Ajtai's theorem for $T^2_2(R)$ and pebble games with backtracking
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866913620217036800 |
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| author | Ken, Eitetsu Narusevych, Mykyta |
| author_facet | Ken, Eitetsu Narusevych, Mykyta |
| contents | We introduce a pebble game extended by backtracking options for one of the two players (called Prover) and reduce the provability of the pigeonhole principle for a generic predicate $R$ in the bounded arithmetic $T^2_2(R)$ to the existence of a particular kind of winning strategy (called oblivious) for Prover in the game. While the unprovability of the said principle in $T^2_2(R)$ is an immediate consequence of a celebrated theorem of Ajtai (which deals with a stronger theory $T_2(R)$), up-to-date no methods working for $T^2_2(R)$ directly (in particular without switching lemma) are known. Although the full analysis of the introduced pebble game is left open, as a first step towards resolving it, we restrict ourselves to a simplified version of the game. In this case, Prover can use only two pebbles and move in an extremely oblivious way. Besides, a series of backtracks can be made only once during a play. Under these assumptions, we show that no strategy of Prover can be winning. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_10924 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Ajtai's theorem for $T^2_2(R)$ and pebble games with backtracking Ken, Eitetsu Narusevych, Mykyta Logic Logic in Computer Science We introduce a pebble game extended by backtracking options for one of the two players (called Prover) and reduce the provability of the pigeonhole principle for a generic predicate $R$ in the bounded arithmetic $T^2_2(R)$ to the existence of a particular kind of winning strategy (called oblivious) for Prover in the game. While the unprovability of the said principle in $T^2_2(R)$ is an immediate consequence of a celebrated theorem of Ajtai (which deals with a stronger theory $T_2(R)$), up-to-date no methods working for $T^2_2(R)$ directly (in particular without switching lemma) are known. Although the full analysis of the introduced pebble game is left open, as a first step towards resolving it, we restrict ourselves to a simplified version of the game. In this case, Prover can use only two pebbles and move in an extremely oblivious way. Besides, a series of backtracks can be made only once during a play. Under these assumptions, we show that no strategy of Prover can be winning. |
| title | Ajtai's theorem for $T^2_2(R)$ and pebble games with backtracking |
| topic | Logic Logic in Computer Science |
| url | https://arxiv.org/abs/2406.10924 |