Abstract
It is well known that quantifier elimination plays a relevant role in proving decidability of theories. Herein the objective is to provide a toolbox that makes it easier to establish quantifier elimination in a semantic way, capitalizing on the fact that a 1-model-complete theory with algebraically prime models has quantifier elimination. Iteration and adjunction are identified as important constructions that can be very helpful, by themselves or composed, in proving that a theory has algebraically prime models. Some guidelines are also discussed towards showing that a theory is 1-model-complete. Illustrations are provided for the theories of the natural numbers with successor, term algebras (having stacks as a particular case) and algebraically closed fields.
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Notes
- 1.
In this work, we consider first-order logic with equality ≅.
- 2.
Recall that, for any first-order theory Θ, Θ ∀ is the set of all sentences entailed by Θ of the form ∀x 1…∀x n φ where φ is a quantifier-free formula.
- 3.
Recall that ∀2 is the smallest class of formulas containing ∃1 and closed under ∧, ∨ and adding universal quantifiers at the front, where ∃1 is the smallest class of formulas containing the quantifier-free formulas and closed under ∧, ∨ and adding existential quantifiers at the front. Observe that every ∀2 formula is equivalent to a ∀2 formula ∀x 1…∀x n ψ with ψ in ∃1 (for more details see, for instance, Sect. 2.4 of [7]). From now on we assume, without loss of generality, that the ∀2 formulas are of this form, i.e. of the form ∀x 1…∀x n ψ with ψ in ∃1.
- 4.
Given a signature Σ, an interpretation structure I=(D,⋅F,⋅P) over Σ, a variable assignment ρ:X→D, and a first-order formula φ, we denote by Iρ⊩ Σ φ the satisfaction of φ by I and ρ. Recall that this relation is inductively defined as follows:
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Iρ⊩ Σ p(t 1,…,t n ) whenever (〚t 1〛Iρ,…,〚t n 〛Iρ)∈p P for every terms t 1,…,t n and n-ary predicate symbol p, where 〚t〛Iρ is the interpretation of term t over I and ρ, inductively defined as follows: (a) 〚x〛Iρ=ρ(x) for every variable x; and (b) 〚f(t 1,…,t n )〛Iρ=f F(〚t 1〛Iρ,…,〚t n 〛Iρ) for every n-ary function symbol f and terms t 1,…,t n ;
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Iρ⊩ Σ ¬φ 1 whenever Iρ⊮ Σ φ 1;
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Iρ⊩ Σ (φ 1⇒φ 2) whenever Iρ⊩ Σ φ 1 implies Iρ⊩ Σ φ 2;
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Iρ⊩ Σ ∀xφ 1 whenever for every ρ′ over I with ρ′(y)=ρ(y) for every y≠x, Iρ′⊩ Σ φ 1.
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- 5.
Given an embedding h:I→I′ and an assignment ρ:X→|I| over I, observe that h∘ρ:X→|I′| is an assignment of values of I′ to the variables of X. So I′h∘ρ⊩∃xφ means that formula ∃xφ is satisfied by model I′ and assignment h∘ρ over I′.
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Acknowledgements
This work was partially supported, under the MCL (Meet-Combination of Logics) and PQDR (Probabilistic, Quantum and Differential Reasoning) initiatives of SQIG at IT, by FCT and EU FEDER, namely via the projects FCT PEst-OE/EEI/LA0008/2013 and AMDSC UTAustin/MAT/0057/2008, as well as by the European Union’s Seventh Framework Programme for Research (FP7), namely through project LANDAUER (GA 318287).
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Rasga, J., Sernadas, C., Sernadas, A. (2015). A Roadmap to Decidability. In: Koslow, A., Buchsbaum, A. (eds) The Road to Universal Logic. Studies in Universal Logic. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-10193-4_20
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