained by considering the semantic conditions involved in being a valid argument in these modal logics. The method is extended from propositional modal logic to predicate modal logic, and issues concerning the domain of quantification and existence in a world's domain are discussed. Finally, (...) we look at the very interesting issues involved with adding identity to the theorem prover in the realm of modal predicate logic. Various alternatives are discussed. (shrink)

We present a method for integrating rippling-based rewriting into matrix-based theorem proving as a means for automating inductive specification proofs. The selection of connections in an inductive matrix proof is guided by symmetries between induction hypothesis and induction conclusion. Unification is extended by decision procedures and a rippling/reverse-rippling heuristic. Conditional substitutions are generated whenever a uniform substitution is impossible. We illustrate the integrated method by discussing several inductive proofs for the integer square root problem as well as the (...) algorithms extracted from these proofs. (shrink)

We investigate several approaches to resolution based automated theoremproving in classical higher-order logic (based on Church's simply typedλ-calculus) and discuss their requirements with respect to Henkincompleteness and full extensionality. In particular we focus on Andrews' higher-order resolution (Andrews 1971), Huet's constrained resolution (Huet1972), higher-order E-resolution, and extensional higher-order resolution(Benzmüller and Kohlhase 1997). With the help of examples we illustratethe parallels and differences of the extensionality treatment of these approachesand demonstrate that extensional higher-order resolution is the sole approach thatcan completely (...) avoid additional extensionality axioms. (shrink)

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)

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)

Smullyan wrote his famous book of puzzles before the boom in automated theorem proving and he solved the puzzles by hand. Hence it is interesting to investigate whether all the puzzles can be solved with one method or not. The paper shows how this can be done with analytic tableaux.

This work provides proof-search algorithms and automated counter-model extraction for a class of STIT logics. With this, we answer an open problem concerning syntactic decision procedures and cut-free calculi for STIT logics. A new class of cut-free complete labelled sequent calculi G3LdmL^m_n, for multi-agent STIT with at most n-many choices, is introduced. We refine the calculi G3LdmL^m_n through the use of propagation rules and demonstrate the admissibility of their structural rules, resulting in auxiliary calculi Ldm^m_nL. In the single-agent case, (...) we show that the refined calculi Ldm^m_nL derive theorems within a restricted class of (forestlike) sequents, allowing us to provide proof-search algorithms that decide single-agent STIT logics. We prove that the proof-search algorithms are correct and terminate. (shrink)

Lean is a new open source theorem prover being developed at Microsoft Research and Carnegie Mellon University, with a small trusted kernel based on dependent type theory. It aims to bridge the gap between interactive and automated theorem proving, by situating automated tools and methods in a framework that supports user interaction and the construction of fully specified axiomatic proofs. Lean is an ongoing and long-term effort, but it already provides many useful components, integrated development (...) environments, and a rich API which can be used to embed it into other systems. It is currently being used to formalize category theory, homotopy type theory, and abstract algebra. We describe the project goals, system architecture, and main features, and we discuss applications and continuing work. (shrink)

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.

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)

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 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)

Thus, despite the di culty of higher-order automated theorem proving, which has to deal with problems like the undecidability of higher-order uni - cation (HOU) and the need for primitive substitution, there are proof problems which lie beyond the capabilities of rst-order theorem provers, but instead can be solved easily by an higher-order theorem prover (HOATP) like Leo. This is due to the expressiveness of higher-order Logic and, in the special case of Leo, due to (...) an appropriate handling of the extensionality principles (functional extensionality and extensionality on truth values). (shrink)

This is the 3rd edition. Although a number of new technological applications require classical deductive computation with non-classical logics, many key technologies still do well—or exclusively, for that matter—with classical logic. In this first volume, we elaborate on classical deductive computing with classical logic. The objective of the main text is to provide the reader with a thorough elaboration on both classical computing – a.k.a. formal languages and automata theory – and classical deduction with the classical first-order predicate calculus with (...) a view to computational implementations, namely in automated theorem proving and logic programming. The present third edition improves on the previous ones by providing an altogether more algorithmic approach: There is now a wholly new section on algorithms and there are in total fourteen clearly isolated algorithms designed in pseudo-code. Other improvements are, for instance, an emphasis on functions in Chapter 1 and more exercises with Turing machines. (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)

Benchmarking automated theorem proving (ATP) systems using standardized problem sets is a well-established method for measuring their performance. However, the availability of such libraries for non-classical logics is very limited. In this work we propose a library for benchmarking Girard's (propositional) intuitionistic linear logic. For a quick bootstrapping of the collection of problems, and for discussing the selection of relevant problems and understanding their meaning as linear logic theorems, we use translations of the collection of Kleene's intuitionistic (...) theorems in the traditional monograph "Introduction to Metamathematics". We analyze four different translations of intuitionistic logic into linear logic and compare their proofs using a linear logic based prover with focusing. In order to enhance the set of problems in our library, we apply the three provability-preserving translations to the propositional benchmarks in the ILTP Library. Finally, we generate a comprehensive set of reachability problems for Petri nets and encode such problems as linear logic sequents, thus enlarging our collection of problems. (shrink)

Automated theorem proving amounts to solving search problems in usually tremendous search spaces. A lot of research therefore focuses on search space reductions. Our approach reduces the search space which arises when using so-called connection tableau calculi for first-order automated theorem proving. It uses disjunctive constraints over first-order equations to compress certain parts of this search space. We present the basics of our constrained-connection-tableau calculi, a constraint extension of connection tableau calculi, and deal with (...) the efficient handling of constraints during the search process. The new techniques are integrated into the automated connection tableau prover Setheo. (shrink)

