Howard A. Blair

The expressiveness of locally stratified programs

This paper completes an investigation of the logical expressibility of finite, locally stratified, general logic programs. We show that every hyperarithmetic set can be defined by a suitably chosen locally stratified logic program (as a set of values of a predicate over its perfect model). This is an optimal result, since the perfect model of a locally stratified program is itself an implicitly definable hyperarithmetic set (under a recursive coding of the Herbrand base); hence, to obtain all hyperarithmetic sets requires something new, in this case selecting one predicate from the model. We find that the expressive power of programs does not increase when one considers the programs which have a unique stable model or a total well-founded model. This shows that all these classes of structures (perfect models of logically stratified logic programs, well-founded models which turn out to be total, and stable models of programs possessing a unique stable model) are all closely connected with Kleenes hyperarithmetical hierarchy. Thus, for general logic programming, negation with respect to two-valued logic is related to the hyperarithmetic hierarchy in the same way as Horn logic is to the class of recursively enumerable sets. In particular, a set is definable in the well-founded semantics by a programP whose well-founded partial model is total iff it is hyperarithmetic.

Paraconsistent logic programming

This paper makes two contributions. Firstly, we give a semantics for sets of clauses of the form L0 ⇐ L1& ... &L n where each L i is a literal. We call such clauses generally-Horn clauses. Any such endeavour has to give a coherent, formal treatment of inconsistency (in the sense of two-valued logic). Thus, as a second contribution, we give a robust semantics for generally-Horn programs that allows us to “make sense” of sets of generally-Horn clauses that are inconsistent (in the two-valued logic sense). This applies to the design of very large knowledge bases where inconsistent information is often present.

Computer aided reasoning

We present a language intended to be a first step in approximating the language of mathematical papers, and a validator; that is, a program that checks the validity of arguments written in this language. The validator approximates the activity of a mathematician in certifying the structure and correctness of the mathematical argument. Both components together constitute a computer aided reasoning (CAR) system. Versions of the system, called MIZAR, have been in use for a decade for discrete mathematics instruction. We are concerned here with the features of the MIZAR language that that are used to diminish the gap between formal natural deduction and mathematical vernacular. The inference checking component of the validator can be easily changed. We demonstrate the influence on the way the expressive power of the language can be exploited by contrasting, for a fixed proposition, two proofs embodying the same idea but for which different checking modules have been used. The more powerful inference checker that we discuss incorporates the formalization of obviousness given by M. Davis.

Set based logic programming

In a previous paper (Blair et al. 2001), the authors showed that the mechanism underlying Logic Programming can be extended to handle the situation where the atoms are interpreted as subsets of a given space X. The view of a logic program as a one-step consequence operator along with the concepts of supported and stable model can be transferred to such situations. In this paper, we show that we can further extend this paradigm by creating a new one-step consequence operator by composing the old one-step consequence operator with a monotonic idempotent operator (miop) in the space of all subsets of X, 2X. We call this extension set based logic programming. We show that such a set based formalism for logic programming naturally supports a variety of options. For example, if the underlying space has a topology, one can insist that the new one-step consequence operator always produces a closed set or always produces an open set. The flexibility inherent in the semantics of set based logic programs is due to both the range of natural choices available for specifying the semantics of negation, as well as the role of monotonic idempotent operators (miops) as parameters in the semantics. This leads to a natural type of polymorphism for logic programming, i.e. the same logic program can produce a variety of outcomes depending on the miop associated with the semantics. We develop a general framework for set based programming involving miops. Among the applications, we obtain integer-based representations of real continuous functions as stable models of a set based logic program.

