Logics for Computation The Story so Far Lecture #9: Putting Tiles I We are working with first-order logic. in an Infinite Bathroom I We discussed soundness and completeness Intuitively, they are the way to synchronize the semantics of a logic with an inferece method like tableaux. Carlos Areces and Patrick Blackburn They are the way to show that we got it right. {carlos.areces,patrick.blackburn}@loria.fr I We also discussed some properties like Compactness and L¨owenhein-Skolem. INRIA Nancy Grand Est I We saw that this properties actually characterize first order Nancy, France logic: Lindstr¨omTheorem. I We’ve come a long way. Think that on Monday we were in ESSLLI 2008 - Hamburg - Germany propositional logic! Areces & Blackburn: Logics for Computation INRIA Nancy Grand Est What do we do Today Tableaux for First Order Logic I Patrick introduced these two rules yesterday: s: x ϕ s: x ϕ I We’ll talk about the satisfiability problem of first-order logic. h i ( x ) ¬h i ( x ) s:ϕ[x n] h i s: ϕ[x o] ¬h i I We will argue that the problem is undecidable ← ¬ ← (wheren is a new name) (whereo is an old name) I That is, there is no algorithm that can answer for any formula of first order logic, whether the formula has a model or not. I How comes that this tableaux do not terminate? I Actually, we are going to show how to tile and infinite I As we already mentioned, bathroom. The SAT problem of FO is undecidable. I Hence, we cannot expect any sound and complete tableaux to also terminate. Areces & Blackburn: Logics for Computation INRIA Nancy Grand Est Areces & Blackburn: Logics for Computation INRIA Nancy Grand Est (Un)Decidability Tiling Problems I A tiling problem is a kind of jigsaw puzzle How can we prove that problem X is undecidable? I a tile T is a 1 1 square, fixed in orientation, with a fixed × One way is color in each side I Ask somebody (more intelligent than us) to prove that some I for example, here we have six different kinds of tiles: @ @ @ @ @ @ problem Y is undecidable @ @ t@t t t @ t@t t t @ @ @ tt@ ttt @ tt@ ttt @ @ @ t@ @ t@ @ I Prove that if X would be decidable then Y would be @ @ @ t @ @ @ ttttt decidable, giving a codification of Y into X. t t t t I a simple tiling problem, could be: The halting problem of Turing machines is the standard example of Is it possible to place tiles of the kind we show above @ @ @ @ an undecidable problem. The behaviour of a Turing machine, and r@ r r r @ @ @ on a grid of 2 4, in such a way that we cover the rrr @ r r @ @ @ the predicate that says that a given turing machine stops on all × rr r r r entire grid and that adjacent tiles have the same @ @r r r r @ @ inputs can be expressed in FO. @ r@ r r r @ @ color on neighboring sides? @ rrr @ r r @ @ r rr r r r r r Areces & Blackburn: Logics for Computation INRIA Nancy Grand Est Areces & Blackburn: Logics for Computation INRIA Nancy Grand Est Tiling Problems Representing Tiling Problems I Notice that every finite set of tiles = t ,..., t can be T { 1 k } I The general form of a tiling problem represented as two binary relations H, and V . Given a finite number of kind of tiles , can we cover a given We put H(ti , tj ) when the right side of ti coincide with the T part of Z Z in such a way that adjacent tiles have the same left side of tj , and similarly for V . × color on the neighboring sides? I For example, for I In some cases, it is also possible to impose certain conditions on what is considered a correct tiling. t t t t t t I Covering N N 1 2 3 4 5 6 × 3 I tiling N N: Given a finite set of tiles , can × T T we will have cover N N? 2 × I this problem is undecidable (It is equivalent to 1 H = (t , t ), (t , t ), (t , t ), (t , t ), (t , t ),... { 1 3 1 6 2 4 2 5 2 1 } the halting problem of Turing machines) V = (t , t ), (t , t ), (t , t ), (t , t ), (t , t ),... 0 1 2 3 { 1 3 1 5 1 6 2 3 2 5 } Areces & Blackburn: Logics for Computation INRIA Nancy Grand Est Areces & Blackburn: Logics for Computation INRIA Nancy Grand Est Coding a Tiling of the Grid in FO Relevant Bibliography I I Theorem: The problem of deciding whether a given FO formula is Undecidable Problems satisfiable is undecidable. Let and be relations, and t1,..., tn be propositinal symbols. → ↑ I Alonzo Church invented lambda calculus and Let ϕ be the conjunction of the formulas: propose it as a model for computation. 1) Total:[ x] y (x: R y) for R , [x] y (x: R y) for He showed then that the problem of satisfiability h i h i ∈ {→ ↑} h i h i I R , of first-order logic could not be coded in the ∈ {→ ↑} 2) Functional:[ x][y][z](x: R y x: R z y:z) for lambda calculus. Hence it was uncomputable. h i ∧ h i → R , [x][y][z](x: R y x: R z y:z) for R , ∈ {→ ↑} h i ∧ h i → ∈ {→ ↑} I Here is a list of some of Church’s doctoral 3) Commuting: students: A. Anderson, P. Andrews, M. Davis, L. [x][y](x: y x: y)[x][y](x: y x: y) h→ih↑i ↔ h↑ih→i h→ih↑i ↔ h↑ih→i Henkin, S. Kleene, M. Rabin, B. Rosser, D. 4) Tiled:[ x](x:t ... x:t )[x](x:t ... x:t ), 1 ∨ ∨ n 1 ∨ ∨ n Scott, R. Smullyan, and A. Turing. 5) But not Twice: Church, Alonzo (1956). Introduction to Mathematical Logic. The Princeton [x](x:t x: t ) for i = j [x](x:t x: t ) for i = j , i → ¬ j 6 i → ¬ j 6 University Press. 6) Horizontal Match:[ x][y]((x:t x: y) i ∧ h→i → ( y:tj ))[x][y]((x:ti x: y) ( y:tj )), Areces & Blackburn:H(ti ,tj ) Logics for Computation∧ h→i → H(ti ,tj ) INRIA Nancy Grand Est Areces & Blackburn: Logics for Computation INRIA Nancy Grand Est 7) Vertical Match:[ x][y]((x:t x: y) W i ∧ h↑i W→ ( y:tj ))[x][y]((x:ti x: y) ( y:tj )) V (ti ,tj ) ∧ h↑i → V (ti ,tj ) ThenW ϕ is satisfiable iff covers N N.W T × Relevant Bibliography II Undecidable Problems I Alan Turing invented Turing Machines and Computer Science. I He showed that the halting problem could not be decided by a Turing Machine. I And that the behavior of a Turing Machine can easily be described in first-order logic, providing an alternative proof that the satisfiability problem of first-order logic is undecidable. Turing, Alan (1936), On Computable Numbers, with an Application to the Entscheidungsproblem. Proceedings of the London Mathematical Society, Series 2, Vol.42, pp 230–265, http://www.thocp.net/biographies/papers/turing_ oncomputablenumbers_1936.pdf Areces & Blackburn: Logics for Computation INRIA Nancy Grand Est.