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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 L2 is at least as expressive as L1 iff for every L1-sentence, there is an L2-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 L1 = (F1, ⊢1) and L2 = (F2, ⊢2): L2 is at least as expressive as L1 if there is a T : L1 − → L2 such that T has P1, P2, ....

◮ many informal statements involving expressive inclusion in this

• framework. E.g.

◮ W´

• jcicki [W´
• j88, 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 L1-formulas. T : L1 − → L2 is a conservative translation when Conservative Translation Γ ⊢L1 φ if and only if T (Γ) ⊢L2 T (φ)

Translational expressiveness

Let Γ ∪ {φ} be L1-formulas. T : L1 − → L2 is a conservative translation when Conservative Translation Γ ⊢L1 φ if and only if T (Γ) ⊢L2 T (φ) Mossakowski et al. gave a formal criterion for translational expressiveness [MDT09, p. 101]:

◮ L2 is at least as expressive as L1 iff there is a conservative

translation T : L1 − → L2

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´

• jcicki [W´
• j88, p. 67]:

Thinking some adequacy criteria for expressiveness

Our first criterion comes from W´

• jcicki [W´
• j88, p. 67]:

◮ Adequacy Criterion 1: L2 is at least as expressive as L1 only

if everything that can be said in terms of the connectives of L1 can also be said in terms of the connectives of L2.

◮ 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 L2 is more

expressive than L1 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 L2 is more

expressive than L1 when

◮ L2 is trivial and L1 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 L2 is more expressive than L1 when

◮ L1 satisfies the standard DT and the language

fragment of L2 purportedly as expressive as L1 does not satisfy (not even) a general formulation of DT;

Thinking some adequacy criteria for expressiveness

[ Adequacy Criterion 2] It cannot hold that L2 is more expressive than L1 when

◮ L2 is trivial and L1 is non trivial; ◮ L2 is decidable and L1 is not decidable; ◮ L1 satisfies the standard DT and the language

fragment of L2 purportedly as expressive as L1 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 L1, L2 such that L2 is not at least as expressive as L1. [ Adequacy Criterion 3]: (taken from [Kui14]) The expressiveness relation should be a non-trivial pre-order.

Adequacy criteria for expressiveness

[ Adequacy Criterion 1]: L2 is at least as expressive as L1 only if everything that can be said in terms of the connectives of L1 can also be said in terms of the connectives of L2. [ Adequacy Criterion 2] It cannot hold that L2 is more expressive than L1 when

◮ L2 is trivial and L1 is non trivial; ◮ L2 is decidable and L1 is not decidable; ◮ L1 satisfies the standard DT and the language

fragment of L2 purportedly as expressive as L1 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 : L1 − → L2 is compositional whenever for every n-ary connective # of L1 there is an L2-formula ψ# such that T (#(φ1, ..., φn)) = ψ#(T (φ1), ..., T (φn)). Many writers require (at least) compositional translations for connective preservation

Capturing adequacy criterion 1

Definition (Compositional)

A translation T : L1 − → L2 is compositional whenever for every n-ary connective # of L1 there is an L2-formula ψ# such that T (#(φ1, ..., φn)) = ψ#(T (φ1), ..., T (φn)). Many writers require (at least) compositional translations for connective preservation (W´

• jcicki) from CPL to Lukasiewicz L3:

T l(pi) = pi T l(¬φ) = T l(φ) → ¬T l(φ) T l(φ → ψ) = T l(φ) → (T l(φ) → T l(ψ)) (G¨

• del) from IPL to S4:

T g(p) = p T g(φ → ψ) = (T g(φ) → T g(ψ)) (...)

Capturing adequacy criterion 1

Modal translation is not compositional!

Capturing adequacy criterion 1

Modal translation is not compositional! (Van Benthem) Standard translation from modal logic to FOL: T x(pi) = Pix T x(♦φ) = ∃y(Rxy ∧ T y(φ)) (...)

