From Hilbert s Program to a Logic Toolbox

Transcription

1 From Hilbert s Program to a Logic Toolbox How to teach Sets and Logic for students in Computer Science Johann A. Makowsky Department of Computer Science Technion Israel Institute of Technology Haifa, Israel janos@cs.technion.ac.il janos : President of the European Association of Computer Science Logic ttl2015.tex 1

2 Previous presentations of similar talks Conference presentations: LPAR 2007, Eriwan International Symposiums on Artificial Intelligence and Mathematics, 2008, Fort Lauderdale, Florida CSR 2008, Moscow CiE 2008, Athens 2009, ETH Zürich Full paper: J.A. Makowsky, From Hilbert s Program to a Logic Toolbox. Annals of Mathematics and Artificial Intelligence, (2008), pp ttl2015.tex 2

3 Logic for the Working Computer Scientist : towards a re-orientation The classical approach: The misleading narrative. Modeling artefacts: Reality, models, and side-effects. Semantics and Formal Methodes: Programming, verification, and querying knowledge bases. Understanding limitations: What cannot be achieved in a given framework. A logic tollbox: From foundations to engineering. ttl2015.tex 3

4 My own background Research: Mathematical Logic, Foundations of Computer Science, Application of Logic to Combinatorics. Teaching: Mathematical Logic and Logic in Computer Science, Foundations of AI and Databases. Industry: European Space Agency, Zurich Financial Services, Co-founder of mental images, GmbH, Berlin, in 1986 (sold to NVIDIA in 2007) Regular columnist for Finanz & Wirtschaft, Zurich ( ) More details, Skip details, ttl2015.tex 4

5 My own research background: : Mathematical Logic, Classical Model Theory, Abstract Model Theory and Generalized Quantifiers; : Application of Logic to Semantics, Logic Programming and Databases; 1995-today: Application of Logic to Algorithmics and Combinatorics. My own undergraduate teaching experience: Courses I have designed Logic for Computer Science, Sets and Logic for Computer Science Database Systems, Database Theory Foundations of Logic Programming, Foundations of Automated Theorem Proving ttl2015.tex 5

6 My own industrial experience: : Büro Dr. Haller subcontracting from the European Space Agency. Exact computation of geostationary orbits of satelites : Zürich Insurance (ZFS). Feasibility study concerning insurability of computer related risk : Finanz und Wirtschaft. Columnist. Writing a regular column on computing related issues : mental images g.m.b.h. Chief scientific consultant. mental images, founded in 1986, is the recognized international leader in providing component and platform software for the creation, manipulation and visualization of 3D content. Its world leading rendering and other technologies are used by the entertainment, computer-aided design, architecture, scientific visualization, and other industries that require sophisticated images primarily as part of their software products and application services. Since 2008, mental images, is a wholly-owned subsidiary of NVIDIA Corporation with headquarters in Berlin, Germany, a subsidiary in the United States, mental images, Inc., and offices in Melbourne, Australia and Stockholm, Sweden. ttl2015.tex 6

7 What to teach from logic? In this lecture I want to examine what we should teach from logic to our non-specialized undergraduate students. I mean, what does every graduate of Computer Science have to learn in/from logic? The current syllabus is often justified more by the traditional narrative than by the practitioner s needs. ttl2015.tex 7

8 Outline of the talk Part 1: The Logical Foundations of Mathematics From Frege to Gödel: the traditional narrative Sets for the Working Mathematician: the practical narrative Part 2: Lessons for the Working Computer Science Graduate My Toolbox Sets as universal data structure Computability Syntax and semantics Definability and interpretability ttl2015.tex 8

9 From Frege to Gödel: The traditional narrative ttl2015.tex 9

10 Logical Foundations of Mathematics Act I: Cantors Paradise Act II: Paradise lost Act III: Hilbert s Program Act IV: Rise and fall of Hilberts Program Act V: Clarifications and repairs Interlude: Sets for the working mathematician Act VI: 100 years later - Fixing Frege Skip details ttl2015.tex 10

