ILP = Logic, CS, ML Stop counting, start reasoning

Transcription

1 ILP = Logic, CS, ML Stop counting, start reasoning Gilles Richard AOC team

2 The story so far Several actors K. Brouwer K. Godel J. Herbrand A. Colmerauer R. Kowalski S. Muggleton L. Brouwer ( ) K. Godel ( ) J. Herbrand( ) H. Colmerauer (1941) R. Kowalski (1941) S. Muggleton (1959)

3 Let us start from PAC PAC model (L. Valiant ) A class of concepts (functions) C = {c i i I } over X with proba. P A target concept (function) c C A training set TS X cardinal n An algo A such that A(TS) = h TS C with disagreement set : e TS = c h TS (true error set)(e TS TS : empirical error) Our hope : ɛ, δ ]0, 1[, P(µ(e TS ) > ɛ) < δ (B ɛ,δ ) C PAC learnable iff A polynomial w.r.t. 1 ɛ, 1 δ, n such that : ɛ, δ, c, n s.t. TS > n = B ɛ,δ Conclusion : Well suited to functional approach

4 Leads to standard proba/stat/optim approaches Yes Choose a class of functions F = {c i i I } over X Pray for the target class to be PAC learnable (VC-dim) Pray for the empirical error e emp to be not too far from the true error e true Get an analytical expression for e emp Minimize via optimization techniques (SGD, batch, etc.)

5 Leads to standard proba/stat/optim approaches Yes Choose a class of functions F = {c i i I } over X Pray for the target class to be PAC learnable (VC-dim) Pray for the empirical error e emp to be not too far from the true error e true Get an analytical expression for e emp Minimize via optimization techniques (SGD, batch, etc.) What if... The target class is not a function BUT a relation between objects R(x, y) - one to many instead one to one The target object is not a number BUT a set of structured objects like lists, trees, etc. (ex. : list permutations) We have no clean analytical expression for e emp

6 Leads to standard proba/stat/optim approaches Yes Choose a class of functions F = {c i i I } over X Pray for the target class to be PAC learnable (VC-dim) Pray for the empirical error e emp to be not too far from the true error e true Get an analytical expression for e emp Minimize via optimization techniques (SGD, batch, etc.) What if... The target class is not a function BUT a relation between objects R(x, y) - one to many instead one to one The target object is not a number BUT a set of structured objects like lists, trees, etc. (ex. : list permutations) We have no clean analytical expression for e emp Conclusion : another approach needed

7 Move to logic Basic material 1 A language to build formulas ; A, a,,,,, etc. 2 A way to give a meaning to formulas : semantics 3 A way to derive new formulas : inference via proof system or syntax

8 Move to logic Basic material 1 A language to build formulas ; A, a,,,,, etc. 2 A way to give a meaning to formulas : semantics 3 A way to derive new formulas : inference via proof system or syntax 3 types of logic (at least) 1 Propositional Logic (0 order) : (A B) ( B A) A, B : propositional variables falsity 2 Predicate Logic (First Order Logic) : x y P(x, y) Q(f (x), a), x individual variable, P, Q : predicate symbols, f : functional symbol, a constant, quantifiers 3 Higher Order Logic (mainly Second Order Logic) : X, x f, X (x, f (x)), X : predicative variable, f functional variable

9 Semantics and Syntax Semantics = Models notion of truth via compositional semantics principle Propositional logic : Boolean interpretation - truth table FOL : more sophisticated with domains (mathematical sets) and interpretations SOL : similar to FOL T F : F is true in any model of T.

10 Semantics and Syntax Semantics = Models notion of truth via compositional semantics principle Propositional logic : Boolean interpretation - truth table FOL : more sophisticated with domains (mathematical sets) and interpretations SOL : similar to FOL T F : F is true in any model of T. Syntax = Inference A computable set of rules to deduce new formulas starting from a set of formulas (theory) T and a relation T F : F is deducible/provable from T T = : F and F is a theorem 2 styles of presentation : Hilbert-style and Gentzen-style

11 Propositional Logic (PL) : proof system Gentzen style : introduction and elimination rules per connector A := A Context A B A B intro A B A elim1 A B B elim2 [A] B A A B A B intro B elim (Modus Ponens) A A B intro1 A A identity B A B intro1 [ A] A A B [A] C C [B] C elim elim(absurd classical logic)

12 Propositional Logic (PL) : proof system Gentzen style : introduction and elimination rules per connector A := A Context A B A B intro A B A elim1 A B B elim2 [A] B A A B A B intro B elim (Modus Ponens) A A B intro1 A A identity B A B intro1 [ A] A A B [A] C C [B] C elim elim(absurd classical logic) How it works A (B A) : to be done A A : to be done

13 First Order Logic (FOL) : proof system The rules PL + 2 rules : A x, A GEN(x non free in proof of A) x, A(x) A([t/x]) elim

