A Structure-preserving Clause Form Translation

Much previous work in artificial intelligence has neglected representing time in all its complexity. In particular, it has neglected continuous change and the indeterminacy of the future. To rectify this, I have developed a first-order temporal logic, in which it is possible to name and prove things about facts, events, plans, and world histories. In particular, the logic provides analyses of causality, continuous change in quantities, the persistence of facts (the frame problem), and the relationship between tasks and actions. It may be possible to implement a temporal-inference machine based on this logic, which keeps track of several “maps” of a time line, one per possible history.

A Temporal Logic for Reasoning about Processes and Plans.

Lifting is one of the most important general techniques for accelerating dii-cult computations. Lifting was used by Alan Robinson in the early 1960's to accelerate rst order inference Robinson, 1965]. A complete inference procedure had been done by Martin Davis and Hillary Putnam a few years prior to Robinson's work Davis and Putnam, 1960]. Unfortunately, it was a ground procedure and suuered from a large search space generated by unguided selection of gound instances of quantiied formulas. Robinson designed a ground procedure for which lifting was particularly simple and demonstrated empirically that lifting greatly improves the performance of this procedure and that the lifted version is far superior to the Davis-Putnam procedure for general purpose theorem proving. Robinson's lifted inference process became known as resolution. Because of the tremendous advantage of a lifted procedure over a ground procedure, resolution immediately became the central paradigm of automated rst order inference. Over the years resolution has been greatly reened. The basic procedure is deened below and several reenements are described. Although lifting is a good technique for accelerating diicult computations, lifting alone has not brought computers near human-level competence in mathematical reasoning. It is still instructive, however, to study the classical resolution theorem proving techniques. The study of resolution theorem proving has resulted in the development of a large variety of general purpose search heuris-tics known as \strategies". In order to using lifting one must rst construct a ground procedure. The rst step in constructing the ground procedure is to convert inference problems to a simple normal form.

Resolution Theorem Proving

We consider the problem of encoding Boolean cardinality constraints in conjunctive normal form (CNF). Boolean cardinality constraints are formulae expressing that at most (resp. at least) k out of n propositional variables are true. We give two novel encodings that improve upon existing results, one which requires only 7n clauses and 2n auxiliary variables, and another one demanding O(n·k) clauses, but with the advantage that inconsistencies can be detected in linear time by unit propagation alone. Moreover, we prove a linear lower bound on the number of required clauses for any such encoding.

/pdf/towards-an-optimal-cnf-encoding-of-boolean-cardinality-5e4buep26z.pdf

Towards an optimal CNF encoding of Boolean cardinality constraints

I. Natural and formal logic.- II. The connection method in propositional logic.- III. The connection method in first-order logic.- IV. Variants and improvements.- V. Applications and extensions.- Mnemonics for use in references.- References.- List of Symbols.

Automated Theorem Proving

Completely non-clausal theorem proving

Path dissolution, a rule of inference that operates on formulas in negation normal form and that employs a representation called semantic graphs is introduced. Path dissolution has several advantages in comparison wdh many other reference technologies. In the ground case, lt preserves equivalence and is strongly complete: Any sequence of dissolution steps applied exhaustively to a semantic grdph G will yield an equivalent linkless graph G. Furthermore, one need not (and cannot) restrict attention to conjunctive normal form (CNF) when employing dlssolutlon: A single application (even to a CNF formula) generally produces a non-CNF formula that is more compact than any of its CNF equivalents. Path dissolution is a global rule: as such, it is employed at the first order level differently from the way locally oriented techmques (such as resolution) are. Two methods for employing dissolution as an inference mechanism for first order logic are presented. Dissolution is related to our theory links mechanism, to the factoring of formulas with the distributive laws. and to analytic tableaux. Some preliminary experimental results are also reported.

Dissolution: making paths vanish

Several methods to compute the prime implicants and the prime implicates of a negation normal form (NNF) formula are developed and implemented. An algorithm PI is introduced that is an extension to negation normal form of an algorithm given by Jackson and Pais. A correctness proof of the PI algorithm is given. The PI algorithm alone is sufficient in a computational sense. However, it can be combined with path dissolution, and it is shown empirically that this is often an advantage. None of these variations rely on conjunctive normal form or on disjunctive normal form. A class of formulas is described for which reliance on CNF or on DNF results in an exponential increase in the time required to compute prime implicants/implicates. The possibility of avoiding this problem with efficient structure preserving clause form translations is examined briefly and appears unfavorable.

CNF and DNF Considered Harmful for Computing Prime Implicants/Implicates

A graphical representation of quantifier-free predicate calculus formulas in negation normal form and a new rule of inference that employs this representation are introduced. The new rule, path resolution, is an amalgamation of resolution and Prawitz analysis. The goal in the design of path resolution is to retain some of the advantages of both Prawitz analysis and resolution methods, and yet to avoid to some extent their disadvantages.Path resolution allows Prawitz analysis of an arbitrary subgraph of the graph representing a formula. If such a subgraph is not large enough to demonstrate a contradiction, a path resolvent of the subgraph may be generated with respect to the entire graph. This generalizes the notions of large inference present in hyperresolution, clash-resolution, NC-resolution, and UR-resolution. A class of subgraphs is described for which deletion of some of the links resolved upon preserves the spanning property.

Inference with path resolution and semantic graphs

The relationship between signed formulas and annotated logics, two approaches that some authors have used to analyze multiple-valued logics (MVLs), is explored. A special case of the signed resolution rule is shown to be equivalent to, and thus to unify, the two inference rules, resolution and reduction, of annotated logic, raising the possibility of an SLD-style resolution rule for annotated logic programs. >

Neil V. Murray

Papers

Completely non-clausal theorem proving

Dissolution: making paths vanish

CNF and DNF Considered Harmful for Computing Prime Implicants/Implicates

Inference with path resolution and semantic graphs

Signed formulas and annotated logics