Introduction to the constructive point of view in the foundations of mathematics, in particular intuitionism due to L.E.J. Brouwer, constructive recursive mathematics due to A.A. Markov, and Bishop’s constructive mathematics. The constructive interpretation and formalization of logic is described. For constructive (intuitionistic) arithmetic, Kleene’s realizability interpretation is given; this provides an example of the possibility of a constructive mathematical practice which diverges from classical mathematics. The crucial notion in intuitionistic analysis, choice sequence, is briefly described and some principles which are valid for choice sequences are discussed. The second half of the article deals with some aspects of proof theory, i.e., the study of formal proofs as combinatorial objects. Gentzen’s fundamental contributions are outlined: his introduction of the so-called Gentzen systems which use sequents instead of formulas and his result on first-order arithmetic showing that (suitably formalized) transfinite induction up to the ordinal "0 cannot be proved in first-order arithmetic.

MetaML is a statically typed functional programming language with special support for program generation. In addition to providing the standard features of contemporary programming languages such as Standard ML, MetaML provides three staging annotations. These staging annotations allow the construction, combination, and execution of object-programs. Our thesis is that MetaML's three staging annotations provide a useful, theoretically sound basis for building program generators. This dissertation reports on our study of MetaML's staging constructs, their use, their implementation, and their formal semantics. Our results include an extended example of where MetaML allows us to produce efficient programs, an explanation of why implementing these constructs in traditional ways can be challenging, two formulations of MetaML's semantics, a type system for MetaML, and a proposal for extending ...

Note: This document accompanies the paper “Practical type inference for arbitrary-rank types ” [6]. Prior reading of the main paper is required. 1 Contents

Let S be some type system. A typing in S for a typable term M is the collection of all of the information other than M which appears in the final judgement of a proof derivation showing that M is typable. For example, suppose there is a derivation in S ending with the judgement A M : # meaning that M has result type # when assuming the types of free variables are given by A. Then (A, #) is a typing for M .

Abstract: As introduced in the Lipari guide, Abstract State Machines (abbreviated as ASMs) are untyped. This is useful for many purposes. However, typed languages have their own advantages. Types structure the data, type checking uncovers errors. Here we propose a typed version of ASMs.

It is well-known that there is an isomorphism between natural deduction derivations and typed lambda terms. Moreover normalising these terms corresponds to eliminating cuts in the equivalent sequent calculus derivations. Several papers have been written on this topic. The correspondence between sequent calculus derivations and natural deduction derivations is, however, not a one-one map, which causes some syntactic technicalities. The correspondence is best explained by two extensionally equivalent type assignment systems for untyped lambda terms, one corresponding to natural deduction (N) and the other to sequent calculus (L). These two systems constitute different grammars for generating the same (type assignment relation for untyped) lambda terms. The second grammar is ambiguous, but the first one is not. This fact explains the many-one correspondence mentioned above. Moreover, the second type assignment system has a `cut-free' fragment (L cf ). This fragment generates exactly the typeable lambda terms in normal form. The cut elimination theorem becomes a simple consequence of the fact that typed lambda terms posses a normal form.

We show that the problems of parsing and surface realization for grammar formalisms with “context-free ” derivations, coupled with Montague semantics (under a certain restriction) can be reduced in a uniform way to Datalog query evaluation. As well as giving a polynomialtime algorithm for computing all derivation trees (in the form of a shared forest) from an input string or input logical form, this reduction has the following complexity-theoretic consequences for all such formalisms: (i) the decision problem of recognizing grammaticality (surface realizability) of an input string (logical form) is in LOGCFL; and (ii) the search problem of finding one logical form (surface string) from an input string (logical form) is in functional LOGCFL. Moreover, the generalized supplementary magic-sets rewriting of the Datalog program resulting from the reduction yields efficient Earley-style algorithms for both parsing and generation. 1

BB 0 IW logic (or T! ) is known to be D-complete. This paper shows that there are infinitely many weaker D-complete logics and it also examines how certain D-incomplete logics can be made complete by altering their axioms using simple substitutions. Keywords: condensed detachment 1 Introduction The condensed detachment rule, first proposed by C. A. Meredith in Lemmon et al [5], is a form of modus ponens preceded by `just enough' substitution to make the modus ponens possible. The substitution mechanism, for implicational formulas, was a precursor to Robinson's unification algorithm [8]. Roughly, a system of implicational logic is D-complete if the system with the same axioms, but with condensed detachment (D) instead of modus ponens and substitution, has the same theorems. To show that a logic is D-complete it is sufficient to show that all the substitution instances of its axioms are deducible in the corresponding condensed logic (i.e. the logic with rule D only). It is well know...

Abstract. We investigate natural deduction proofs of the Lambek calculus from the point of view of tree automata. The main result is that the set of proofs of the Lambek calculus cannot be accepted by a finite tree automaton. The proof is extended to cover the proofs used by grammars based on the Lambek calculus, which typically use only a subset of the set of all proofs. While Lambek grammars can assign regular tree languages as structural descriptions, there exist Lambek grammars that assign non-regular structural descriptions, both when considering normal and non-normal proof trees. Combining the results of Pentus (1993) and Thatcher (1967), we can conclude that Lambek grammars, although generating only context-free languages, can extend the strong generative capacity of context-free grammars. Furthermore, we show that structural descriptions that disregard the use of introduction rules cannot be used for a compositional semantics following the Curry-Howard isomorphism. 1

We give a polymorphic account of the relational algebra. We introduce a formalism of “type formulas ” specifically tuned for relational algebra expressions, and present an algorithm that computes the “principal ” type for a given expression. The principal type of an expression is a formula that specifies, in a clear and concise manner, all assignments of types (sets of attributes) to relation names, under which a given relational algebra expression is well-typed, as well as the output type that expression will have under each of these assignments. Topics discussed include complexity and polymorphic expressive power. 1