Automated theorem proving amounts to solving search problems in usually tremendous search spaces. A lot of research therefore focuses on search space reductions. Our approach reduces the search space which arises when using so-called connection tableau calculi for first-order automated theorem proving. It uses disjunctive constraints over first-order equations to compress certain parts of this search space. We present the basics of our constrained-connection-tableau calculi, a constraint extension of connection tableau calculi, and deal with (...) the efficient handling of constraints during the search process. The new techniques are integrated into the automated connection tableau prover Setheo. (shrink)

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)

In this paper we focus on theorem proving for conditional logics. First, we give a detailed description of CondLean, a theorem prover for some standard conditional logics. CondLean is a SICStus Prolog implementation of some labeled sequent calculi for conditional logics recently introduced. It is inspired to the so called “lean” methodology, even if it does not fit this style in a rigorous manner. CondLean also comprises a graphical interface written in Java. Furthermore, we introduce a goal-directed (...) proof search mechanism, derived from the above mentioned sequent calculi based on the notion of uniform proofs. Finally, we describe GOALDUCK, a simple SICStus Prolog implementation of the goal-directed calculus mentioned here above. Both the programs CondLean and GOALDUCK, together with their source code, are available for free download at. (shrink)

Computers may help us to better understand (not just verify) arguments. In this article we defend this claim by showcasing the application of a new, computer-assisted interpretive method to an exemplary natural-language ar- gument with strong ties to metaphysics and religion: E. J. Lowe’s modern variant of St. Anselm’s ontological argument for the existence of God. Our new method, which we call computational hermeneutics, has been particularly conceived for use in interactive-automated proof assistants. It aims at shedding light on (...) the meanings of words and sentences by framing their inferential role in a given argument. By employing automated theorem reasoning technology within interactive proof assistants, we are able to drastically reduce (by several orders of magnitude) the time needed to test the logical validity of an argu- ment’s formalization. As a result, a new approach to logical analysis, inspired by Donald Davidson’s account of radical interpretation, has been enabled. In computational hermeneutics, the utilization of automated reasoning tools ef- fectively boosts our capacity to expose the assumptions we indirectly commit ourselves to every time we engage in rational argumentation and it fosters the explicitation and revision of our concepts and commitments. (shrink)

This book contains an introduction to symbolic logic and a thorough discussion of mechanical theorem proving and its applications. The book consists of three major parts. Chapters 2 and 3 constitute an introduction to symbolic logic. Chapters 4–9 introduce several techniques in mechanical theorem proving, and Chapters 10 an 11 show how theorem proving can be applied to various areas such as question answering, problem solving, program analysis, and program synthesis.

Higher-Order Logic (HOL) is a proof development system intended for applications to both hardware and software. It is principally used in two ways: for directly proving theorems, and as theorem-proving support for application-specific verification systems. HOL is currently being applied to a wide variety of problems, including the specification and verification of critical systems. Introduction to HOL provides a coherent and self-contained description of HOL containing both a tutorial introduction and most of the material that is needed (...) for day-to-day work with the system. After a quick overview that gives a "hands-on feel" for the way HOL is used, there follows a detailed description of the ML language. The logic that HOL supports and how this logic is embedded in ML, are then described in detail. This is followed by an explanation of the theorem-proving infrastructure provided by HOL. Finally two appendices contain a subset of the reference manual, and an overview of the HOL library, including an example of an actual library documentation. (shrink)

\ is a 7000 line Prolog parser/theorem-prover for logical categorial grammar. In such logical categorial grammar syntax is universal and grammar is reduced to logic: an expression is grammatical if and only if an associated logical statement is a theorem of a fixed calculus. Since the syntactic component is invariant, being the logic of the calculus, logical categorial grammar is purely lexicalist and a particular language model is defined by just a lexical dictionary. The foundational logic of continuity (...) was established by Lambek while a corresponding extension including also logic of discontinuity was established by Morrill and Valentín :167–192, 2010). \ implements a logic including as primitive connectives the continuous and discontinuous connectives of the displacement calculus, additives, 1st order quantifiers, normal modalities, bracket modalities, and universal and existential subexponentials. In this paper we review the rules of inference for these primitive connectives and their linguistic applications, and we survey the principles of Andreoli’s focusing, and of a generalisation of van Benthem’s count-invariance, on the basis of which \ is implemented. (shrink)

Uniform proofs systems have recently been proposed [Mi191j as a proof-theoretic foundation and generalization of logic programming. In [Mom92a] an extension with constructive negation is presented preserving the nature of abstract logic programming language. Here we adapt this approach to provide a complete theorem proving technique for minimal, intuitionistic and classical logic, which is totally goal-oriented and does not require any form of ancestry resolution. The key idea is to use the Godel-Gentzen translation to embed those logics in (...) the syntax of Hereditary Harrop formulae, for which uniform proofs are complete. We discuss some preliminary implementation issues. (shrink)

ABSTRACT We are concerned with the theorem proving in annotated logics. By using annotated polynomials to express knowledge, we develop an inference rule superposition. A proof procedure is thus presented, and an improvement named M- strategy is mainly described. This proof procedure uses single overlaps instead of multiple overlaps, and above all, both the proof procedure and M-strategy are refutationally complete.

A system for the modal logic K furnishes a simple mechanical process for proving theorems.