11 Logical Foundations of Mathematics Act I: Cantors Paradise First G. Cantor ( ) created the Paradise of Sets Then G. Frege (1879) created the modern Logical Formalisms, including the correct binding rules for quantification, and set out to lay the Foundations of Mathematics with his Die Grundgesetze der Arithmetik, Volume1 (1893). G. Peano, author of The principles of arithmetic, presented by a new method (1889), wrote a positive review of it. Frege s Program: Explain all of Mathematics using Logic (and Set Theory). ttl2015.tex 11

12 Logical Foundations of Mathematics Act II: Paradise lost On 16 June 1902, Bertrand Russell pointed out, with great modesty, that the Russell paradox gave a contradiction in Frege s system of axioms. And with Russel s paradox started the crisis of the Foundations of Mathematics, G. Cantor had sensed this, when he noticed trouble with the set of all sets. Let V be the set of all sets. Then its power set P (V ) V. But V < P (V ), a contradiction. ttl2015.tex 12

13 Logical Foundations of Mathematics Act III: Hilbert s Program (quoted from Wikipedia) D. Hilbert around 1920 designs a program to provide secure foundations for all mathematics. In particular this should include: Formalization of all mathematics: all mathematical statements should be written in a precise formal language, and manipulated according to well defined rules. Completeness: a proof that all true mathematical statements can be proved in the formalism. Consistency: a proof that no contradiction can be obtained in the formalism of mathematics. This consistency proof should preferably use only finitistic reasoning about finite mathematical objects. ttl2015.tex 13

14 Logical Foundations of Mathematics Hilbert s Program (contd) Conservation: a proof that any result about real objects obtained using reasoning about ideal objects (such as uncountable sets) can be proved without using ideal objects. Decidability: there should be an algorithm for deciding the truth or falsity of any mathematical statement. In 1928, D. Hilbert and W. Ackermann publish Grundzüge der theoretischen Logik. The Logic in question is Second Order Logic. What we call First Order Logic, is called there the restricted calculus. They prove soundness of the calculus, and ask the question of completeness. ttl2015.tex 14

15 Logical Foundations of Mathematics Act IV: Rise and Fall of Hilberts Program Initial successes: Leopold Löwenheim (1915), Thoralf Skolem (1920), Mojżesz Pressburger (1929), Alfred Tarski (1930), Frank Plumpton Ramsey (1930), László Kalmár (1939) and many others prove partial decidability results for fragments of Logic, and for Arithmetic, Algebra, Geometry. In 1929 Kurt Gödel proves the completeness of the Hilbert-Ackermann axiomatization of the the restricted (first order) calculus. ttl2015.tex 15

16 Logical Foundations of Mathematics Rise and Fall of Hilberts Program (contd) Final blows: 1931 K. Gödel proves that every recursive theory which contains arithmetic is incomplete K. Gödel proves that every recursive consistent theory which contains arithmetic cannot prove its own consistency Alonzo Church and Alain Turing show that already for the restricted calculus with free relation variables the set of tautologies is not computable (but is semi-computable). ttl2015.tex 16

17 Logical Foundations of Mathematics Act V: Clarifications and repairs Proof Theory arises from work by W. Ackermann, G. Gentzen, J. Herbrand, D. Hilbert and P. Bernays. Set Theory arises from work by E. Zermelo, D. Mirimanoff, J. von Neumann, A. Fränkel, K. Gödel and P. Bernays. Alternative approaches are developed by, among others, W. Quine, W. Ackermann, and J.L. Kelley and A.P. Morse Recursion Theory arises from work by E. Post, J. Herbrand, K. Gödel, A. Church, A. Turing, H. Curry. Model theory arises from work by T. Skolem, A. Tarski, A. Robinson, R. Fraïssé and A. Mal cev, And for long this remained the classical divide of Mathematical Logic. ttl2015.tex 17