14 First Order Logic (FOL) : proof system The rules PL + 2 rules : A x, A GEN(x non free in proof of A) x, A(x) A([t/x]) elim How it works ( x, P(x)) x, (P(x)) Horn clauses : x, y, z, p(x, y) q(f (y), z) r(x, z) equivalent to : p(x, y) q(f (y), z) r(x, z) x, y, p(x, y) is a fact HORNSAT (in PL) : linear time

15 Second Order Logic (SOL) : proof system only 1 connector needed then := X, X - X := X A B := X, ((A (B X )) X ) A B := X, (A (X (B X ))) xa := X, ( x, (A X ) X ) x = y := X, (X (x) X (y))

16 Second Order Logic (SOL) : proof system only 1 connector needed then := X, X - X := X A B := X, ((A (B X )) X ) A B := X, (A (X (B X ))) xa := X, ( x, (A X ) X ) x = y := X, (X (x) X (y)) The rules FOL reduced to 5 rules + 2 rules for on predicate variables A F, A GEN(X non free in proof of A) elim X, A A([F /X ])

17 Second Order Logic (SOL) : proof system only 1 connector needed then := X, X - X := X A B := X, ((A (B X )) X ) A B := X, (A (X (B X ))) xa := X, ( x, (A X ) X ) x = y := X, (X (x) X (y)) The rules FOL reduced to 5 rules + 2 rules for on predicate variables A F, A GEN(X non free in proof of A) elim How it works X, A A([F /X ]) P, P(0) x(p(x) P(x + 1)) x P(x) x, y, P(x, y) f, P(x, f (x)) (AC)

18 What do we expect? 1 soundness : if T A (provable), then T A (true) 2 completeness : if T A (true), T A (provable) 3 inconsistency : A s.t.t A and T A 4 consistency : the opposite! 5 satisfiability : T has a model (SAT ) 6 A (closed) decidable : T A or T A (K(s) = n, Goldbach conjecture?, etc.) 7 T decidable iff Th(T ) is a recursive set.

19 What do we expect? 1 soundness : if T A (provable), then T A (true) 2 completeness : if T A (true), T A (provable) 3 inconsistency : A s.t.t A and T A 4 consistency : the opposite! 5 satisfiability : T has a model (SAT ) 6 A (closed) decidable : T A or T A (K(s) = n, Goldbach conjecture?, etc.) 7 T decidable iff Th(T ) is a recursive set. Examples PL : sound - complete - decidable FOL : sound - complete (Gödel 1929) - only semi-decidable (Church-Turing) SOL : sound - not complete - not decidable (Gödel) For any consistent theory which is capable of expressing arithmetic, there is a true non decidable formula A (Gödel )

20 (Classical logic versus intuitionnistic logic) A standard example Proof : to be done + analysis Issues a, b R \ Q s.t. a b Q No clue about 2 2 Q! Rule of excluded middle has to be fired! Leading to intuitionnistic logic Hilbert problem and Gödel theorem 2 words only

21 Automatic deduction in FOL... why not? 2 ideas at least Bad : - Reduce non determinism by reducing the number of rules - No way to keep completeness Good : Consider a fragment of FOL where one rule is enough

22 Automatic deduction in FOL... why not? 2 ideas at least Prolog Bad : - Reduce non determinism by reducing the number of rules - No way to keep completeness Good : Consider a fragment of FOL where one rule is enough Fragment : Definite Horn clauses only x, y, z, p(x, y) q(f (y), z) r(x, z) Inference rule : Resolution (M. Davis - H. Putnam ) c1 a c2 σ(a) σ(c1 c2) where σ = MGU(a, b) Main tool to be implemented : Unification (A. Robinson )

23 Unification example : integration by parts The big deal b a b u(x)v (x)dx = [u(x)v(x)] a b u (x)v(x)dx a

24 Unification example : integration by parts The big deal Example b a b u(x)v (x)dx = [u(x)v(x)] a b u (x)v(x)dx b 2 options at least (substitutions) a xln(x)dx u(x) x and v (x) ln(x) : Then u (x) = 1 and v(x) = xln(x) x u(x) ln(x) and v (x) x : Then u (x) = 1 x2 x and v(x) = 2 a

25 Resolution example The big deal Resolve a b c Against d e f a Leads to : d e f c

26 Resolution example The big deal Resolve a b c Against d e f a Leads to : d e f c Lesson learned To prove c (c is our goal) Try to prove a and b Then try to prove d, e, f and b When do we stop? If we have a fact like d (asserting d is true), d is removed, But e, f and b remain And we have to carry on with the remaining clauses until nothing else to be proved

27 Logic as a programming language : Prolog Main characteristic - A. Colmerauer (1972) Non procedural : problem description only No type Reversibility : no input/output Non determinism Backtracking Syntax : d e f c written c : d, e, f.

