Eugenio G. Omodeo

{log}: A language for programming in logic with finite sets

Abstract An extended logic programming language is presented, that embodies the fundamental form of set designation based on the (nesting) element insertion operator. The kind of sets to be handled is characterized both by adaptation of a suitable Herbrand universe and via axioms. Predicates ϵ and = designating set membership and equality are included in the base language, along with their negative counterparts ∉ and ≠. A unification algorithm that can cope with set terms is developed and proved correct and terminating. It is proved that by incorporating this new algorithm into SLD resolution and providing suitable treatment of ϵ, ≠, and ∉ as constraints, one obtains a correct management of the distinguished set predicates. Restricted universal quantifiers are shown to be programmable directly in the extended language and thus are added to the language as a convenient syntactic extension. A similar solution is shown to be applicable to intensional set-formers, provided either a built-in set collection mechanism or some form of negation in goals and clause bodies is made available.

The automation of syllogistic. II. optimization and complexity issues

In the first paper of this series it was shown that any unquantified formula p in the collection MLSSF (multilevel syllogistic extended with the singleton operator and the predicate Finite) can be decomposed as a disjunction of set-theoretic formulae called syllogistic schemes. The syllogistic schemes are satisfiable and no two of them have a model in common, therefore the previous result already implied the decidability of the class MLSSF by simply checking if the set of syllogistic schemes associated with the given formula is empty.In the first section of this paper a new and improved searching algorithm for syllogistic schemes is introduced, based on a proof of existence of a ‘minimum effort’ scheme for any given satisfiable formula in MLSF. The algorithm addressed above can be piloted quite effectively even though it involves backtracking.In the second part of the paper, complexity issues are studied by showing that the class of (∀)o1-simple prenex formulae (an extension of MLS) has a decision problem which is NP-complete. The decision algorithm that proves the membership of this decision problem to NP can be seen as a different decision algorithm for MLS.

Notes from the Logbook of a Proof-Checker’s Project*

We present three newsletters drawn from the documentation of a project aimed at developing a software system which ingests proofs formalized within Zermelo-Fraenkel set theory and checks their compliance with mathematical rigor. Our verifier will accept trivial steps as obvious and will be able to process large proof scripts (say dozens of thousands of proofware lines written on persistent files). To test our prototype proof-checker we are developing, starting from the bare rudiments of set theory, various “proof scenarios,” the largest of which concerns the Cauchy Integral Theorem.

Computational Logic and Set Theory

This must-read text presents the pioneering work of the late Professor Jacob (Jack) T. Schwartz on computational logic and set theory and its application to proof verification techniques, culminating in the AEtnaNova system, a prototype computer program designed to verify the correctness of mathematical proofs presented in the language of set theory. Topics and features: describes in depth how a specific first-order theory can be exploited to model and carry out reasoning in branches of computer science and mathematics; presents a unique system for automated proof verification in large-scale software systems; integrates important proof-engineering issues, reflecting the goals of large-scale verifiers; includes an appendix showing formalized proofs of ordinals, of various properties of the transitive closure operation, of finite and transfinite induction principles, and of Zorns lemma.

An Equational Re-engineering of Set Theories

New successes in dealing with set theories by means of stateof-the-art theorem-provers may ensue from terse and concise axiomatizations, such as can be moulded in the framework of the (fully equational) Tarski-Givant map calculus. In this paper we carry out this task in detail, setting the ground for a number of experiments.

Set Graphs. III. Proof Pearl: Claw-Free Graphs Mirrored into Transitive Hereditarily Finite Sets

We report on the formalization of two classical results about claw-free graphs, which have been verified correct by Jacob T. Schwartz’s proof-checker Referee. We have proved formally that every connected claw-free graph admits (1) a near-perfect matching, (2) Hamiltonian cycles in its square. To take advantage of the set-theoretic foundation of Referee, we exploited set equivalents of the graph-theoretic notions involved in our experiment: edge, source, square, etc. To ease some proofs, we have often resorted to weak counterparts of well-established notions such as cycle, claw-freeness, longest directed path, etc.