18 Interlude: The Foundations of Modern Analysis A pragmatic Frege program It used to be customary to teach the foundations of calculus by: Starting with sets. Defining the number systems N, Z, Q and their arithmetic operations inductively and using quotient structures. Defining the reals R, using Dedekind cuts. Defining structures, say, groups, fields, Banach spaces, Lie algebras, axiomatically. Existence of axiomatically defined objects had to be established by an explicit sequence of set construction steps within the cumulative hierarchy. ttl2015.tex 18

19 Logical Foundations of Mathematics Act VI: 100 years later - Fixing Frege C. Wright, P. Geach and H. Hodes suggested, and G. Boolos proved (1987) that the modified Frege program actually is feasible. G. Boolos, On the Consistency of Frege s Foundation of Mathematics, reprinted in: G. Boolos, Logic, Logic and Logic, Harvard University Press, So we have: Frege: The Peano Postulates can be deduced in dyadic second order logic from Hume s principle and suitable definitions of the natural numbers (Frege s Arithmetic). Boolos: Frege s arithmetic is interpretable in second order Peano Arithmetic. J.P. Burgess, Fixing Frege, Princeton University Press, 2005 That much for the big crisis. ttl2015.tex 19

20 The classical textbooks in Logic and Naive Set Theory The Classical Texts follow this narrative: Logic is needed to resolve the paradoxes of set theory. First Order Logic is THE LOGIC due to its completeness theorem. The main theorems of logic are the Completeness Theorem and the Compactness Theorem The tautologies of First Order Logic are not recursive. Arithmetic Truth is not recursive enumerable. One cannot prove CONSISTENCY within rich enough systems. Naive Set Theory is used for the Foundations of Analysis and Topology. ttl2015.tex 20

21 This is NOT what a Practitioner of Computing Sciences NEEDS! ttl2015.tex 21

22 So WHAT aspects of Logic does a Practitioner of Computing Sciences NEED? ttl2015.tex 22

23 Knowledge of theoretical orientation vs practical knowledge, I Theoretical orientation: awareness that his domain of discourse is an idealized world of artefacts which models fairly accurately the artefacts which allow us to run and interact with computing machinery. awareness of the different levels of abstractions. awareness that in this world of artefacts there are a priori limitations. Not everything is realizable, computable, etc. ttl2015.tex 23

24 Knowledge of theoretical orientation vs practical knowledge, II Practical knowledge: tools which allow him to model new artefacts, whenever they arise; languages which allow him to describe properties of the modeled artefacts in Second Order Logic. tools which allow him to manipulate quantifiers. including: Ehrenfeucht-Fraïssé Games. tools which allow him to prove properties of the modeled artefacts in the language of sets. ttl2015.tex 24

25 He needs a carefully adapted blend of the practical Frege program, with the knowledge of its limitations. He needs both proficiency and performance in his practical knowledge. ttl2015.tex 25

26 He does NOT need to see (in a first course) A a proof of the completenss theorem for Propositional Order Logic for Hilbert-style or Gentzen-style proof systems. A a proof of the completenss theorem for First Order Logic, (Hilbert-Ackermann: the restricted calculus) the Compactness Theorem. In the sequel of this talk, as time permits, I elaborate. ttl2015.tex 26

27 My Logic Toolbox The Fundamental Property of Transductions and Interpretations. The Ehrenfeucht-Fraïssé Theorem and its refinements. The Feferman-Vaught Decomposition Theorem and its many variations. Note that all these tools were already developed before 1960, and used by specialists, but not included in introductory textbooks. I have surveyed how to use these tools in Computer Science in my papers J.A. Makowsky and E.V. Ravve, Dependency Preserving Refinements and the Fundamental Problem of Database Design, Data & Knowledge Engineering, vol (1998) pp J.A. Makowsky, Algorithmic uses of the Feferman-Vaught theorem, Annals of Pure and Applied Logic, vol. 126 (2004) pp ttl2015.tex 27