28 Logic as a programming language : Prolog Main characteristic - A. Colmerauer (1972) Non procedural : problem description only No type Reversibility : no input/output Non determinism Backtracking Syntax : d e f c written c : d, e, f. Learning by doing! list membership and concatenation live

29 Logic as a programming language : Prolog Main characteristic - A. Colmerauer (1972) Non procedural : problem description only No type Reversibility : no input/output Non determinism Backtracking Syntax : d e f c written c : d, e, f. Learning by doing! list membership and concatenation live Well suited to AI problem

30 N-queens problem First setting : 1848 First solution : 1850 Researcher : C. F. Gauss, E. Dijkstra (1972) n = 8 : 4,426,165,368 combinations reduced to 8 8 > 16M with background knowledge - 92 solutions Reduce to 12 fundamental solutions (symmetry)

31 N-queens solution with Prolog : 10lines! queen at position (i,j) represented as q(i,j) sol ([ q(1, J1),q(2, J2),q(3, J3),q(4, J4),q(5, J5),q(6, J6),q(7, J7),q(8, J8 )]) :- permut ([1,2,3,4,5,6,7,8],[ J1,J2,J3,J4,J5,J6,J7,J8 ]), safe ([ q(1, J1),q(2, J2),q(3, J3),q(4, J4),q(5, J5),q(6, J6),q(7, J7),q(8, J8 )]). safe ([]). safe ([Q L]) :- noattack (Q,L), safe (L). noattack (Q, []). noattack (Q,[ X L]) :- peace (Q,X), noattack (Q,L). peace (q(i,j),q(k,l)): - I =\=K,J =\=L,I-K =\=J-L,I-K =\=L-J. DEMO?

32 N-queens solution with Prolog : 10lines! queen at position (i,j) represented as q(i,j) sol ([ q(1, J1),q(2, J2),q(3, J3),q(4, J4),q(5, J5),q(6, J6),q(7, J7),q(8, J8 )]) :- permut ([1,2,3,4,5,6,7,8],[ J1,J2,J3,J4,J5,J6,J7,J8 ]), safe ([ q(1, J1),q(2, J2),q(3, J3),q(4, J4),q(5, J5),q(6, J6),q(7, J7),q(8, J8 )]). safe ([]). safe ([Q L]) :- noattack (Q,L), safe (L). noattack (Q, []). noattack (Q,[ X L]) :- peace (Q,X), noattack (Q,L). peace (q(i,j),q(k,l)): - I =\=K,J =\=L,I-K =\=J-L,I-K =\=L-J. DEMO? BUT WE DO ML NOT AI!

33 Progol : learn a Prolog program Logical framework Input : TS = TS + TS (examples), a set of facts Background B, a consistent set of formulas B /TS + Find H such that B H consistent, B H TS +, B H /TS and H "simpler than" TS. How to do it Equivalent to B TS + H (contraposition) Let be, the conjunction of all ground literals s.t. B TS + Necessarily H i.e. H THEN H can be chosen as a subset of clauses subsuming

34 Progol : internal mechanism Algorithm init : H = While TS + { Choose e TS + - construct for e then select h from H = H {h} TS + = TS + \ {e TS + s.t. B H e} }

35 Progol : internal mechanism Algorithm init : H = While TS + { Choose e TS + - construct for e then select h from H = H {h} TS + = TS + \ {e TS + s.t. B H e} } The magic : tuning Introducing types Introducing input/output via mode declaration Limit the number of resolution step r Limit the stack depth h before backtracking No theoretical results regarding these numbers (core of good practices)

36 Progol : learning by seeing...

37 Progol : learning by seeing...

38 Progol : learning by seeing...

39 Progol : learning by doing... LEARNING SET MEMBERSHIP OR CONCAT OR...

40 Progol : learning by doing... LEARNING SET MEMBERSHIP OR CONCAT OR... Some options Progol is not alone! FOIL ( R. Quinlan - http :// Aleph ( A Learning Engine for Proposing Hypotheses - A. Srinivasan - Oxford) quickfoil ( Univ. of Wisconsin-Madison)

41 Bottou intro? crystal clear now Success of optim/proba/stat based methods 1 Very old domain 2 Very large community of mathematicians 3 Nvidia with linear algebra as core business

42 Bottou intro? crystal clear now Success of optim/proba/stat based methods 1 Very old domain 2 Very large community of mathematicians 3 Nvidia with linear algebra as core business (relative) failure of logic-based methods 1 Recent domain (20th century for FOL and SOL) 2 Sparse community of mathematicians (especially France) 3 No Nvidia with unification as core business

43 Bottou intro? crystal clear now Success of optim/proba/stat based methods 1 Very old domain 2 Very large community of mathematicians 3 Nvidia with linear algebra as core business (relative) failure of logic-based methods 1 Recent domain (20th century for FOL and SOL) 2 Sparse community of mathematicians (especially France) 3 No Nvidia with unification as core business BUT the future is...

44 XAI! darpa.mil/program/explainable-artificial-intelligence

45 Thank you for your time