It is generally accepted that in principle it’s possible to formalize completely almost all of present-day mathematics. The practicability of actually doing so is widely doubted, as is the value of the result. But in the computer age we believe that such formalization is possible and desirable. In contrast to the QED Manifesto however, we do not offer polemics in support of such a project. We merely try to place the formalization of mathematics in its historical perspective, as well as looking at existing praxis and identifying what we regard as the most interesting issues, theoretical and practical.

In this note, we reprove a theorem of Statman that deciding the fij-equality of two firstorder typable -terms is not elementary recursive [Sta79]. The basic idea of our proof, like that of Statman's, is the Henkin quantifier elimination procedure [Hen63]. However, our coding is much simpler, and easy to understand. 1 Introduction A well known theorem of Richard Statman states that if we have two -terms that are first-order typable, deciding whether the terms reduce to the same normal form is not Kalmar elementary: namely, it cannot be decided in f k (n) steps for any fixed integer k 0, where n is the length of the two terms, and f 0 (n) = n, f t+1 (n) = 2 f t (n) . The theorem is often cited, but in contrast, its proof is not well understood. In this note, we give a simple proof of the theorem. The key idea that vastly simplifies the technical details of the proof is to use list iteration as a quantifier elimination procedure. 2 Preliminaries 2.1 Deciding truth of formulas in high...

We investigate complexity of the SUCCESS problem for logic query languages with complex values: check whether a query defines a nonempty set. The SUCCESS problem for recursive query languages with complex values is undecidable, so we study the complexity of nonrecursive queries. By complex values we understand values such as trees, finite sets, and multisets. Due to the well-known correspondence between relational query languages and datalog, our results can be considered as results about relational query languages with complex values. The paper gives a complete complexity classification of the SUCCESS problem for nonrecursive logic programs over trees depending on the underlying signature, presence of negation, and range restrictedness. We also prove several results about finite sets and multisets. 1 Introduction A number of complexity results have been established for logic query languages. They are surveyed in [49, 18]. The major themes in these results are the complexity and expr...

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We prove that any decision procedure for a modest fragment of L. Henkin's theory of pure propositional types [7, 12, 15, 11] requires time exceeding a tower of 2's of height exponential in the length of input. Until now the highest known lower bounds for natural decidable theories were at most linearly high towers of 2's and since mid-seventies it was an open problem whether natural decidable theories requiring more than that exist [12, 2]. We give the affirmative answer. As an application of this today's strongest lower bound we improve known and settle new lower bounds for several problems in the simply typed lambda calculus. 1. Introduction In his survey paper [12] A. Meyer mentioned (p. 479), as a curious empirical observation, that all known natural decidable non-elementary problems require at most (upper bound) F (1; n) = exp 1 (n) = 2 2 \Delta \Delta \Delta 2 oe n Turing machine steps on inputs of length n to decide 1 . Until now the highest known lower bounds for natu...

ion is in a precise sense a converse operation to application. Given 49 50 CHAPTER 5. PRIMITIVE BASIS OF HOL LIGHT a variable x and a term t, which may or may not contain x, one can construct the so-called lambda-abstraction x: t, which means `the function of x that yields t'. (In HOL's ASCII concrete syntax the backslash is used, e.g. \x. t.) For example, x: x + 1 is the function that adds one to its argument. Abstractions are not often seen in informal mathematics, but they have at least two merits. First, they allow one to write anonymous function-valued expressions without naming them (occasionally one sees x 7! t[x] used for this purpose), and since our logic is avowedly higher order, it's desirable to place functions on an equal footing with rstorder objects in this way. Secondly, they make variable dependencies and binding explicit; by contrast in informal mathematics one often writes f(x) in situations where one really means x: f(x). We should give some idea of how ordina...

Abstract. Classical logic predicts that everything (thus nothing useful at all) follows from inconsistency. A paraconsistent logic is a logic where an inconsistency does not lead to such an explosion, and since in practice consistency is difficult to achieve there are many potential applications of paraconsistent logics in knowledge-based systems, logical semantics of natural language, etc. Higher order logics have the advantages of being expressive and with several automated theorem provers available. Also the type system can be helpful. We present a concise description of a paraconsistent higher order logic with countable infinite indeterminacy, where each basic formula can get its own indeterminate truth value (or as we prefer: truth code). The meaning of the logical operators is new and rather different from traditional many-valued logics as well as from logics based on bilattices. The adequacy of the logic is examined by a case study in the domain of medicine. Thus we try to build a bridge between the HOL and MVL communities. A sequent calculus is proposed based on recent work by Muskens. Many non-classical logics are, at the propositional level, funny toys which work quite good, but when one wants to extend them to higher levels to get a real logic that would enable one to do mathematics or other more sophisticated reasonings, sometimes dramatic troubles appear.

This paper presents an extended version of Church's simple type theory called Basic Extended Simple Type Theory (bestt). By adding type variables and support for reasoning with tuples, lists, and sets to simple type theory, it is intended to be a practical logic for formalized mathematics. 1

In the tradition of Denotational Semantics one usually lets program constructs take their denotations in reflexive domains, i.e. in domains where self-application is possible. For the bulk of programming constructs, however, working with reflexive domains is an unnecessary complication. In this paper we shall use the domains of ordinary classical type logic to provide the semantics of a simple programming language containing choice and recursion. We prove that the rule of Scott Induction holds in this new setting, prove soundness of a Hoare calculus relative to our semantics, give a direct calculus C on programs, and prove that the denotation of any program P in our semantics is equal to the union of the denotations of all those programs L such that P follows from L in our calculus and L does not contain recursion or choice. 1

ion is in a precise sense a converse operation to application. Given 49 50 CHAPTER 5. PRIMITIVE BASIS OF HOL LIGHT a variable x and a term t, which may or may not contain x, one can construct the so-called lambda-abstraction x: t, which means `the function of x that yields t'. (In HOL's ASCII concrete syntax the backslash is used, e.g. "x. t.) For example, x: x + 1 is the function that adds one to its argument. Abstractions are not often seen in informal mathematics, but they have at least two merits. First, they allow one to write anonymous function-valued expressions without naming them (occasionally one sees x 7! t[x] used for this purpose), and since our logic is avowedly higher order, it's desirable to place functions on an equal footing with firstorder objects in this way. Secondly, they make variable dependencies and binding explicit; by contrast in informal mathematics one often writes f(x) in situations where one really means x: f(x). We should give some idea of how ordinary...