Translational expressiveness between logics: giving adequacy criteria Diego P. Fernandes PhD student at University of Salamanca, research funded by CAPES/Brazil May 23, 2017
Basic intuition for relative expressiveness between logics A logic L 2 is at least as expressive as L 1 iff for every L 1 -sentence, there is an L 2 -sentence with the same meaning.
Three frameworks for expressiveness ◮ single-class (for model-theoretic logics) ◮ translations of sentences of logics in the same class of structures ◮ multi-class (for model-theoretic logics) ◮ translations of sentences and structures ◮ translational expressiveness (for logics in general) ◮ translations of sentences
Translational expressiveness General idea, for logics L 1 = ( F 1 , ⊢ 1 ) and L 2 = ( F 2 , ⊢ 2 ): L 2 is at least as expressive as L 1 if there is a T : L 1 − → L 2 such that T has P 1 , P 2 , ... . ◮ many informal statements involving expressive inclusion in this framework. E.g. ◮ W´ ojcicki [W´ oj88, p. 67] It is worth noticing that the expressive power of ⊢ 2 can be greater than that of ⊢ 1 even if ⊢ 2 ⊆ ⊢ 1 . This, for instance, is the case of (...) K � and (...) L � 3 ;
Translational expressiveness Let Γ ∪ { φ } be L 1 -formulas. T : L 1 − → L 2 is a conservative translation when Conservative Translation Γ ⊢ L 1 φ if and only if T (Γ) ⊢ L 2 T ( φ )
Translational expressiveness Let Γ ∪ { φ } be L 1 -formulas. T : L 1 − → L 2 is a conservative translation when Conservative Translation Γ ⊢ L 1 φ if and only if T (Γ) ⊢ L 2 T ( φ ) Mossakowski et al. gave a formal criterion for translational expressiveness [MDT09, p. 101]: ◮ L 2 is at least as expressive as L 1 iff there is a conservative translation T : L 1 − → L 2
Mossakowski et al.’s expressiveness: Problems!! ◮ Jeˇ r´ abek [Jeˇ r12] has shown that there are conservative translations between: ◮ classical propositional logic, ◮ intuitionistic logics, minimal logics, and intermediate logics, ◮ modal logics (classical or intuitionistic), ◮ substructural logics, ◮ first-order (or higher-order) extensions of the above logics.
Mossakowski et al.’s expressiveness: Problems!! ◮ Jeˇ r´ abek [Jeˇ r12] has shown that there are conservative translations between: ◮ classical propositional logic, ◮ intuitionistic logics, minimal logics, and intermediate logics, ◮ modal logics (classical or intuitionistic), ◮ substructural logics, ◮ first-order (or higher-order) extensions of the above logics. ◮ Intuitively these logics do not have the same expressiveness!
What is the problem with conservative translations? ◮ The mappings are not required to ◮ preserve the structure of the formulas in any way ◮ preserve the properties of the logic
What is the problem with conservative translations? ◮ The mappings are not required to ◮ preserve the structure of the formulas in any way ◮ preserve the properties of the logic ◮ A stricter notion of translation is needed in a criterion for expressiveness
What is the problem with conservative translations? ◮ The mappings are not required to ◮ preserve the structure of the formulas in any way ◮ preserve the properties of the logic ◮ A stricter notion of translation is needed in a criterion for expressiveness ◮ Think first on some adequacy criteria
Thinking some adequacy criteria for expressiveness Our first criterion comes from W´ ojcicki [W´ oj88, p. 67]:
Thinking some adequacy criteria for expressiveness Our first criterion comes from W´ ojcicki [W´ oj88, p. 67]: ◮ Adequacy Criterion 1 : L 2 is at least as expressive as L 1 only if everything that can be said in terms of the connectives of L 1 can also be said in terms of the connectives of L 2 . ◮ Connectives are the basic tools for expressing things in a logic
Thinking some adequacy criteria for expressiveness ◮ There are some meta-properties of logics that are intuitively known to limit or increase expressiveness
Thinking some adequacy criteria for expressiveness ◮ There are some meta-properties of logics that are intuitively known to limit or increase expressiveness ◮ Adequacy Criterion 2 It cannot hold that L 2 is more expressive than L 1 when
Thinking some adequacy criteria for expressiveness ◮ There are some meta-properties of logics that are intuitively known to limit or increase expressiveness ◮ Adequacy Criterion 2 It cannot hold that L 2 is more expressive than L 1 when ◮ L 2 is trivial and L 1 is non trivial; ◮ A trivial logic cannot be more expressive than any logic;
Thinking some adequacy criteria for expressiveness ◮ van Benthem’s Golden Rule of Logic [vB06, p. 119]: “Gains in expressive power are lost in higher complexity”