28 Where these tools work Linear Algebra in Math is pervasive. Logic, as traditionally tought, is NOT pervasive in the CS curriculum. It is important that the student sees the tools work in the basic courses, and in the follow-up courses. My toolbox is (or could/should be) at work in Automata Theory and Computability Database Systems Graph Algorithms and Complexity Formal Methods and Verification, in all their ramifications Decision procedures in Automated Theorem Proving ttl2015.tex 28

29 Lessons from 150 years of Modern Logic and the Foundations of Mathematics As much as time permits..., or read the full paper! Modeling the world Modeling computability Modeling syntax and semantics Limitations of formalisms Intepretatbility and Reducibility ttl2015.tex 29

30 1. Modeling the world Our scientific language: Natural Language enhanced by precise use of boolean operations, quantification and the use of naive language of sets. Our universal data structure: A cumulative world of sets. Modeling the world: We model ALL artefacts of our computing world by constructed objects in the world of sets. Modeling involves side effects: Modeled artefact have properties not intended. Ordered pairs: N. Wiener: (x, y) := {{{x}, }, {{y}}}, K. Kuratowski {{x}, {x, y}}, Simplified {a, {a, b}}. Fixing levels of abstraction: Introducing structures, and fixing which sets are not further to be analyzed. A graph is a pair < V, E >. A finite automaton is a tuple < S, Σ, R, I, T >. ttl2015.tex 30

31 Lesson 1 (contd) Like in the foundations of Analysis, as practiced by R. Dedekind, E. Landau and N. Bourbaki, we need the precise language mix of normalized natural language augmented by the language of sets to model the idealized artefacts of computer science. Artefacts: strings, concatenation, natural numbers, graphs, relational structures stacks, arrays; circuits, Turing machines, register machines; specification and programming languages, Tools: Inductive definitions, proofs by induction; enumerations, proving countability and uncountability; well-orderings (for termination) ttl2015.tex 31

32 Is this not too denotational? a worried friend asked me once. It does map everything into sets. But truth does not presuppose a world of sets. Truth in the sense of Frege s world is defined by the laws (introduction and elimination rules) of logic and of the Fregean constructs. It leaves your foundational options open... ttl2015.tex 32

33 2. Modeling Computability and its limitations (when modeled) Computability is modeled over different domains, computing operations, resource restrictions. Natural numbers and recursion: The original definition of the set of recursive functions. Natural numbers and register machines: Close to early programming languages. Turing machines and words: Close to assembly languages. Other models: Logic programs, Lambda calculus, cellular automata, quantum computing Showing their equivalence involves Translation between the domains. Translations between programs (interpreters and compilers). ttl2015.tex 33

34 Lesson 2 (contd) Computability may be taught before logic in a more naive way. Here I want to stress The different basic structures involved, and their bi-interpretability. Orientation: Tools: Not everything is computable. Precise statement of various versions of the the Church Turing Hypothesis. Separating slogans from precise definitions Effectively computable = P, NP, RP, QP,...? Proving non-computability. Establishing simulations. ttl2015.tex 34

35 3. Modeling Syntax and Semantics We look at Propositional, First Order, Second Order Logic, or any other logic of assertions. Syntax: Semantics: The syntax is an inductively defined set of words, the well formed expressions. Structures are interpretations of the basic non-logical symbols. Assignments are interpretations of the variables. The meaning function associates with structures, assignments, and formulas a truth value. What is the meaning of an assertion? ttl2015.tex 35

36 Lesson 3 (contd) The meaning of an assertion is: Without free variables: A truth value. But this is misleading! With free variables: The set of interpretations of its free variables. Only first order variables: A relation We define usually logical validity via truth values. It would be preferable to define validity and logical consequence directly for formulas with free variables. ttl2015.tex 36

37 Do we need the Completeness Theorem? For the practical knowledge we need: The semantic notion of logical consequence. Enough basic logical equivalences to to prove the Prenex Normal Form Theorem (PNF). Introduction and elimination rules for quantifiers (via constants). A game theoretic interpretation of formulas in PNF. For the knowledge of orientation we might state (but not prove) the Completeness Theorem for our redundant set of manipulation rules. ttl2015.tex 37