Theory-specific automated reasoning

In designing a large-scale computerized proof system, one is often confronted with issues of two kinds: issues regarding an underlying logical calculus, and issues that refer to theories, either specified axiomatically or characterized by indication of either a privileged model or a family of intended models. Proof services related to the theories most often take the form of satisfiability decision or semi-decision procedures (in a sense, polyadic inference rules), while some of the services offered by the calculus (e.g., the Davis-Putnam propositional satisfiability checker) provide low-level mechanisms for integrating services of the former kind. Integration among services can ensure speed-up (i.e., lower number of steps) in the proofs, but it must always be legitimatized by a conservativeness result. Interoperability among proof checkers and autonomous theorem provers is another key point of integration. In discussing these and related issues, this paper refers to Set Theory as the unifying background, and to a specific proof-checker based on a slightly unorthodox formalization of it as an arena for experimentation.

Layered map reasoning: An experimental approach put to trial on sets

Abstract New successes in dealing with set-theories by means of state-of-the-art theorem-provers may ensue from terse and concise axiomatic systems, such as can be moulded in the framework of the (fully equational) Tarski-Givant formalism of dyadic relations, here named ‘maps’. This paper sets the ground for systematic experimentation based on such axiomatic systems. On top of a kernel axiomatization of map algebra, we develop a layered formalization of basic set-theoretic concepts. A number of concrete experiments have been carried out in this framework, as the paper reports, with the assistance of a first-order theorem-prover. The aim is to assess the potential usefulness of the proposed layered architecture and, to the extent it reveals promising, to best tune it.

A graphical approach to relational reasoning

Abstract Relational reasoning is concerned with relations over an unspecified domain of discourse. Two limitations to which it is customarily subject are: only dyadic relations are taken into account; all formulas are equations, having the same expressive power as first-order sentences in three variables. The relational formalism inherits from the Peirce-Schroder tradition, through contributions of Tarski and many others. Algebraic manipulation of relational expressions (equations in particular) is much less natural than developing inferences in first-order logic; it may in fact appear to be overly machine-oriented for direct hand-based exploitation. The situation radically changes when one resorts to a convenient representation of relations based on labeled graphs. The paper provides details of this representation, which abstracts w.r.t. inessential features of expressions. Formal techniques illustrating three uses of the graph representation of relations are discussed: one technique deals with translating first-order specifications into the calculus of relations; another one, with inferring equalities within this calculus with the aid of convenient diagram-rewriting rules; a third one with checking, in the specialized framework of set theory, the definability of particular set operations. Examples of use of these techniques are produced; moreover, a promising approach to mechanization of graphical relational reasoning is outlined.

Infinity, in short

It is shown that within the language of Set Theory, if membership is assumed to be non-well-founded a la Aczel, then one can state the existence of infinite sets by means of an ∃∃∀∀ prenex sentence. Somewhat surprisingly, this statement of infinity is essentially the one which was proposed in 1988 for well-founded sets, and it is satisfied exclusively by well-founded sets. Stating infinity inside the BSR (Bernays–Schonfinkel–Ramsey) class of the ∃*∀*-sentences becomes more challenging if no commitment is taken as whether membership is well-founded or not: for this case, we produce an ∃∃∀∀∀ -sentence, thus lowering the complexity of the quantificational prefix with respect to earlier prenex formulations of infinity. We also show that no prenex specification of infinity can have a prefix simpler than ∃∃∀∀. The problem of determining whether a BSR-sentence involving an uninterpreted predicate symbol and = can be satisfied over a large domain is then reduced to the satisfiability problem for the set theoretic class BSR subject to the ill-foundedness assumption. Envisaged enhancements of this reduction, cleverly exploiting the expressive power of the set theoretic BSR-class, add to the motivation for tackling the satisfaction problem for this class, which appears to be anything but unchallenging.