Eugenio Omodeo | Università degli Studi di Udine / University of Udine (original) (raw)

Papers by Eugenio Omodeo

On Sets and Graphs

On Sets and Graphs, 2017

This treatise presents an integrated perspective on the interplay of set theory and graph theory,... more This treatise presents an integrated perspective on the interplay of set theory and graph theory, providing an extensive selection of examples that highlight how methods from one theory can be used to better solve problems originated in the other. Features: explores the interrelationships between sets and graphs and their applications to finite combinatorics; introduces the fundamental graph-theoretical notions from the standpoint of both set theory and dyadic logic, and presents a discussion on set universes; explains how sets can conveniently model graphs, discussing set graphs and set-theoretic representations of claw-free graphs; investigates when it is convenient to represent sets by graphs, covering counting and encoding problems, the random generation of sets, and the analysis of infinite sets; presents excerpts of formal proofs concerning graphs, whose correctness was verified by means of an automated proof-assistant; contains numerous exercises, examples, definitions, problems and insight panels.

This paper enriches a pre-existing decision algorithm, which in its turn augmented a fragment of ... more This paper enriches a pre-existing decision algorithm, which in its turn augmented a fragment of Tarski's elementary algebra with one-argument real functions endowed with continuous first derivative. In its present (still quantifier-free) version, our decidable language embodies addition of functions; the issue we address is the one of satisfiability. As regards real numbers, individual variables and constructs designating the basic arithmetic operations are available, along with comparison relators. As regards functions, we have another sort of variables, out of which compound terms are formed by means of constructs designating addition and-outermostly-differentiation. An array of predicates designate various relationships between functions, as well as function properties, that may hold over intervals of the real line; those are: function comparisons, strict and non-strict monotonicity / convexity / concavity, comparisons between the derivative of a function and a real term. With respect to results announced in earlier papers of the same stream, a significant effort went into designing the family of interpolating functions so that it could meet the new constraints stemming from the presence of function addition (along with differentiation) among the constructs of our fragment of mathematical analysis.

We present a variant of the Davis-Fechter's technique for eliminating quantifiers in first-order ... more We present a variant of the Davis-Fechter's technique for eliminating quantifiers in first-order logic, aimed at reducing the incidence of irrelevant dependencies in the construction of Skolem terms. The basic idea behind this contribution is to treat as a single syntactic unit every maximal 'quantifier batch', i.e., group of contiguous alike quantifiers, whose internal order (which has no significance) is thereby prevented from entangling the final result. Through concrete cross-translations, our version of the free-variable predicate calculus turns out to be equipollent-in means of expression and of proof-to the one originally proposed by Davis and Fechter and hence to a relatively conventional version of quantified first-order logic.

On existentially quantified conjunctions of atomic formulae of L

Sufficient conditions for an 9, 89, or 889prenex L+-sentence to be translatableinto the variable-... more Sufficient conditions for an 9, 89, or 889prenex L+-sentence to be translatableinto the variable-free formalism L\Thetawill be singled out in what follows. An efficienttest based on such conditions will also be described. Through minor modifications ofthis testing algorithm, one can obtain the translation when the sufficient conditionsare met.1 IntroductionL\Thetais a ground equational formalism that can compete with first-order predicate logicas

ACM Transactions on Computational Logic, 2006

Formal set theory is traditionally concerned with pure sets; consequently, the satisfiability pro... more Formal set theory is traditionally concerned with pure sets; consequently, the satisfiability problem for fragments of set theory was most often addressed (and in many cases positively solved) in the pure framework. In practical applications, however, it is common to assume the existence of a number of primitive objects (sometimes called atoms ) that can be members of sets but behave differently from them. If these entities are assumed to be devoid of members, the standard extensionality axiom must be revised; then decidability results can sometimes be achieved via reduction to the pure case and sometimes can be based on direct goal-driven algorithms. An alternative approach to modeling atoms that allows one to retain the original formulation of extensionality was proposed by Quine: atoms are self-singletons. In this article we adopt this approach in coping with the satisfiability problem: We show the decidability of this problem relativized to ∃*∀-sentences, and develop a goal-driv...

We continue our investigation aimed at spotting small fragments of Set Theory (in this paper, sub... more We continue our investigation aimed at spotting small fragments of Set Theory (in this paper, sublanguages of Boolean Set Theory) that might be of use in automated proof-checkers based on the set-theoretic formalism. Here we propose a method that leads to a cubictime satisfiability decision test for the language involving, besides variables intended to range over the von Neumann set-universe, the Boolean operator ∪ and the logical relators = and =. It can be seen that the dual language involving the Boolean operator ∩ and, again, the relators = and =, also admits a decidable cubic-time satisfiability test; noticeably, the same algorithm can be used for both languages. Suitable pre-processing can reduce richer Boolean languages to the said two fragments, so that the same cubic satisfiability test can be used to treat the relators ⊆, and the predicates ' = ∅' and 'Disj(,)', meaning 'the argument is empty' and 'the arguments are disjoint sets', along with their opposites ' = ∅' and '¬Disj(,)'. Those richer languages are 'polynomial maximal', in the sense that all languages strictly containing them have an NP-hard satisfiability problem.