Thinking some adequacy criteria for expressiveness ◮ van Benthem’s Golden Rule of Logic [vB06, p. 119]: “Gains in expressive power are lost in higher complexity” ◮ The complexity levels decidability/undecidability can be useful for expressiveness comparisons
Thinking some adequacy criteria for expressiveness ◮ van Benthem’s Golden Rule of Logic [vB06, p. 119]: “Gains in expressive power are lost in higher complexity” ◮ The complexity levels decidability/undecidability can be useful for expressiveness comparisons ◮ If a logic is decidable, then it cannot describe Turing machines, Post’s normal systems, or semi-Thue systems,
Thinking some adequacy criteria for expressiveness ◮ van Benthem’s Golden Rule of Logic [vB06, p. 119]: “Gains in expressive power are lost in higher complexity” ◮ The complexity levels decidability/undecidability can be useful for expressiveness comparisons ◮ If a logic is decidable, then it cannot describe Turing machines, Post’s normal systems, or semi-Thue systems, ◮ Thus it is reasonable that ◮ a decidable logic cannot be more expressive than an undecidable logic
Thinking some adequacy criteria for expressiveness ◮ A logic has a deduction theorem (DT) when it is able to express in the object language its deductibility relation
Thinking some adequacy criteria for expressiveness ◮ A logic has a deduction theorem (DT) when it is able to express in the object language its deductibility relation ◮ Other things being equal, a logic having DT is more expressive than another one lacking it.
Thinking some adequacy criteria for expressiveness ◮ A logic has a deduction theorem (DT) when it is able to express in the object language its deductibility relation ◮ Other things being equal, a logic having DT is more expressive than another one lacking it. ◮ DT is formulation-sensitive: ◮ A less expressive logic might have the standard DT while the more expressive has only a general version of DT
Thinking some adequacy criteria for expressiveness ◮ A logic has a deduction theorem (DT) when it is able to express in the object language its deductibility relation ◮ Other things being equal, a logic having DT is more expressive than another one lacking it. ◮ DT is formulation-sensitive: ◮ A less expressive logic might have the standard DT while the more expressive has only a general version of DT ◮ For example take Menselson’s FOL [Men97, p. 76]: ◮ the propositional fragment satisfies the standard DT, while FOL satisfies only a general version of it.
Thinking some adequacy criteria for expressiveness [ Adequacy Criterion 2 ] It cannot hold that L 2 is more expressive than L 1 when ◮ L 1 satisfies the standard DT and the language fragment of L 2 purportedly as expressive as L 1 does not satisfy (not even) a general formulation of DT;
Thinking some adequacy criteria for expressiveness [ Adequacy Criterion 2 ] It cannot hold that L 2 is more expressive than L 1 when ◮ L 2 is trivial and L 1 is non trivial; ◮ L 2 is decidable and L 1 is not decidable; ◮ L 1 satisfies the standard DT and the language fragment of L 2 purportedly as expressive as L 1 does not satisfy (not even) a general formulation of DT;
Thinking some adequacy criteria for expressiveness The expressiveness relation should be transitive and reflexive and there must be logics L 1 , L 2 such that L 2 is not at least as expressive as L 1 . [ Adequacy Criterion 3 ]: (taken from [Kui14]) The expressiveness relation should be a non-trivial pre-order.
Adequacy criteria for expressiveness [ Adequacy Criterion 1 ]: L 2 is at least as expressive as L 1 only if everything that can be said in terms of the connectives of L 1 can also be said in terms of the connectives of L 2 . [ Adequacy Criterion 2 ] It cannot hold that L 2 is more expressive than L 1 when ◮ L 2 is trivial and L 1 is non trivial; ◮ L 2 is decidable and L 1 is not decidable; ◮ L 1 satisfies the standard DT and the language fragment of L 2 purportedly as expressive as L 1 does not satisfy (not even) a general formulation of DT; [ Adequacy Criterion 3 ]: ([Kui14]) The expressiveness relation should be a non-trivial pre-order.
Capturing adequacy criterion 1 Definition (Compositional) A translation T : L 1 − → L 2 is compositional whenever for every n -ary connective # of L 1 there is an L 2 -formula ψ # such that T (#( φ 1 , ..., φ n )) = ψ # ( T ( φ 1 ) , ..., T ( φ n )). Many writers require (at least) compositional translations for connective preservation
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