(Aka “Geometric”) Iff It Is Axiomatised by “Coherent Implications”
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Skolemization Modulo Theories
Skolemization Modulo Theories Konstantin Korovin1 and Margus Veanes2 1 The University of Manchester [email protected] 2 Microsoft Research [email protected] Abstract. Combining classical automated theorem proving techniques with theory based reasoning, such as satisfiability modulo theories, is a new approach to first-order reasoning modulo theories. Skolemization is a classical technique used to transform first-order formulas into eq- uisatisfiable form. We show how Skolemization can benefit from a new satisfiability modulo theories based simplification technique of formulas called monadic decomposition. The technique can be used to transform a theory dependent formula over multiple variables into an equivalent form as a Boolean combination of unary formulas, where a unary for- mula depends on a single variable. In this way, theory specific variable dependencies can be eliminated and consequently, Skolemization can be refined by minimizing variable scopes in the decomposed formula in order to yield simpler Skolem terms. 1 The role of Skolemization In classical automated theorem proving, Skolemization [8, 3] is a technique used to transform formulas into equisatisfiable form by replacing existentially quanti- fied variables by Skolem terms. In resolution based methods using clausal normal form (CNF) this is a necessary preprocessing step of the input formula. A CNF represents a universally quantified conjunction of clauses, where each clause is a disjunction of literals, a literal being an atom or a negated atom. The arguments of the atoms are terms, some of which may contain Skolem terms as subterms where a Skolem term has the form f(¯x) for some Skolem function symbol f and a sequencex ¯ of variables; f may also be a constant (¯x is empty). -
Chapter 1 Preliminaries
Chapter 1 Preliminaries 1.1 First-order theories In this section we recall some basic definitions and facts about first-order theories. Most proofs are postponed to the end of the chapter, as exercises for the reader. 1.1.1 Definitions Definition 1.1 (First-order theory) —A first-order theory T is defined by: A first-order language L , whose terms (notation: t, u, etc.) and formulas (notation: φ, • ψ, etc.) are constructed from a fixed set of function symbols (notation: f , g, h, etc.) and of predicate symbols (notation: P, Q, R, etc.) using the following grammar: Terms t ::= x f (t1,..., tk) | Formulas φ, ψ ::= t1 = t2 P(t1,..., tk) φ | | > | ⊥ | ¬ φ ψ φ ψ φ ψ x φ x φ | ⇒ | ∧ | ∨ | ∀ | ∃ (assuming that each use of a function symbol f or of a predicate symbol P matches its arity). As usual, we call a constant symbol any function symbol of arity 0. A set of closed formulas of the language L , written Ax(T ), whose elements are called • the axioms of the theory T . Given a closed formula φ of the language of T , we say that φ is derivable in T —or that φ is a theorem of T —and write T φ when there are finitely many axioms φ , . , φ Ax(T ) such ` 1 n ∈ that the sequent φ , . , φ φ (or the formula φ φ φ) is derivable in a deduction 1 n ` 1 ∧ · · · ∧ n ⇒ system for classical logic. The set of all theorems of T is written Th(T ). Conventions 1.2 (1) In this course, we only work with first-order theories with equality. -
Automated ZFC Theorem Proving with E
Automated ZFC Theorem Proving with E John Hester Department of Mathematics, University of Florida, Gainesville, Florida, USA https://people.clas.ufl.edu/hesterj/ [email protected] Abstract. I introduce an approach for automated reasoning in first or- der set theories that are not finitely axiomatizable, such as ZFC, and describe its implementation alongside the automated theorem proving software E. I then compare the results of proof search in the class based set theory NBG with those of ZFC. Keywords: ZFC · ATP · E. 1 Introduction Historically, automated reasoning in first order set theories has faced a fun- damental problem in the axiomatizations. Some theories such as ZFC widely considered as candidates for the foundations of mathematics are not finitely axiomatizable. Axiom schemas such as the schema of comprehension and the schema of replacement in ZFC are infinite, and so cannot be entirely incorpo- rated in to the prover at the beginning of a proof search. Indeed, there is no finite axiomatization of ZFC [1]. As a result, when reasoning about sufficiently strong set theories that could no longer be considered naive, some have taken the alternative approach of using extensions of ZFC that admit objects such as proper classes as first order objects, but this is not without its problems for the individual interested in proving set-theoretic propositions. As an alternative, I have programmed an extension to the automated the- orem prover E [2] that generates instances of parameter free replacement and comprehension from well formuled formulas of ZFC that are passed to it, when eligible, and adds them to the proof state while the prover is running. -