This paper reports on using theÆtnaNova/Referee proof-verification system to formalize issues reg... more This paper reports on using theÆtnaNova/Referee proof-verification system to formalize issues regarding the satisfiability of CNF-formulae of propositional logic. We specify an “archetype ” version of the Davis-Putnam-Logemann-Loveland algorithm through the THEORY of recur-sive functions based on a well-founded relation, and prove it to be correct. Within the same framework, and by resorting to the Zorn lemma, we de-velop a straightforward proof of the compactness theorem. 1.

Entity-Relationship modeling is a popular technique for data modeling. Despite its popularity and... more Entity-Relationship modeling is a popular technique for data modeling. Despite its popularity and wide spread use, it lacks a rm semantic foundation. We propose a translation of an ERmodel into mapalgebra, suggesting that mapalgebra does provide suitable mechanisms for establishing a formal semantics of entity-relationship modeling. This report deals with the techniques necessary for the translation and provides a static view of an ER-model in its mapalgebraic disguise.

Meta-interpreting SETL

Le Matematiche, 1990

This paper describes a SETL interpreter written in SETL. This module may be reused as a basis to ... more This paper describes a SETL interpreter written in SETL. This module may be reused as a basis to build debuggers, type checkers, symbolic executers, tracers, and many other general purpose programming tools. Other more advanced uses include experimenting with altered semantics for SETL and building interpreters for multi-paradigm languages, as in SetLog project, which aims at constructing a language integrating logic programming and set-oriented programming.

Dedicated to Martin Davis and Yuri Matiyasevich for their respective 90th and 70th birthdays, and... more Dedicated to Martin Davis and Yuri Matiyasevich for their respective 90th and 70th birthdays, and to the memory of Julia Robinson, for her centennial. The Davis-Putnam-Robinson theorem showed that every partially computable m-ary function f (a 1 ,. .. , a m) = c on the natural numbers can be specified by means of an exponential Diophantine formula involving, along with parameters a 1 ,. .. , a m , c, some number κ of existentially quantified variables. Yuri Matiyasevich improved this theorem in two ways: on the one hand, he proved that the same goal can be achieved with no recourse to exponentiation and, thereby, he provided a negative answer to Hilbert's 10th problem; on the other hand, he showed how to construct an exponential Diophantine equation specifying f which, once a 1 ,. .. , a m have been fixed, is solved by at most one tuple v 0 ,. .. , v κ of values for the remaining variables. This latter property is called single-foldness. Whether there exists a single-(or, at worst, finite-) fold polynomial Diophantine representation of any partially computable function on the natural numbers is as yet an open problem. This work surveys relevant results on this subject and tries to draw a route towards a hoped-for positive answer to the finite-fold-ness issue.

Properties of a few familiar structures (natural numbers, nestedlists, lattices) are formally spe... more Properties of a few familiar structures (natural numbers, nestedlists, lattices) are formally specified in Tarski-Givant's map calculus, withthe aim of bringing to light new translation techniques that may bridge thegap between first-order predicate calculus and the map calculus. It is alsohighlighted to what extent a state-of-the-art theorem-prover for first-orderlogic, namely Otter, can be exploited not only to emulate, but also

Lecture Notes in Computer Science, 2002

We often need to associate some highly compound meaning with a symbol. Such a symbol serves us as... more We often need to associate some highly compound meaning with a symbol. Such a symbol serves us as a kind of container carrying this meaning, always with the understanding that it can be opened if we need its content.

In a first-order theory Θ, the decision problem for a class of formulae Φ is solvable if there is... more In a first-order theory Θ, the decision problem for a class of formulae Φ is solvable if there is an algorithmic procedure that can assess whether or not the existential closure φ∃ of φ belongs to Θ, for any φ ∈ Φ. In 1988, Parlamento and Policriti already showed how to apply Gödel-like arguments to a very weak axiomatic set theory, with respect to the class of Σ1-formulae with (∀∃∀)0-matrix, i.e., existential closures of formulae that contain just restricted quantifiers of the kind (∀x ∈ y) and (∃x ∈ y) and are writeable in prenex form with at most two alternations of restricted quantifiers (the outermost quantifier being a ‘∀’). While revisiting their work, we show slightly stronger theories under which incompleteness for recursively axiomatizable extensions holds with respect to existential closures of (∀∃)0-matrices, namely formulae with at most one alternation of restricted quantifiers.

