Subatomic systems need not be subatomic
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Subatomic systems were recently introduced to identify the structural principles underpinning the normalization of proofs. "Subatomic" means that we can reformulate logical systems in accordance with two principles. Their atomic formulas become instances of sub-atoms, non-commutative self-dual relations among logical constants, and their rules are derivable by means of a unique deductive scheme, the medial shape. One of the results is that the cut-elimination of subatomic systems implies the cut-elimination of every standard system we can represent sub-atomically. We here introduce Subatomic systems-1.1. They relax and widen the properties that the sub-atoms of Subatomic systems can satisfy while maintaining the use of the medial shape as their only inference principle. Since sub-atoms can operate directly on variables we introduce P. The cut-elimination of P is a corollary of the cut-elimination that we prove for Subatomic systems-1.1. Moreover, P is sound and complete with respect to the clone at the top of Post's Lattice. I.e. P proves all and only the tautologies that contain conjunctions, disjunctions and projections. So, P extends Propositional logic without any encoding of its atoms as sub-atoms of P. This shows that the logical principles underpinning Subatomic systems also apply outside the sub-atomic level which they are conceived to work at. We reinforce this point of view by introducing the set R of medial shapes. The formulas that the rules in R deal with belong to the union of two disjoint clones of Post's Lattice. The SAT-problem of the first clone is in P-Time. The SAT-problem of the other is NP-Time complete. So, R and the proof technology of Subatomic systems could help to identify proof-theoretical properties that highlight the phase transition from P-Time to NP-Time complete satisfiability.
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