1998 European Summer Meeting of the Association for Symbolic

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Meanings are given in term of actions and are adapted to states using the view of state formulae given above. B , actions: to do A or to do B but one does not know which one (outer non-determinism), states: one of A and B is true but one does not know which one; (Dependent sequentiality) A > B, actions: to do B by beginning to do A , states: B is true but one must first prove A in order to prove B: (Independent sequentiality) A >> B , actions: to do A and then to do B, states: A and B are true but A is proved before B; (Logical implication) A + B, actions: if you know how to do A , you can do B, states: B can be logically deduced from A: (Causality) A --+ B, action: doing A causes doing B, states: nothing; (Universality)V x .

E-mail: daquinomunina. it. ) Let M be a model of IAo + and K be the residue field of M for a nonstandard prime pin M. e.. it satisfies the following axioms of Ax (1) there exists a unique extension of each degree n ; (2) every absolutely irreducible curve has a point in the field. In the case of Open Induction. Macintyre and Marker proved in [2] that for any field L of characteristic 0 there is a model M of Open Induction and a prime p E M such that the residue field K is elementary equivalent to L.

Prove() does A. Axiomatisation. Our logic is a sequent calculus. The axiomatisation is made difficult because of the need of linearity. Moreover, realizations are imperative programs which cannot be combined in any way. In the sequent, the 8 connective intuitively links formulae on both sides of the entailment sign t-. Therefore, our structural rules are unusual: These rules make the proofs much more natural than classical sequent calculus rules by allowing a linear structure of proofs. They also allow a more human-like automatic research of proofs.

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