Bisimilarity, Hypersets, and Stable Partitioning: a Survey

Since Hopcroft proposed his celebrated $n \log n$ algorithm for minimizing states in a finite automaton, the race for efficient partition refinement methods has inspired much research in algorithmics. In parallel, the notion of bisimulation has gained ground in theoretical investigations not less than in applications, till it even pervaded the axioms of a variant Zermelo-Fraenkel set theory. As is well-known, the coarsest stable partitioning problem and the determination of bisimilarity (i.e., the largest partition stable relative to finitely many dyadic relations) are two faces of the same coin. While there is a tendency to refer these topics to varying frameworks, we will contend that the set-theoretic view not only offers a clear conceptual background (provided stability is referred to a non-well-founded membership), but is leading to new insights on the algorithmic complexity issues

Six equations in search of a finite-fold-ness proof

By following the same construction pattern which Martin Davis proposed in a 1968 paper of his, we have obtained six quaternary quartic Diophantine equations that candidate as `rule-them-all' equations: proving that one of them has only a finite number of integer solutions would suffice to ensure that each recursively enumerable set admits a finite-fold polynomial Diophantine representation

Layered map reasoning

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Stating Infinity in Set/Hyperset Theory

It is known that the Infinity Axiom can be expressed, even if the Axiom of Foundation is not assumed, in a logically simple form, by means of a formula involving only restricted universal quantifiers. Moreover, with Aczel's Anti-Foundation Axiom superseding von Neumann's Axiom of Foundation, a similar formula has recently emerged, which enjoys the additional property that it is satisfied only by (infinite) ill-founded sets. We give here new short proofs of both results

Solvable (and unsolvable) cases of the decision problem for fragments of analysis

We survey two series of results concerning the decidability of fragments of Tarksi’s elementary algebra extended with one-argument functions which meet significant properties such as continuity, differentiability, or analyticity. One series of results regards the initial levels of a hierarchy of prenex sentences involving a single function symbol: in a number of cases, the decision problem for these sentences was solved in the positive by H. Friedman and A. Seress, who also proved that beyond two quantifier alternations decidability gets lost. The second series of results refers to merely existential sentences, but it brings into play an arbitrary number of functions, which are requested to be, over specified closed intervals, monotone increasing or decreasing, concave, or convex; any two such functions can be compared, and in one case, where each function is supposed to own continuous first derivative, their derivatives can be compared with real constants

Set-syllogistics meet combinatorics

This paper considers 03* 00* prenex sentences of pure first-order predicate calculus with equality. This is the set of formulas which Ramsey's treated in a famous article of 1930. We demonstrate that the satisfiability problem and the problem of existence of arbitrarily large models for these formulas can be reduced to the satisfiability problem for 03* 00* prenex sentences of Set Theory (in the relators 08, =). We present two satisfiability-preserving (in a broad sense) translations \u3a6 \u21a6 (Formula presented.) and \u3a6 \u21a6 \u3a6\u3c3 of 03* 00* sentences from pure logic to well-founded Set Theory, so that if (Formula presented.) is satisfiable (in the domain of Set Theory) then so is \u3a6, and if \u3a6\u3c3 is satisfiable (again, in the domain of Set Theory) then \u3a6 can be satisfied in arbitrarily large finite structures of pure logic. It turns out that |(Formula presented.)| = (Formula presented.)(|\u3a6|) and |\u3a6\u3c3| = (Formula presented.)(|\u3a6|2). Our main result makes use of the fact that 03* 00* sentences, even though constituting a decidable fragment of Set Theory, offer ways to describe infinite sets. Such a possibility is exploited to glue together infinitely many models of increasing cardinalities of a given 03* 00* logical formula, within a single pair of infinite sets

Mapping Sets and Hypersets into Numbers

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

USING ÆTNANOVA TO FORMALLY PROVE THAT THE DAVIS-PUTNAM SATISFIABILITY TEST IS CORRECT

This paper reports on using the ÆtnaNova/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 recursive 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 develop a straightforward proof of the compactness theorem.This paper reports on using the ÆtnaNova/Referee proof-verificationsystem to formalize issues regarding the satisfiability of CNF-formulaeof propositional logic. We specify an “archetype” version of the Davis-Putnam-Logemann-Loveland algorithm through the THEORY of recursive 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 develop a straightforward proof of the compactness theorem

USING ÆTNANOVA TO FORMALLY PROVE THAT THE DAVIS-PUTNAM SATISFIABILITY TEST IS CORRECT

This paper reports on using the ÆtnaNova/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 recursive 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 develop a straightforward proof of the compactness theorem.This paper reports on using the ÆtnaNova/Referee proof-verificationsystem to formalize issues regarding the satisfiability of CNF-formulaeof propositional logic. We specify an “archetype” version of the Davis-Putnam-Logemann-Loveland algorithm through the THEORY of recursive 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 develop a straightforward proof of the compactness theorem