INF551 Computational Logic: Artificial Intelligence in Mathematical Reasoning Stephane´ Graham-Lengrand [email protected] Practical information Timetable: 9 (lectures & practicals) Fridays [14:00-16:00] & [16:15-18:15], from 21st September Room: Nicole-Reine Lepaute, Turing building Website (slides, links to practical sheets, . ): http: //www.enseignement.polytechnique.fr/informatique/INF551/ Evaluation: 25%: Class participation (once in the 8 remaining weeks: showing to the class what you have done / tried to do from one week to the next) 75%: Exam on 21st December (using paper+laptop) . If you need to be evaluated by a project, come and see me. 2 Course notes Distributed course notes = G. Dowek’s book Available in English! Course notes are self-contained. Some of its material should have / may have been covered in one of your undergraduate courses, typically INF412 at Polytechnique. This year in INF551 we shall: treat parts of book not treated in undergrad. curriculum develop other parts further make TD more practical / machine-based. 3 Lecture 0 Introduction 4 Reasoning and computing You reason since the first words you spoke You compute since kindergarten You encountered the definition of such mechanisms at undergrad. level (INF412) Similar situation in the History of Science: reasoning and computing since Stone Age . properly defined during XIXth or XXth century 5 Reasoning vs. computing Easier to compute than to reason (especially since computers) Follow some computing rules. No intelligence needed, just time and space. Can be automated (done by a machine). Question addressed by this course: Can we use computing to reason? 6 What do we mean by “reasoning”? “The art of establishing truth.” Since ancient times, the question is: Is a particular mathematical statement true or false? What do we mean by “true”? Since ancient times: confronting with reality First mathematical fields = arithmetic and geometry c.f. Plato (e.g. Meno) Since then, the fields of mathematics have. diversified got further and further away from witnessable reality 7 XIXth century: Logic crisis Most notable example: The notion of infinity How can truth be checked against reality? Triggered by works of Cantor identifying 6= notions of infinity R ' converging sequences in Q (1870-1872) 2 Bijections & jQj 6= jRj (1874), jP(A)j > jAj, jRj = R to Dedekind in 1877: “Je le vois, mais je ne le crois pas!” Continuum Hypothesis (1878) 1874-1884: developped ideas into theory of ordinals and cardinals 1883: Cantor’s set premisses of Mandelbrot’s fractal theory Huge impact on topology& measure theory (Borel-Lebesgue,. ) Criticised by e.g. Kronecker for not being able to produce sets in finite number of “steps” from natural numbers First ideas of constructivism. To Lindemann proving that π is transcendental: “Why study such problems when irrational numbers do not exist?” Fear of Axioms, Fear of paradoxes/inconsistencies (lots found between 1885-1905) 8 A standard example Suppose a and b are strictly positive integers. a = b a2 = ab a2 − b2 = ab − b2 (a − b)(a + b) = b(a − b) a + b = b b + b = b 2b = b 2 = 1 Teacher: “This is wrong! You divide by 0!” Pupil: “Can I not?” Reasoning is about rules (applying them correctly), more than about truth. ) Computation? 9 Computation & the foundations of Mathematics From the 1870-1940 research period: rules of logical inference formalised Combined with axioms to form the notion of proof , they shifted the problem of truth to the problem of provability: Is a statement true or false? ) Is a statement provable or not? suggested that rigourous reasoning could be reduced to a mechanical task: “If controversies were to arise, there would be no more need of disputation between two philosophers than between two calculators. For it would suffice for them to take their pencils in their hands and to sit down at the abacus, and say to each other: Let us calculate.” Leibniz (1677) Inference rules & mathematical axioms: incredibly tied together “more-or-less” interchangeable, but rules are usually more “computer-friendly” More generally: Computation intrinsically tied to the foundations of mathematics (Which axioms & rules can be used to develop all mathematics & describe any problem that we would like a machine to solve?) 10 Logic gave birth to computers and A.I. General-purpose computers invented on paper (30s) before built in real life! . for the very purpose of automating mathematical reasoning . by the very people who cleared up the foundations of mathematics (Goedel, Church, Turing) “I propose to consider the question, ‘Can machines think?’ ” (1950) “Artificial Intelligence” coined by John McCarthy at Dartmouth Conference (1956) 11 A very brief & approximate overview of A.I. “Can machines think?” Goals ranging Deduction, reasoning, to Interaction with human / from problem solving environment, NLP McCarthy & Minsky found A.I. lab at MIT (1958) then McCarthy found Stanford A.I. lab (SAIL) (1963) Approaches: Symbolic A.I. Logic-based A.I. Neural nets Cognitive simul. “The neats” “The scruffies” (McCarthy, championing (Minsky) mathematical logic for A.I.) Logic programming Probabilistic methods Automated reasoning Uncertain reasoning Constraint solving Machine learning 12 In this course: Logic-based approach to A.I. Several algorithmics disciplines related to proofs: Inference rules { # Algorithmics of Algorithmics of Algorithmics of Proof-checking Proof-searching Proof-reducing Easy Lect. 2-5 Lect. 9 Two questions remain central: Termination of such algorithms Determinism of such algorithms and computational cost Some unexpected results: While proof-search can be implemented (under minimalistic hypotheses) as computational process, the converse is true: All computation can be seen as the search for a mathematical proof Proof-search not just 1 small algorithmic domain ) ubiquitous in CS 13 The plan Week 1: Introduction and review of undergrad. material Week 2,3,4,5: Algorithmics of proof-search Week 6,7,8: Modelling all mathematical problems in universal framework Week 9: Constructivism 14 Questions? 15 Lecture I What do you remember from your undergraduate programme? 16 Contents I. Predicate logic II. Meta-mathematics III. Some theories IV. The notion of proof & the soundness/completeness theorem V. Computability 17 I. Predicate logic 18 Stating the obvious Logic is about statements, statements require language. As usual, language = syntax + semantics Predicate logic on two levels: terms (whose semantics are the mathematical objects you want to talk about) formulae (the aforementioned statements) The rest of this section presents chapter 5 of INF412 course notes. 19 More generally, today’s slides indicate section numbers in course notes: Section in INF412 Slide title Section in INF551 http://www.enseignement.polytechnique.fr/informatique/ INF412/i.php/Main/Poly 20 5.1 Syntax of terms 1.2-1.3 Syntax of terms depends on: a term signature, i.e., a set of elements called term symbols, equiped with a function mapping term symbols to natural numbers called their arities a denumerable set of elements called variables Terms are defined inductively as follows: a variable is a term for every term symbol f of arity n in the signature, if t1,. ,tn are terms, then f(t1; : : : ; tn) is a term Do not see f(t1; : : : ; tn) as a string with parentheses and f commas: even though I use this as concrete syntax on my slides, what I am talking about is really a tree, with the root ... t labelled with symbol f and n direct sub-trees t1,. ,tn: t1 n 21 5.1 Syntax of terms 1.2-1.3 That was long and tedious. Here’s a more synthetic presentation of the same thing: Let Σ be a term signature. The set of terms is defined by t; t1; t2;::: ::= x j f(t1; : : : ; tn) with x ranging over variables and f=n ranging over Σ 22 5.1 Syntax of formulae 1.2-1.3 Similarly: predicate signature = set of elements called predicate symbols, equiped with function mapping predicate symbols to natural numbers called their arities Let Ψ be a predicate signature. The set of (pre-)formulae is defined by A; B; C; : : : ::= p(t1; : : : ; tn) j > j ? j :A j A ^ B j A _ B j A ) B j 8xA j 9xA with x ranging over variables and p=n ranging over Ψ Again, we are talking about trees But this is not finished! 23 5.1 Syntax of formulae 1.2-1.3 Formulae are not exactly trees, because 8x p(x) is the same formula as 8y p(y) (though technically 6= trees) R 1 x is a bound variable, as is x in 0 f(x)dx Set of formulae = set of such trees quotiented by the renaming of bound variables The next two slides formalise this. 24 5.2.2 Free variables 1.2-1.3 Bound= not free Free variables of terms, defined by induction on terms: FV(x) := x FV(f(t1; : : : ; tn)) := FV(t1) [···[ FV(ti) Free variables of formulae, defined by induction on formulae: FV(p(t1; : : : ; tn)) := FV(t1) [···[ FV(tn) FV(>) = FV(?) := ; FV(:A) := FV(A) FV(A ^ B) = FV(A _ B) = FV(A ) B) := FV(A) [ FV(B) FV(8xA) = FV(9xA) := FV(A)nfxg t (resp. A) is closed if FV(t) = ; (resp. FV(A) = ;) 25 α-equivalence 1.2-1.3 Define the swap of 2 variables in (all terms, and then) all formulae P : (xy)P everywhere where x is written (free or bound), write y, and vice versa (easy definition by induction on terms and formulae) 8xP is identified with 8y (xy)P if y 62 FV(P ) 9xP is identified with 9y (xy)P if y 62 FV(P ) α-equivalence := smallest equivalence relation on trees identifying the above at the root of trees or in their sub-trees Example: p(w) ^ 8x p(x + w) and p(w) ^ 8y p(y + w) are α-equivalent Why “if y 62 FV(P )” (i.e.