Dynamic reasoning without variables

A variable free notation for dynamic logic is proposed which takes its cue from De Bruijn's variable free notation for lambda calculus. De Bruijn indexing replaces variables by indices which indicate the distance to their binders. We propose to use reverse De Bruijn indexing, which works almost the same, only now the indices refer to the depth of the binding operator in the formula. The resulting system is analysed at length and applied to a new rational reconstruction of discourse representation theory. It is argued that the present system of dynamic logic without variables provides an explicit account of anaphoric context and yields new insight into the dynamics of anaphoric linking in reasoning. A calculus for dynamic reasoning with anaphora is presented and its soundness and completeness are established

Type-omega DPLs

Type-omega DPLs (Denotational Proof Languages) are languages for proof presentation and search that offer strong soundness guarantees. LCF-type systems such as HOL offer similar guarantees, but their soundness relies heavily on static type systems. By contrast, DPLs ensure soundness dynamically, through their evaluation semantics; no type system is necessary. This is possible owing to a novel two-tier syntax that separates deductions from computations, and to the abstraction of assumption bases, which is factored into the semantics of the language and allows for sound evaluation. Every type-omega DPL properly contains a type-alpha DPL, which can be used to present proofs in a lucid and detailed form, exclusively in terms of primitive inference rules. Derived inference rules are expressed as user-defined methods, which are "proof recipes" that take arguments and dynamically perform appropriate deductions. Methods arise naturally via parametric abstraction over type-alpha proofs. In that light, the evaluation of a method call can be viewed as a computation that carries out a type-alpha deduction. The type-alpha proof "unwound" by such a method call is called the "certificate" of the call. Certificates can be checked by exceptionally simple type-alpha interpreters, and thus they are useful whenever we wish to minimize our trusted base. Methods are statically closed over lexical environments, but dynamically scoped over assumption bases. They can take other methods as arguments, they can iterate, and they can branch conditionally. These capabilities, in tandem with the bifurcated syntax of type-omega DPLs and their dynamic assumption-base semantics, allow the user to define methods in a style that is disciplined enough to ensure soundness yet fluid enough to permit succinct and perspicuous expression of arbitrarily sophisticated derived inference rules. We demonstrate every major feature of type-omega DPLs by defining and studying NDL-omega, a higher-order, lexically scoped, call-by-value type-omega DPL for classical zero-order natural deduction---a simple choice that allows us to focus on type-omega syntax and semantics rather than on the subtleties of the underlying logic. We start by illustrating how type-alpha DPLs naturally lead to type-omega DPLs by way of abstraction; present the formal syntax and semantics of NDL-omega; prove several results about it, including soundness; give numerous examples of methods; point out connections to the lambda-phi calculus, a very general framework for type-omega DPLs; introduce a notion of computational and deductive cost; define several instrumented interpreters for computing such costs and for generating certificates; explore the use of type-omega DPLs as general programming languages; show that DPLs do not have to be type-less by formulating a static Hindley-Milner polymorphic type system for NDL-omega; discuss some idiosyncrasies of type-omega DPLs such as the potential divergence of proof checking; and compare type-omega DPLs to other approaches to proof presentation and discovery. Finally, a complete implementation of NDL-omega in SML-NJ is given for users who want to run the examples and experiment with the language

Certified Computation

This paper introduces the notion of certified computation. A certified computation does not only produce a result r, but also a correctness certificate, which is a formal proof that r is correct. This can greatly enhance the credibility of the result: if we trust the axioms and inference rules that are used in the certificate,then we can be assured that r is correct. In effect,we obtain a trust reduction: we no longer have to trust the entire computation; we only have to trust the certificate. Typically, the reasoning used in the certificate is much simpler and easier to trust than the entire computation. Certified computation has two main applications: as a software engineering discipline, it can be used to increase the reliability of our code; and as a framework for cooperative computation, it can be used whenever a code consumer executes an algorithm obtained from an untrusted agent and needs to be convinced that the generated results are correct. We propose DPLs (Denotational Proof Languages)as a uniform platform for certified computation. DPLs enforce a sharp separation between logic and control and over versatile mechanicms for constructing certificates. We use Athena as a concrete DPL to illustrate our ideas, and we present two examples of certified computation, giving full working code in both cases

On the proper treatment of context in NL

The proper treatment of quantification in Natural Language proposed by Richard Montague some thirty years ago does not do proper justice to the fact that interpretation of texts both uses context and sets up new contexts. The dynamic turn in NL semantics is the attempt to model this basic fact, but the use of dynamically quantified variables introduces an undesirable element into this attempt. By extending a variable free `incremental dynamics' with a flexible system of type scheme patterns and type scheme pattern matching, we arrive at a Montague style architecture for NL semantics that provides a proper treatment both of quantification and of context use and context change

Perturbations and Stability of Black Ellipsoids

We study the perturbations of two classes of static black ellipsoid solutions of four dimensional vacuum Einstein equations. Such solutions are described by generic off--diagonal metrics which are generated by anholonomic transforms of diagonal metrics. The analysis is performed in the approximation of small eccentricity deformations of the Schwarzschild solution. We conclude that such anisotropic black hole objects may be stable with respect to the perturbations parametrized by the Schrodinger equations in the framework of the one--dimensional inverse scattering theory.Comment: Published variant in IJMD with small modifications in formulas and new reference

Denotational proof languages

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2000.Includes bibliographical references (p. [417]-421).by Konstantinos Arkoudas.Ph.D

A graph-theoretic account of logics

A graph-theoretic account of logics is explored based on the general notion of m-graph (that is, a graph where each edge can have a finite sequence of nodes as source). Signatures, interpretation structures and deduction systems are seen as m-graphs. After defining a category freely generated by a m-graph, formulas and expressions in general can be seen as morphisms. Moreover, derivations involving rule instantiation are also morphisms. Soundness and completeness theorems are proved. As a consequence of the generality of the approach our results apply to very different logics encompassing, among others, substructural logics as well as logics with nondeterministic semantics, and subsume all logics endowed with an algebraic semantics