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.

Dependency Directed Reasoning for Complex Program Understanding

Artificial Intelligence research involves the creation of extremely complex programs which must possess the capability to introspect, learn, and improve their expertise. Any truly intelligent program must be able to create procedures and to modify them as it gathers information from its experience. [Sussman, 1975] produced such a system for a 'mini-world'; but truly intelligent programs must be considerably more complex. A crucial stepping stone in AI research is the development of a system which can understand complex programs well enough to modify them. There is also a complexity barrier in the world of commercial software which is making the cost of software production and maintenance prohibitive. Here too a system which is capable of understanding complex programs is a necessary step. The Programmer's Apprentice Project [Rich and Shrobe, 76] is attempting to develop an interactive programming tool which will help expert programmers deal with the complexity involved in engineering a large software system. This report describes REASON, the deductive component of the programmer's apprentice. REASON is intended to help expert programmers in the process of evolutionary program design. REASON utilizes the engineering techniques of modelling, decomposition, and analysis by inspection to determine how modules interact to achieve the desired overall behavior of a program. REASON coordinates its various sources of knowledge by using a dependency-directed structure which records the justification for each deduction it makes. Once a program has been analyzed these justifications can be summarized into a teleological structure called a plan which helps the system understand the impact of a proposed program modification.

SIR: A Computer Program for Semantic Information Retrieval

SIR is a computer system, programmed in the LISP language, which accepts information and answers questions expressed in a restricted form of English. This system demonstrates what can reasonably be called an ability to "understand" semantic information. SIR's semantic and deductive ability is based on the construction of an internal model, which uses word associations and property lists, for the relational information normally conveyed in conversational statements. A format-matching procedure extracts semantic content from English sentences. If an input sentence is declarative, the system adds appropriate information to the model. If an input sentence is a question, the system searches the model until it either finds the answer or determines why it cannot find the answer. In all cases SIR reports its conclusions. The system has some capacity to recognize exceptions to general rules, resolve certain semantic ambiguities, and modify its model structure in order to save computer memory space. Judging from its conversational ability, SIR, is a first step toward intelligent man-machine communication. The author proposes a next step by describing how to construct a more general system which is less complex and yet more powerful than SIR. This proposed system contains a generalized version of the SIR model, a formal logical system called SIR1, and a computer program for testing the truth of SIR1 statements with respect to the generalized model by using partial proof procedures in the predicate calculus. The thesis also describes the formal properties of SIR1 and how they relate to the logical structure of SIR.

Description and Theoretical Analysis (Using Schemata) of Planner: A Language for Proving Theorems and Manipulating Models in a Robot

Planner is a formalism for proving theorems and manipulating models in a robot. The formalism is built out of a number of problem-solving primitives together with a hierarchical multiprocess backtrack control structure. Statements can be asserted and perhaps later withdrawn as the state of the world changes. Under BACKTRACK control structure, the hierarchy of activations of functions previously executed is maintained so that it is possible to revert to any previous state. Thus programs can easily manipulate elaborate hypothetical tentative states. In addition PLANNER uses multiprocessing so that there can be multiple loci of changes in state. Goals can be established and dismissed when they are satisfied. The deductive system of PLANNER is subordinate to the hierarchical control structure in order to maintain the desired degree of control. The use of a general-purpose matching language as the basis of the deductive system increases the flexibility of the system. Instead of explicitly naming procedures in calls, procedures can be invoked implicitly by patterns of what the procedure is supposed to accomplish. The language is being applied to solve problems faced by a robot, to write special purpose routines from goal oriented language, to express and prove properties of procedures, to abstract procedures from protocols of their actions, and as a semantic base for English.

A Multiple-Context Equality-Based Reasoning System

Expert systems are too slow. This work attacks that problem by speeding up a useful system component that remembers facts and tracks down simple consequences. The redesigned component can assimilate new facts more quickly because it uses a compact, grammar-based internal representation to deal with whole classes of equivalent expressions at once. It can support faster hypothetical reasoning because it remembers the consequences of several assumption sets at once. The new design is targeted for situations in which many of the stored facts are equalities. The deductive machinery considered here supplements stored premises with simple new conclusions. The stored premises include permanently asserted facts and temporarily adopted assumptions. The new conclusions are derived by substituting equals for equals and using the properties of the logical connectives AND, Or, and NOT. The deductive system provides supporting premises for its derived conclusions. Reasoning that involves quantifiers is beyond the scope of its limited and automatic operation. The expert system of which the reasoning system is a component is expected to be responsible for overall control of reasoning.