Download presentation

Presentation is loading. Please wait.

Published byDeon Redburn Modified over 3 years ago

1
Semantic of (Ongoing work with S. Hayashi, T. Coquand) Jouy-en-Josas, France, December 2004 Stefano Berardi, Semantic of Computation group C.S. Dept., Turin University, http://www.di.unito.it/~stefanohttp://www.di.unito.it/~stefano

2
Abstract of the talk Backtracking is the possibility, for a given computation, to come back to a previous state and restart from it, forgetting everything took place after it. We introduce a new notion, 1-backtracking. It is a particular case of backtracking in which forgetting is irreversible. If we forget a state we can never restore it back. We introduce a game theoretical model for 1- backtracking. 2

3
Games in Set Theory A game G = (T, R, turn, W E, W A ) between two players, E (Eloise) and A (Abelard), consists of: 1.a tree T with a father/child relation R. 2.A map turn :T { E, A }. 3.A partition (W E, W A ) of infinite branches of T. A play is any finite or infinite branch of T, starting from the root of T. In each node x of the branch, the player turn (x) must select a child of x in T, otherwise his opponent wins. If a play continues forever, the winner is E if the play is in W E, and it is A if the play is in W A.

4
Games with 1-backtracking Given a game G, we define a game bck(G). In bck(G), the player p moving from the last position z of the play can come back to some previous position x, provided: x is before z in the tree of positions of G, and p moved from x. Then p can change the move y he did from x. xzyy … 1-backtracking x before z in G y previous movey new move

5
Infinite 1-backtracking is loosing A player is allowed to backtrack to a given position x i of the play only finitely many times. A player backtracking infinitely many times to the same position x i looses. If E backtracks infinitely many times to some x i and A infinitely many times to some x j, the looser is the player backtracking infinitely many times to a position with smaller index.

6
Removing 1-backtracking Fix any (finite or infinite) play = of bck(G). We can remove 1-backtracking from, by waiting that both players stop backtracking. The result is some canonical (finite or infinite) backtracking-free play (1) = of G. Definition of (1) runs as follows: 1.t 0 = initial position of G. 2.t n+1 = last child of t n in, provided t n has a last child in. Otherwise (1) ends.

7
A formal definition of bck(G) Positions of bck(G). All finite successions over G, such that: (i) s 0 initial position of G (ii) any s i+1 is a child in G of some s j, with j i, s j ancestor in G of s i, and turn (s j ) = turn (s i ). Turn. The player on turn on a position of bck(G) is the player on turn on s n. Winner of a infinite play. The winner of an infinite play of bck(G) is the winner, in G, of the backtracking-free play (1).

8
1-Backtracking and Limit Computable Mathematic Let A be any arithmetical formula in the connectives,,,. Let Tarski (A) be the Tarski game associated to A. Let LCM be Hayashis Limit Computable Mathematic (or Arithmetic with incremental learning). Theorem. A is true in LCM if and only E if has a recursive winning strategy on bck( Tarski (A)). 1-backtracking characterizes the set of formulas we can learn

9
1-Backtracking and Recursive Degrees Let G any game either with alternating players, or with no infinite plays. Let p any player. Theorem. p has a winning strategy of recursive degree 1 for G if and only if p has a winning strategy of recursive degree 0 for bck(G). Assume we can define a winning strategy for G using an oracle for the Halting problem. Then: if we allow 1-backtracking, we can define a recursive winning strategy for the same G

10
1- Backtracking and Excluded Middle Let A be any arithmetical formula in the connectives,,,. Let Tarski (A) be the Tarski game associated to A. Let HA be Intuitionistic Arithmetic. Let 1-EM be Excluded Middle for degree 1 formulas. Theorem. E has a recursive winning strategy for bck( Tarski (A)) if, and only if: HA + -rule + 1-EM |- A 1-backtracking characterizes the set of intuitionistic consequences of 1-EM.

11
Cut-free 1-Backtracking As done by Coquand for general backtracking, we can define a cut-free version bck CF (G) of bck(G). bck CF (G) is the subgame of bck(G) in which Abelard cannot backtrack (cannot answer to a move which is not the previous one). It is much easier to define winning strategies for Eloise on the cut-free version of bck(G), because Abelard has an handicap. Every winning strategy for Eloise on bck CF (G) can be raised, in a canonical way, to a winning strategy for Eloise on bck(G).

12
Iterating 1-Backtracking We defined the maps G| bck(G),bck CF (G) from games of Set Theory to games of Set Theory. By iteration, for any n N we can define bck n (G), bck CF n (G), the games with n-backtracking over G, with and without cuts. The difference between 1-backtracking and 2- backtracking is that, in 2-backtracking, forgetting is sometimes reversible (we may recover a previous state of the computation forgotten by 1- backtracking). By direct limit we can define bck (G), bck CF (G), for all ordinal.

13
1-Backtracking and unlimited Backtracking For any game G with alternating players and all plays of length n, Coquand defined a game Coq(G). In Coq(G), Eloise has an unlimited backtracking over G. Abelard, instead, cannot backtrack: Coq(G) is cut-free. Theorem. bck CF (G) is a subgame of Coq(G), and conversely, any winning strategy in Coq(G) is a winning strategy in some bck CF (G). Unlimited backtracking can be obtained by iterating 1-backtracking.

14
References T. Coquand, A semantics of evidence for classical arithmetic, Journal of Symbolic Logic 60, pp. 325-337, 1995. S. Hayashi and M. Nakata, Towards Limit Computable Mathematics, Types for Proofs and Programs, LNCS 2277, pp. 125-144, 2001.

Similar presentations

OK

Recursion Chapter 7. Chapter 7: Recursion2 Chapter Objectives To understand how to think recursively To learn how to trace a recursive method To learn.

Recursion Chapter 7. Chapter 7: Recursion2 Chapter Objectives To understand how to think recursively To learn how to trace a recursive method To learn.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Games we play ppt on ipad Ppt on 10 figures of speech Ppt on beer lambert law and its function Ppt on cross-sectional study psychology Ppt on polynomials of 9 Ppt on limits and derivatives examples Ppt on android os software Mba project ppt on imf Ppt on power system earthing Ppt on nature and human quotes