1.4 Theories
1.4 Theories Before we start to work with one of the most central notions of mathematical logic – that of a theory – it will be useful to make some observations about countable sets. Lemma 1.3. 1. A set X is countable iff there is a surjection ψ : N X. 2. If X is countable and Y X then Y is countable. 3. If X and Y are countable, then X Y is countable. 4. If X and Y are countable, then X ¢ Y is countable. k 5. If X is countable and k 1 then X is countable. ä 6. If Xi is countable for each i È N, then Xi is countable. iÈN 7. If X is countable, then ä Xk is countable. k ÈN Ø Ù Proof. 1. If X is countably infinite note that every bijection is a surjection. If X x0,...,xn Õ Ô Õ is finite, let ψ Ôi xi for 0 i n and ψ j xn for j n. For the other direction, Ô Õ Ô Õ Ô Õ Ù let ψ be a surjection, then X is Øψ 0 , ψ 1 , ψ 2 ,... and removing duplicates from the Ô ÕÕ Ô N sequence ψ i i¥0 yields a bijective ϕ : X. È 2. Obvious if Y is finite. Otherwise, let ϕ : N X be a bijection and let y0 Y . Define " Õ Ô Õ È ϕÔn if ϕ n Y ψ : N Y,n y0 otherwise which is a surjection. 3. Let ϕ : N X and ψ : N Y be bijections. -
Lipics-FSCD-2021-29.Pdf (0.9
On the Logical Strength of Confluence and Normalisation for Cyclic Proofs Anupam Das ! Ï University of Birmingham, UK Abstract In this work we address the logical strength of confluence and normalisation for non-wellfounded typing derivations, in the tradition of “cyclic proof theory” . We present a circular version CT of Gödel’ s system T, with the aim of comparing the relative expressivity of the theories CT and T. We approach this problem by formalising rewriting-theoretic results such as confluence and normalisation for the underlying “coterm” rewriting system of CT within fragments of second-order arithmetic. We establish confluence of CT within the theory RCA0, a conservative extension of primitive recursive arithmetic and IΣ1. This allows us to recast type structures of hereditarily recursive operations as “coterm” models of T. We show that these also form models of CT, by formalising a totality argument for circular typing derivations within suitable fragments of second-order arithmetic. Relying on well-known proof mining results, we thus obtain an interpretation of CT into T; in fact, more precisely, we interpret level-n-CT into level-(n + 1)-T, an optimum result in terms of abstraction complexity. A direct consequence of these model-theoretic results is weak normalisation for CT. As further results, we also show strong normalisation for CT and continuity of functionals computed by its type 2 coterms. 2012 ACM Subject Classification Theory of computation → Equational logic and rewriting; Theory of computation → Proof theory; Theory of computation → Higher order logic; Theory of computation → Lambda calculus Keywords and phrases confluence, normalisation, system T, circular proofs, reverse mathematics, type structures Digital Object Identifier 10.4230/LIPIcs.FSCD.2021.29 Related Version This work is based on part of the following preprint, where related results, proofs and examples may be found. -
Geometry and Categoricity∗
Geometry and Categoricity∗ John T. Baldwiny University of Illinois at Chicago July 5, 2010 1 Introduction The introduction of strongly minimal sets [BL71, Mar66] began the idea of the analysis of models of categorical first order theories in terms of combinatorial geometries. This analysis was made much more precise in Zilber’s early work (collected in [Zil91]). Shelah introduced the idea of studying certain classes of stable theories by a notion of independence, which generalizes a combinatorial geometry, and characterizes models as being prime over certain independent trees of elements. Zilber’s work on finite axiomatizability of totally categorical first order theories led to the development of geometric stability theory. We discuss some of the many applications of stability theory to algebraic geometry (focusing on the role of infinitary logic). And we conclude by noting the connections with non-commutative geometry. This paper is a kind of Whig history- tying into a (I hope) coherent and apparently forward moving narrative what were in fact a number of independent and sometimes conflicting themes. This paper developed from talk at the Boris-fest in 2010. But I have tried here to show how the ideas of Shelah and Zilber complement each other in the development of model theory. Their analysis led to frameworks which generalize first order logic in several ways. First they are led to consider more powerful logics and then to more ‘mathematical’ investigations of classes of structures satisfying appropriate properties. We refer to [Bal09] for expositions of many of the results; that’s why that book was written. It contains full historical references. -
The Development of Mathematical Logic from Russell to Tarski: 1900–1935
