Combinatorial Integer Labeling Thorems on Finite Sets with an Application to Discrete Systems of Nonlinear Equations

Tucker's well-known combinatorial lemma states that for any given symmetric triangulation of the n-dimensional unit cube and for any integer labeling that assigns to each vertex of the triangulation a label from the set f§1;§2; ¢ ¢ ¢ ;§ng with the property that antipodal vertices on the boundary of the cube are assigned opposite labels, the triangulation admits a 1-dimensional simplex whose two vertices have opposite labels. In this paper we are concerned with an arbitrary finite set D of integral vectors in the n-dimensional Euclidean space and an integer labeling that assigns to each element of D a label from the set f§1;§2; ¢ ¢ ¢ ;§ng. Using a constructive approach we prove two combinatorial theorems of Tucker type, stating that under some mild conditions there exists two integral vectors in D having opposite labels and being cell-connected in the sense that both belong to the set f0; 1gn+q for some integral vector q. These theorems will be used to show in a constructive way the existence of an integral solution to a system of nonlinear equations under certain natural conditions.Sperner lemma;Tucker lemma;integer labeling;simplicial algorithm;discrete nonlinear equations

Intuitionistic Completeness of First-Order Logic

We establish completeness for intuitionistic first-order logic, iFOL, showing that is a formula is provable if and only if it is uniformly valid under the Brouwer Heyting Kolmogorov (BHK) semantics, the intended semantics of iFOL. Our proof is intuitionistic and provides an effective procedure Prf that converts uniform evidence into a formal first-order proof. We have implemented Prf . Uniform validity is defined using the intersection operator as a universal quantifier over the domain of discourse and atomic predicates. Formulas of iFOL that are uniformly valid are also intuitionistically valid, but not conversely. Our strongest result requires the Fan Theorem; it can also be proved classically by showing that Prf terminates using K¨onig’s Theorem. The fundamental idea behind our completeness theorem is that a single evidence term evd witnesses the uniform validity of a minimal logic formula F. Finding even one uniform realizer guarantees validity because Prf (F, evd) builds a first-order proof of F, establishing its uniform validity and providing a purely logical normalized realizer. We establish completeness for iFOL as follows. Friedman showed that iFOL can be embedded in minimal logic (mFOL). By his transformation, mapping formula A to F r(A). If A is uniformly valid, then so is F r(A), and by our Basic Completeness result, we can find a proof of F r(A) in minimal logic. Then we prove A from F r(A) in intuitionistic logic by a proof procedure fixed in advance. Our result resolves an open question posed by Beth in 1947

Combinatorial theorems on the simplotope that generalize results on the simplex and cube

"April 1984."Bibliography: leaf i.by Robert M. Freund

How Much Change is Too Much Change? Rethinking the Reasons Behind the Lack of Reception to Brouwer’s Intuitionism

The paper analyzes Brouwer’s intuitionistic attempt to reform mathematics through the prism of Leo Corry’s philosophical model of “body” and “image” of knowledge. Such an analysis sheds new light on the question of whether Brouwer’s intuitionism could at all be attractive to broader groups of mathematicians. It focuses on three characteristics that are unique to Bouwer’s reformation attempt and suggests that when considered together, they combine to provide a more complex understanding of the reasons behind the lack of reception to Brouwer’s intuitionism than any of the three can offer alone

Balanced Simplices on Polytopes

The well known Sperner lemma states that in a simplicial subdivision of a simplex with a properly labeled boundary there is a completely labeled simplex. We present two combinatorial theorems on polytopes which generalize Sperner's lemma.Using balanced simplices, a generalized concept of completely labeled simplices, a uni ed existence result of balanced simplices in any simplicial subdivision of a polytope is given.This theorem implies the well-known lemmas of Sperner, Scarf, Shapley, and Garcia as well as some other results as special cases.A second theorem which imposes no restrictions on the integer labeling rule is established; this theorem implies several results of Freund.

Formalizing Constructive Analysis: A comparison of minimal systems and a study of uniqueness principles.