38 Arguments for/ against proving the Completeness Theorem The classical argument pro: Completeness and its corollary, Compactness is at the heart of logic. My arguments contra: None of these are part of the practical knowledge we aim at. The proof of the Completeness Theorem is a waste of time at the cost of teaching more important skill of understanding the manipulation and meaning of formulas. First Order Logic is not privileged in our context. We deal very often with finite structures, where the Completeness Theorem is not true. Second Order Logic anyhow is the natural logic we work in, and not taking that seriously confuses the student. ttl2015.tex 38

39 Lesson 3 (contd) We should concentrate on understanding quantification: Tools: Read, write and understand the meaning of First Order and Second Order formulas. Understand the relationship between projection of relations and quantification. Understand that Relational Calculus and First Order Logic are really the same (i.e., bi-interpretable). Play with the game interpretation of quantifiers to analyze the amount of quantification needed to express, say there exists at least n elements x such that φ(x). ttl2015.tex 39

40 4. Limitations of formalisms: Definability Before we prove the Completeness Theorem I would like the students to understand the difference between First Order (FO) and Second Order (SO) Logic. There are an equal number of x with P (x) and Q(x) where P, Q are unary predicate symbols, is expressible in SO but not in FO. We can prove this having the games available. In the natural numbers N, multiplication is SO-definable using addition only, but not FO-definable. Multiplications is FO-definable using addition and squaring. The negative result we can not prove in an undergraduate course, as we need the decidability of FO Pressburger Arithmetic. But we can explain it. ttl2015.tex 40

41 5. Interpretability and Reducibility Before we prove the Completeness Theorem I would like the students to understand what it means that The integers Z with their arithmetic are interpretable inside the natural numbers N with their arithmetic. We define a transduction T (N) = Z as follows. The new universe consists of equivalence classes of pairs of natural numbers such that (x, y) (x, y ) iff x + y = x + y. The new equality is this equivalence. The new addition is the old addition on representatives. Same for multiplication. ttl2015.tex 41

42 Lesson 5 (contd) We define the interpretation S : F ormulas F ormulas as follows: For any SO-formula φ we let S(φ) be the result of substituting the new definitions of addition and multiplication and equality for the corresponding symbols. S and T are intimately related: Z = T (N) = φ iff N = S(φ) which is the Fundamental Property of Transductions and Interpretations. ttl2015.tex 42

43 Lesson 5 (contd) In the same way we can see that The Cartesian product is interpretable in the disjoint union. Many graph transformations are given as transductions. All implementations of one data structure in another are of this form. Transductions and interpretations are everywhere ttl2015.tex 43

44 The Fundamental Properties of SO Isomorphic structures satisfy the same SO sentences That Fundamental Property of Transductions and Interpretations. The Prenex Normal Form Theorem and its visualization as a two person game. ttl2015.tex 44

45 The Fundamental Properties of FO Besides the properties of SO we have The Ehrenfeucht-Fraïssé Theorem: Two structures can be distinguished by a sentences of quantifier depth k iff Player I can force a win in the EF-game of length k. or, equivalently Two structures cannot be distinguished by a sentences of quantifier depth k iff they are k-isomorphic. Furthermore: k-isomorphism is preserved under the formation of disjoint unions of structures. Modified versions also hold for Monadic Second Order Logic, but not for SO. ttl2015.tex 45

46 Combining EF-Games and Transductions Combining games and transductions gives a very powerful tool to compute the meaning function of a FO formula in a complex structure by reducing this computation to simpler structures. If G is obtained from graphs H 1, H 2 by applying disjoint unions, Cartesian products, and first order definable transductions T 1, T 2, say G = T 1 (H 1 T 2 (H 2 )) then the truth of the formulas of quantifier rank k in G is uniquely and effectively determined by the the truth of the formulas of quantifier rank k which hold in H 1 and H 2. This is the Feferman-Vaught Theorem. ttl2015.tex 46

47 THANK YOU FOR YOUR ATTENTION! ttl2015.tex 47