The Development of Mathematical Logic from Russell to Tarski: 1900–1935 Paolo Mancosu Richard Zach Calixto Badesa The Development of Mathematical Logic from Russell to Tarski: 1900–1935 Paolo Mancosu (University of California, Berkeley) Richard Zach (University of Calgary) Calixto Badesa (Universitat de Barcelona) Final Draft—May 2004 To appear in: Leila Haaparanta, ed., The Development of Modern Logic. New York and Oxford: Oxford University Press, 2004 Contents Contents i Introduction 1 1 Itinerary I: Metatheoretical Properties of Axiomatic Systems 3 1.1 Introduction . 3 1.2 Peano’s school on the logical structure of theories . 4 1.3 Hilbert on axiomatization . 8 1.4 Completeness and categoricity in the work of Veblen and Huntington . 10 1.5 Truth in a structure . 12 2 Itinerary II: Bertrand Russell’s Mathematical Logic 15 2.1 From the Paris congress to the Principles of Mathematics 1900–1903 . 15 2.2 Russell and Poincar´e on predicativity . 19 2.3 On Denoting . 21 2.4 Russell’s ramified type theory . 22 2.5 The logic of Principia ......................... 25 2.6 Further developments . 26 3 Itinerary III: Zermelo’s Axiomatization of Set Theory and Re- lated Foundational Issues 29 3.1 The debate on the axiom of choice . 29 3.2 Zermelo’s axiomatization of set theory . 32 3.3 The discussion on the notion of “definit” . 35 3.4 Metatheoretical studies of Zermelo’s axiomatization . 38 4 Itinerary IV: The Theory of Relatives and Lowenheim’s¨ Theorem 41 4.1 Theory of relatives and model theory . 41 4.2 The logic of relatives . -
Inseparability and Conservative Extensions of Description Logic Ontologies: a Survey
Inseparability and Conservative Extensions of Description Logic Ontologies: A Survey Elena Botoeva1, Boris Konev2, Carsten Lutz3, Vladislav Ryzhikov1, Frank Wolter2, and Michael Zakharyaschev4 1 Free University of Bozen-Bolzano, Italy fbotoeva,[email protected] 2 University of Liverpool, UK fkonev,[email protected] 3 University of Bremen, Germany [email protected] 4 Birkbeck, University of London, UK [email protected] Abstract. The question whether an ontology can safely be replaced by another, possibly simpler, one is fundamental for many ontology engi- neering and maintenance tasks. It underpins, for example, ontology ver- sioning, ontology modularization, forgetting, and knowledge exchange. What `safe replacement' means depends on the intended application of the ontology. If, for example, it is used to query data, then the answers to any relevant ontology-mediated query should be the same over any relevant data set; if, in contrast, the ontology is used for conceptual reasoning, then the entailed subsumptions between concept expressions should coincide. This gives rise to different notions of ontology insep- arability such as query inseparability and concept inseparability, which generalize corresponding notions of conservative extensions. We survey results on various notions of inseparability in the context of descrip- tion logic ontologies, discussing their applications, useful model-theoretic characterizations, algorithms for determining whether two ontologies are inseparable (and, sometimes, for computing the difference between them if they are not), and the computational complexity of this problem. 1 Introduction Description logic (DL) ontologies provide a common vocabulary for a domain of interest together with a formal modeling of the semantics of the vocabulary items (concept names and role names). -
The Strength of Mac Lane Set Theory
The Strength of Mac Lane Set Theory A. R. D. MATHIAS D´epartement de Math´ematiques et Informatique Universit´e de la R´eunion To Saunders Mac Lane on his ninetieth birthday Abstract AUNDERS MAC LANE has drawn attention many times, particularly in his book Mathematics: Form and S Function, to the system ZBQC of set theory of which the axioms are Extensionality, Null Set, Pairing, Union, Infinity, Power Set, Restricted Separation, Foundation, and Choice, to which system, afforced by the principle, TCo, of Transitive Containment, we shall refer as MAC. His system is naturally related to systems derived from topos-theoretic notions concerning the category of sets, and is, as Mac Lane emphasizes, one that is adequate for much of mathematics. In this paper we show that the consistency strength of Mac Lane's system is not increased by adding the axioms of Kripke{Platek set theory and even the Axiom of Constructibility to Mac Lane's axioms; our method requires a close study of Axiom H, which was proposed by Mitchell; we digress to apply these methods to subsystems of Zermelo set theory Z, and obtain an apparently new proof that Z is not finitely axiomatisable; we study Friedman's strengthening KPP + AC of KP + MAC, and the Forster{Kaye subsystem KF of MAC, and use forcing over ill-founded models and forcing to establish independence results concerning MAC and KPP ; we show, again using ill-founded models, that KPP + V = L proves the consistency of KPP ; turning to systems that are type-theoretic in spirit or in fact, we show by arguments of Coret -