Αυτή η διατριβή εξετάζει ορισμένες πλευρές της τυποποίησης και της αξιωματικοποίησης της κατασκευαστικής ανάλυσης. Η έρευνα στους κλάδους της κατασκευαστικής ανάλυσης που αντιστοιχούν στις διάφορες εκδοχές κατασκευαστικότητας διεξάγεται σε μια πλειάδα τυπικών ή όχι συστημάτων, των οποίων οι σχέσεις είναι ασαφείς. Αυτό το πρόβλημα αποβαίνει κρίσιμο για την ανάπτυξη της σχετικά νέας περιοχής των κατασκευαστικών ανάστροφων μαθηματικών. Η εργασία αυτή συμβάλλει σε μια πιο καθαρή εικόνα. Το Μέρος 1 περιέχει μία ακριβή σύγκριση των δύο ευρύτερα χρησιμοποιούμενων συστημάτων που τυποποιούν τον κοινό πυρήνα της κατασκευαστικής, της ενορατικής, της αναδρομικής και της κλασικής ανάλυσης, των Μ και EL, των Kleene και Troelstra, αντιστοίχως. Αποδεικνύεται ότι το EL είναι ασθενέστερο από το M και ότι η διαφορά τους αποτυπώνεται από μια αρχή η οποία εγγυάται την ύπαρξη χαρακτηριστικής συνάρτησης για κάθε αποκρίσιμο κατηγόρημα φυσικών αριθμών. Με παρόμοια επιχειρήματα προκύπτουν συγκρίσεις για τα περισσότερα από τα χρησιμοποιούμενα ελαχιστικά συστήματα. Στην κατασκευαστική ανάλυση χρησιμοποιούνται διάφορες αρχές επιλογής, συνέχειας και άλλες. Στο Μέρος 2, μελετώνται σχέσεις μεταξύ πολλών από αυτές, στις εκδοχές τους με μία συνθήκη μοναδικότητας, ένα χαρακτηριστικό από το οποίο απορρέουν ενδιαφέρουσες ιδιότητες, καθώς και σχέσεις μεταξύ αυτών των αρχών και μη κατασκευαστικών λογικών αρχών, στο πνεύμα των ανάστροφων μαθηματικών.This dissertation investigates certain aspects of the formalization and axiomatization of constructive analysis. The research in the branches of constructive analysis corresponding to the various forms of constructivism is carried out in a multitude of formal or informal systems, whose relations are unclear. This problem becomes quite crucial for the development of the relatively new field of constructive reverse mathematics. This work contributes to a clearer picture. Part 1 contains a precise comparison of the two most widely used systems which formalize the common core of constructive, intuitionistic, recursive and classical analysis, namely Kleene's M and Troelstra's EL. It is shown that EL is weaker than M and that their difference is captured by a function existence principle asserting that every decidable predicate of natural numbers has a characteristic function. Applying similar arguments, comparisons of most of the used minimal systems are obtained. In constructive analysis, various forms of choice principles, continuity principles and many others are used. Part 2 studies relations between many of them, in their versions having a uniqueness condition, a feature from which interesting properties follow, as well as relations between these principles and non-constructive logical principles, in the spirit of reverse mathematics

Combinatorial analogs of Brouwer's fixed point theorem on a bounded polyhedron

Bibliography: p.61-62.by Robert M. Freund

A Dempster-Shafer theory inspired logic.

Issues of formalising and interpreting epistemic uncertainty have always played a prominent role in Artificial Intelligence. The Dempster-Shafer (DS) theory of partial beliefs is one of the most-well known formalisms to address the partial knowledge. Similarly to the DS theory, which is a generalisation of the classical probability theory, fuzzy logic provides an alternative reasoning apparatus as compared to Boolean logic. Both theories are featured prominently within the Artificial Intelligence domain, but the unified framework accounting for all the aspects of imprecise knowledge is yet to be developed. Fuzzy logic apparatus is often used for reasoning based on vague information, and the beliefs are often processed with the aid of Boolean logic. The situation clearly calls for the development of a logic formalism targeted specifically for the needs of the theory of beliefs. Several frameworks exist based on interpreting epistemic uncertainty through an appropriately defined modal operator. There is an epistemic problem with this kind of frameworks: while addressing uncertain information, they also allow for non-constructive proofs, and in this sense the number of true statements within these frameworks is too large. In this work, it is argued that an inferential apparatus for the theory of beliefs should follow premises of Brouwer's intuitionism. A logic refuting tertium non daturìs constructed by defining a correspondence between the support functions representing beliefs in the DS theory and semantic models based on intuitionistic Kripke models with weighted nodes. Without addional constraints on the semantic models and without modal operators, the constructed logic is equivalent to the minimal intuitionistic logic. A number of possible constraints is considered resulting in additional axioms and making the proposed logic intermediate. Further analysis of the properties of the created framework shows that the approach preserves the Dempster-Shafer belief assignments and thus expresses modality through the belief assignments of the formulae within the developed logic

A Dempster-Shafer theory inspired logic

Issues of formalising and interpreting epistemic uncertainty have always played a prominent role in Artificial Intelligence. The Dempster-Shafer (DS) theory of partial beliefs is one of the most-well known formalisms to address the partial knowledge. Similarly to the DS theory, which is a generalisation of the classical probability theory, fuzzy logic provides an alternative reasoning apparatus as compared to Boolean logic. Both theories are featured prominently within the Artificial Intelligence domain, but the unified framework accounting for all the aspects of imprecise knowledge is yet to be developed. Fuzzy logic apparatus is often used for reasoning based on vague information, and the beliefs are often processed with the aid of Boolean logic. The situation clearly calls for the development of a logic formalism targeted specifically for the needs of the theory of beliefs. Several frameworks exist based on interpreting epistemic uncertainty through an appropriately defined modal operator. There is an epistemic problem with this kind of frameworks: while addressing uncertain information, they also allow for non-constructive proofs, and in this sense the number of true statements within these frameworks is too large. In this work, it is argued that an inferential apparatus for the theory of beliefs should follow premises of Brouwer's intuitionism. A logic refuting tertium non daturìs constructed by defining a correspondence between the support functions representing beliefs in the DS theory and semantic models based on intuitionistic Kripke models with weighted nodes. Without addional constraints on the semantic models and without modal operators, the constructed logic is equivalent to the minimal intuitionistic logic. A number of possible constraints is considered resulting in additional axioms and making the proposed logic intermediate. Further analysis of the properties of the created framework shows that the approach preserves the Dempster-Shafer belief assignments and thus expresses modality through the belief assignments of the formulae within the developed logic.EThOS - Electronic Theses Online ServiceGBUnited Kingdo