ArXiv, 2021

As a contribution to quantitative set-theoretic inferencing, a translation is proposed of conjunc... more As a contribution to quantitative set-theoretic inferencing, a translation is proposed of conjunctions of literals of the forms x = y \ z, x 6= y \ z, and z = {x }, where x, y, z stand for variables ranging over the von Neumann universe of sets, into unquantified Boolean formulae of a rather simple conjunctive normal form. The formulae in the target language involve variables ranging over a Boolean ring of sets, along with a difference operator and relators designating equality, non-disjointness and inclusion. Moreover, the result of each translation is a conjunction of literals of the forms x = y \ z, x 6= y \ z and of implications whose antecedents are isolated literals and whose consequents are either inclusions (strict or non-strict) between variables, or equalities between variables. Besides reflecting a simple and natural semantics, which ensures satisfiability-preservation, the proposed translation has quadratic algorithmic time-complexity, and bridges two languages both of w...

Plan of Activities on the Map Calculus

Tarski-Givant's map calculus is briefly reviewed and a plan of research is outlined, aimed a... more Tarski-Givant's map calculus is briefly reviewed and a plan of research is outlined, aimed at investigating applications of this formalism in the theorem-proving field. The connections between first-order logic and the map calculus are investigated, focusing on techniques for translating single sentences from one context to the other as well as on the translation of entire set theories. Issues regarding 'safe' forms of map reasoning are singled out, in sight of possible generalizations to the database area. Keywords: Algebraic logic, Relation algebras, First-order theorem-proving. Introduction Everybody remembers that Boole's Laws of thought (1854), Frege's Begriffsschrift (1879), and the Whitehead-Russell's Principia Mathematica (1910) have been three major milestones in the development of contemporary logic (cf. [3, 10, 21, 4]). Only a few people are aware that very important pre-Principia milestones were laid down by C.S. Peirce and E. Schroder and c...

New successes in dealing with set theories by means of state-of-the-art theoremprovers may ensue ... more New successes in dealing with set theories by means of state-of-the-art theoremprovers 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. Key words: Set theory, relation algebras, first-order theorem-proving, algebraic logic. 1 Introduction Like other mature fields of mathematics, Set Theory deserves sustained efforts that bring to light richer and richer decidable fragments of it [5], general inference rules for reasoning in it [23, 2], effective proof strategies based on its domain-knowledge, and so forth. Advances in this specialized area of automated reasoning tend, in spite of their steadiness, to be slow compared to the overall progress in the field. Many experiments with set theories have hence been carried out with standard theorem-proving systems. Still today such experiments pose conside...

Sets are a ubiquitous data structure which gets employed, in Computer Science as well as in Mathe... more Sets are a ubiquitous data structure which gets employed, in Computer Science as well as in Mathematics, since the most fundamental level. A key feature of standard sets is the acyclicity of the membership relation upon which they are based. All in all, this is the feature enabling one to provide an inductive description of the

As a by-product of the negative solution of Hilbert’s 10 problem, various prime-generating polyno... more As a by-product of the negative solution of Hilbert’s 10 problem, various prime-generating polynomials were found. The best known upper bound for the number of variables in such a polynomial, to wit 10, was found by Yuri V. Matiyasevich in 1977. We show that this bound could be lowered to 8 if the converse of Wolstenholme’s theorem (1862) holds, as conjectured by James P. Jones. This potential improvement is achieved through a Diophantine representation of the set of all integers p > 5 that satisfy the congruence ( 2 p p ) ≡ 2 mod p. Our specification, in its turn, relies upon a terse polynomial representation of exponentiation due to Matiyasevich and Julia Robinson (1975), as further manipulated by Maxim Vsemirnov (1997). We briefly address the issue of also determining a lower bound for the number of variables in a prime-representing polynomial, and discuss the autonomous significance of our result about Wostenholme’s pseudoprimality, independently of Jones’s conjecture.

Global Skolemization with grouped quantifiers ( Abstract )

Elimination of quantifiers from formulae of classical first-order logic is a process with many im... more Elimination of quantifiers from formulae of classical first-order logic is a process with many implications in automated deduction [6, 1] and in foundational issues [4, 7]. When no particular theory is considered, quantifiers are usually eliminated by adopting Skolemization or the ε-operator. Traditionally, Skolemization and the ε-symbol have different, if not complementary, employments. Skolemization is one of the most widespread techniques to eliminate existential quantifiers in automated proofs [6]. This is motivated by the fact that Skolem terms can easily be manipulated in deductions. In fact, they can be treated (both at the syntactical and at the semantical level) in the same way as the terms of the initial language. Skolem terms are generally unrelated to the formulae that generate them and to the context they are introduced in. This may become a drawback because some information which could be crucial for the automatic discovery of shorter automatic proofs might rely on rel...

Fundamenta Informaticae, 2015

We introduce and prove the basic properties of encodings that generalize to non-well-founded here... more We introduce and prove the basic properties of encodings that generalize to non-well-founded hereditarily finite sets the bijection defined by Ackermann in 1937 between hereditarily finite sets and natural numbers.