Fundamental Theorems in Mathematics
SOME FUNDAMENTAL THEOREMS IN MATHEMATICS OLIVER KNILL Abstract. An expository hitchhikers guide to some theorems in mathematics. Criteria for the current list of 243 theorems are whether the result can be formulated elegantly, whether it is beautiful or useful and whether it could serve as a guide [6] without leading to panic. The order is not a ranking but ordered along a time-line when things were writ- ten down. Since [556] stated “a mathematical theorem only becomes beautiful if presented as a crown jewel within a context" we try sometimes to give some context. Of course, any such list of theorems is a matter of personal preferences, taste and limitations. The num- ber of theorems is arbitrary, the initial obvious goal was 42 but that number got eventually surpassed as it is hard to stop, once started. As a compensation, there are 42 “tweetable" theorems with included proofs. More comments on the choice of the theorems is included in an epilogue. For literature on general mathematics, see [193, 189, 29, 235, 254, 619, 412, 138], for history [217, 625, 376, 73, 46, 208, 379, 365, 690, 113, 618, 79, 259, 341], for popular, beautiful or elegant things [12, 529, 201, 182, 17, 672, 673, 44, 204, 190, 245, 446, 616, 303, 201, 2, 127, 146, 128, 502, 261, 172]. For comprehensive overviews in large parts of math- ematics, [74, 165, 166, 51, 593] or predictions on developments [47]. For reflections about mathematics in general [145, 455, 45, 306, 439, 99, 561]. Encyclopedic source examples are [188, 705, 670, 102, 192, 152, 221, 191, 111, 635]. -
More Model Theory Notes Miscellaneous Information, Loosely Organized
More Model Theory Notes Miscellaneous information, loosely organized. 1. Kinds of Models A countable homogeneous model M is one such that, for any partial elementary map f : A ! M with A ⊆ M finite, and any a 2 M, f extends to a partial elementary map A [ fag ! M. As a consequence, any partial elementary map to M is extendible to an automorphism of M. Atomic models (see below) are homogeneous. A prime model of T is one that elementarily embeds into every other model of T of the same cardinality. Any theory with fewer than continuum-many types has a prime model, and if a theory has a prime model, it is unique up to isomorphism. Prime models are homogeneous. On the other end, a model is universal if every other model of its size elementarily embeds into it. Recall a type is a set of formulas with the same tuple of free variables; generally to be called a type we require consistency. The type of an element or tuple from a model is all the formulas it satisfies. One might think of the type of an element as a sort of identity card for automorphisms: automorphisms of a model preserve types. A complete type is the analogue of a complete theory, one where every formula of the appropriate free variables or its negation appears. Types of elements and tuples are always complete. A type is principal if there is one formula in the type that implies all the rest; principal complete types are often called isolated. A trivial example of an isolated type is that generated by the formula x = c where c is any constant in the language, or x = t(¯c) where t is any composition of appropriate-arity functions andc ¯ is a tuple of constants. -
Set-Theoretic Foundations
Contemporary Mathematics Volume 690, 2017 http://dx.doi.org/10.1090/conm/690/13872 Set-theoretic foundations Penelope Maddy Contents 1. Foundational uses of set theory 2. Foundational uses of category theory 3. The multiverse 4. Inconclusive conclusion References It’s more or less standard orthodoxy these days that set theory – ZFC, extended by large cardinals – provides a foundation for classical mathematics. Oddly enough, it’s less clear what ‘providing a foundation’ comes to. Still, there are those who argue strenuously that category theory would do this job better than set theory does, or even that set theory can’t do it at all, and that category theory can. There are also those who insist that set theory should be understood, not as the study of a single universe, V, purportedly described by ZFC + LCs, but as the study of a so-called ‘multiverse’ of set-theoretic universes – while retaining its foundational role. I won’t pretend to sort out all these complex and contentious matters, but I do hope to compile a few relevant observations that might help bring illumination somewhat closer to hand. 1. Foundational uses of set theory The most common characterization of set theory’s foundational role, the char- acterization found in textbooks, is illustrated in the opening sentences of Kunen’s classic book on forcing: Set theory is the foundation of mathematics. All mathematical concepts are defined in terms of the primitive notions of set and membership. In axiomatic set theory we formulate . axioms 2010 Mathematics Subject Classification. Primary 03A05; Secondary 00A30, 03Exx, 03B30, 18A15.