## Can proofs be animated by games? (2005)

Venue: | . URZYCZYN ED., TLCA 2005, LNCS 3461 |

Citations: | 4 - 0 self |

### BibTeX

@INPROCEEDINGS{Hayashi05canproofs,

author = {Susumu Hayashi},

title = {Can proofs be animated by games?},

booktitle = {. URZYCZYN ED., TLCA 2005, LNCS 3461},

year = {2005},

pages = {11--22},

publisher = {}

}

### OpenURL

### Abstract

Proof animation is a way of executing proofs to find errors in the formalization of proofs. It is intended to be "testing in proof engineering". Although the realizability interpretation as well as the functional interpretation based on limit-computations were introduced as means for proof animation, they were unrealistic as an architectural basis for actual proof animation tools. We have found game theoretical semantics corresponding to these interpretations, which is likely to be the right architectural basis for proof animation.

### Citations

88 |
Software Engineering. 6 th Edition
- Sommerville
- 2001
(Show Context)
Citation Context ...tion by executing it. If the system is correct w.r.t. a speci cation, then we cancheck speci cations against our intention through validating the system. This kind of activities are called validation =-=[16]-=-. Veri cation is to ask \Did we build the system right?". Validation is to ask \Did we build the right system?". We may build a wrong system which isright relative to wrong speci cations. Can we do va... |

43 | A semantics of evidence for classical arithmetic - Coquand - 1995 |

34 |
PX: A Computational Logic
- Hayashi, Nakano
- 1988
(Show Context)
Citation Context .... Nonetheless, it has not been known if such learning algorithms can be fully automatically extracted from formalized versions of such informal proofs. According to our experiences with the PX system =-=[6]-=-, algorithms which are automatically extracted from the proofs based on the mathematical soundness theorem or the original Curry-Howard isomorphism are much more complicated and illegible than the one... |

30 |
The Game of Language
- Hintikka
- 1983
(Show Context)
Citation Context ...have togive an game theoretical interpretation of implication which is equivalent to LCM-semantics of implication. There are at least two ways to handle implication in game theoretical semantics (see =-=[12]-=-). The standard way is to regard A ! B as A ? _ B, where A ? is the dual game. Another way is to use the notion of the subgame. Although some modi cations are necessary, it is basically easy to extend... |

15 | Proof mining: a systematic way of analysing proofs in mathematics
- Kohlenbach, Oliva
- 2003
(Show Context)
Citation Context ...non-trivial e orts. We call this di culty global ilegibility. 1 . If proof animation is for nding useful information such as bounds for solutions and algorithms in classical proofs as proof mining in =-=[14]-=-, global ilegibility is not a real obstacle. However, our aim is to test proofs to our intentions just as engineers test systems. Proof executions must be light and legible as test runs of programs. T... |

10 |
Game-theoretical semantics. In: Handbook of Logic and Language
- Hintikka, Sandu
- 1997
(Show Context)
Citation Context ...pect that it will give aright framework to solve this problem. 3.1 1-backtracking game Game theoretical semantics of logical formulas are known to be a good substitute for Tarskian semantics of logic =-=[13]-=-. It is said that game semantics is easier to learn than Tarski semantics. Coquand [5] introduced a game theoretical semantics of classical rst order arithmetic. It allows Eloise, the player for exist... |

8 |
An Arithmetical Hierarchy of the Law of Excluded Middle and Related Principles
- Akama, Berardi, et al.
(Show Context)
Citation Context ...as hold. The fragment of classical mathematics valid by this interpretation was named LCM, Limit-Computable Mathematics. It has been proved that there exists a ne hierarchy of classical principles in =-=[1]-=-. According to the results of [1], LCM corresponds to the lower part of the hierarchy. We cannot therefore derive all the classical theorems in LCM, but it is known that quite a large variety of nonco... |

7 |
Systems that Learn - 2nd Edition, An Introduction to Learning Theory
- Jain, Osherson, et al.
- 1999
(Show Context)
Citation Context ...NP. There is no recursive realizer for MNP. However, there is a 0 2 -function computing x. It is known that 0 2 - functions represent learning algorithms called inductive inference in Learning theory =-=[17]-=-. An inductive inference is a try-and-error algorithmic process to nd aright solution in nite time. Here is an inductive inference for MNP. At the beginning, we temporarily assume that f(0) is the min... |

5 |
Classical logic as Limit Completion, -a constructive model for non-recursive maps-, submitted
- Berardi
- 2001
(Show Context)
Citation Context ...d. After these di erences, proof animation tools based on these two frameworks would be rather di erent. 3 Berardi has introduced a series of limit-interpretations whose indexes are sets of conditions=-=[2]-=-. It is expected that these notions are closely related.s3.5 Why is 1-game legible? We will close this section by a remark on legibility of the 1-games. Since the full backtracking game needs only rec... |

4 |
Mathematics based on incremental learning, excluded middle and inductive inference
- Hayashi
(Show Context)
Citation Context ...istic. When formal veri cation technologies become a reality technology, the last problem left would be \how toshow correctness of formalization." Let us illustrate this problem by an example used in =-=[9]-=-. Assume that we are developing a formal theory of a metric jjxjj on the interval [m� n] oftheset of integers by the distance from n. For example, jjnjj is 0 and jjmjj is n ; m. A linear order is de n... |

4 | die Theorie der algebraische Formen - Hilbert |

3 |
Realizability Interpretation for Limit Computable
- Nakata, Hayashi
- 1991
(Show Context)
Citation Context ...ofs to our intentions just as engineers test systems. Proof executions must be light and legible as test runs of programs. Thus, the global ilegibility is an essential defect for proof animations. In =-=[7, 15]-=-, we introduced a new realizability interpretation to overcome the global ilegibility. The de nition of our new realizability interpretation of logical connectives is the same as the original one by K... |

3 |
A Calibration of Ine ective Theorems of Analysis in a Hierarchy of SemiClassical Logical Principles, submitted
- Toftdal
- 2004
(Show Context)
Citation Context ...ponds to the lower part of the hierarchy. We cannot therefore derive all the classical theorems in LCM, but it is known that quite a large variety of nonconstructive theorems belong to LCM: see, e.g. =-=[18]-=-. For example, the minimal number principle for the natural numbers (MNP) 9x:8y:(f(x) f(y))� 1 Local and global legibility are terminologies due to Stefano Berardi 3s4 where x and y are natural number... |

2 |
Towards Limit Computable Mathematics, in Types for Proofs and Programs
- Hayashi, Nakata
- 2001
(Show Context)
Citation Context ...ofs to our intentions just as engineers test systems. Proof executions must be light and legible as test runs of programs. Thus, the global ilegibility is an essential defect for proof animations. In =-=[7, 15]-=-, we introduced a new realizability interpretation to overcome the global ilegibility. The de nition of our new realizability interpretation of logical connectives is the same as the original one by K... |

1 |
Theory of Algebraic Invariants, translated by Laubenbacher
- Hilbert
- 1993
(Show Context)
Citation Context ... Hilbert's original proof is much more \learning theoretic" than the contemporary counterparts. Especially, the discussions in his course at Gottingen July 5th 1897 shows its learning theoretic nature=-=[11]-=-. 5s6 moves alternatively. This restriction is not essential, and makes things easier. If the last move of a position is played by a player A, we say that A played the position. EndOfDef Let us note t... |