A continuum of discrete systems

We show how to regard covered logic programs as cellular automata. Covered logic programs are ones for which every variable occurring in the body of a given clause also occurs in the head of the same clause. We generalize the class of register machine programs to permit negative literals and characterize the members of this class of programs as n-state 2-dimensional cellular automata. We show how monadic covered programs, the class of which is computationally universal, can be regarded as 1-dimensional cellular automata. We show how to continuously (and differentiably) deform 1-dimensional cellular automata from one to another and understand the arrangement of these cellular automata in a separable Hilbert space over the real numbers. The embedding of the cellular automata of fixed radius r is a linear mapping into R22r+1 in which a cellular automatons transition function is the attractor of a state-governed iterated function system of affine contraction mappings. The class of covered monadic programs having a particular fixed point has a uniform arrangement in an affine subspace of the Hilbert space l2. Furthermore, these programs are construable as almost everywhere continuous functions from the unit interval {x | 0 ≤ x ≤ 1} to the real numbers R. As one consequence, in particular, we can define a variety of natural metrics on the class of these programs. Moreover, for each program in this class, the set of initial segments of the programs fixed points, with respect to an ordering induced by the programs dependency relation, is a regular set.

Simulations between Programs as Cellular Automata

We present cellular automata on appropriate digraphs and show that any covered normal logic program is a cellular automaton. Seeing programs as cellular automata shifts attention from classes of Herbrand models to orbits of Herbrand interpretations. Orbits capture both the declarative, model-theoretic meaning of programs as well as their inferential behavior. Logically and intentionally different programs can produce orbits that simulate each other. Simple examples of such behavior are compellingly exhibited with space-time diagrams of the programs as cellular automata. Construing a program as a cellular automaton leads to a general method for simulating any covered program with a Horn clause program. This means that orbits of Horn programs are completely representative of orbits of covered normal programs.

Definite clause programs are canonical (over a suitable domain)

For each first-order languageL with a nonempty Herbrand universe, we construct an algebraC interpreting the function symbols ofL that is a model of the Clark equality theory with languageL and is canonical in the sense that for every definite clause programP in the languageL,TPC ↓ ω is the greatest fixed point ofTPC. The universe of individuals inC is a quotient of the set of terms ofL and is, a fortiori, countable ifL is countable. If ℒ contains at least one function symbol of arity at least 2, then the graphs of partial recursive functions onC, suitably defined, are representable in a natural way as individuals inC.

Continuous Models of Computation for Logic Programs: Importing Continuous Mathematics into Logic Programming’s Algorithmic Foundations

Logic programs may be construed as discrete-time and continuous-time dynamical systems with continuous states. Techniques for obtaining explicit formulations of such dynamical systems are presented and the computational performance of examples is presented. Extending 2-valued and n-valued logic to continuous-valued logic is shown to be unique, up to choosing the representations of the individual truth values as elements of a continuous field, provided that lowest degree polynomials are selected. In the case of 2-valued logic, the constraint that enables the uniqueness of the continualization is that the Jacobian matrices of the continualizations of the Boolean connectives have only affine entries. This property of the Jacobian matrix facilitates computation via gradient descent methods.

General Model Theoretic Semantics for Higher-Order Horn Logic Programming

We introduce model-theoretic semantics [6] for Higher-Order Horn logic programming language. One advantage of logic programs over conventional non-logic programs has been that the least fixpoint is equal to the least model, therefore it is associated to logical consequence and has a meaningful declarative interpretation. In simple theory of types [9] on which Higher-Order Horn logic programming language is based, domain is dependent on interpretation [10]. To define T P operator for a logic program P, we need a fixed domain without regard to interpretation which is usually taken to be a set of atomic propositions. We build a semantics where we can fix a domain while changing interpretations. We also develop a fixpoint semantics based on our model, and show that we can get the least fixpoint which is the least model. Using this fixpoint we prove the completeness of the interpreter of our language in [14].

Topological Signals of Singularities in Ricci Flow

We implement methods from computational homology to obtain a topological signal of singularity formation in a selection of geometries evolved numerically by Ricci flow. Our approach, based on persistent homology, produces precise, quantitative measures describing the behavior of an entire collection of data across a discrete sample of times. We analyze the topological signals of geometric criticality obtained numerically from the application of persistent homology to models manifesting singularities under Ricci flow. The results we obtain for these numerical models suggest that the topological signals distinguish global singularity formation (collapse to a round point) from local singularity formation (neckpinch). Finally, we discuss the interpretation and implication of these results and future applications.