Capturing adequacy criterion 1

Modal translation is not compositional! (Van Benthem) Standard translation from modal logic to FOL: T x(pi) = Pix T x(♦φ) = ∃y(Rxy ∧ T y(φ)) (...) The usual meaning given to ♦φ seems successfully replicated in FOL by the translation. Thus a wider notion of translation is needed!

Capturing adequacy criterion 1

Adequacy Criterion 1: L2 is at least as expressive as L1 only if everything that can be said in terms of the connectives of L1 can also be said in terms of the connectives of L2.

◮ what would be a general but reasonable notion of preservation

• f connectives via a translation T ?

Capturing adequacy criterion 1

Adequacy Criterion 1: L2 is at least as expressive as L1 only if everything that can be said in terms of the connectives of L1 can also be said in terms of the connectives of L2.

◮ what would be a general but reasonable notion of preservation

• f connectives via a translation T ?

(α) for each n-ary (composite) connective ⊗ in L1 and L1-formulas φ1, ..., φn, there must be L2-formulas δ⊗(p1, ..., pm) (possibly m = n) and ψ1, ..., ψm such that ⊗(φ1, ..., φn) has a similar deductive behaviour with δ⊗(ψ1/p1, ..., ψm/pm).

Capturing adequacy criterion 1

Adequacy Criterion 1: L2 is at least as expressive as L1 only if everything that can be said in terms of the connectives of L1 can also be said in terms of the connectives of L2.

◮ what would be a general but reasonable notion of preservation

• f connectives via a translation T ?

(α) for each n-ary (composite) connective ⊗ in L1 and L1-formulas φ1, ..., φn, there must be L2-formulas δ⊗(p1, ..., pm) (possibly m = n) and ψ1, ..., ψm such that ⊗(φ1, ..., φn) has a similar deductive behaviour with δ⊗(ψ1/p1, ..., ψm/pm).

◮ Refine (α):

◮ how the formula δ⊗(ψ1, ..., ψm) shall be obtained from

⊗(φ1, ..., φn),

Capturing adequacy criterion 1

Adequacy Criterion 1: L2 is at least as expressive as L1 only if everything that can be said in terms of the connectives of L1 can also be said in terms of the connectives of L2.

◮ what would be a general but reasonable notion of preservation

• f connectives via a translation T ?

(α) for each n-ary (composite) connective ⊗ in L1 and L1-formulas φ1, ..., φn, there must be L2-formulas δ⊗(p1, ..., pm) (possibly m = n) and ψ1, ..., ψm such that ⊗(φ1, ..., φn) has a similar deductive behaviour with δ⊗(ψ1/p1, ..., ψm/pm).

◮ Refine (α):

◮ how the formula δ⊗(ψ1, ..., ψm) shall be obtained from

⊗(φ1, ..., φn),

◮ what it means for them to have a similar deductive behavior.

Capturing adequacy criterion 1

Adequacy Criterion 1: L2 is at least as expressive as L1 only if everything that can be said in terms of the connectives of L1 can also be said in terms of the connectives of L2.

◮ what would be a general but reasonable notion of preservation

• f connectives via a translation T ?

(α) for each n-ary (composite) connective ⊗ in L1 and L1-formulas φ1, ..., φn, there must be L2-formulas δ⊗(p1, ..., pm) (possibly m = n) and ψ1, ..., ψm such that ⊗(φ1, ..., φn) has a similar deductive behaviour with δ⊗(ψ1/p1, ..., ψm/pm).

◮ Refine (α):

◮ how the formula δ⊗(ψ1, ..., ψm) shall be obtained from

⊗(φ1, ..., φn),

◮ what it means for them to have a similar deductive behavior.

◮ should (α) hold for T (L1) or for the entire L2 in order that

criterion 1 be captured properly?

◮ Apparently, only for T (L1) would be enough.

Capturing adequacy criterion 2

◮ the clauses on triviality and decidability are clear enough ◮ the one on the deduction theorem needs elucidation

Capturing adequacy criterion 2

◮ the clauses on triviality and decidability are clear enough ◮ the one on the deduction theorem needs elucidation ◮ Definition (standard deduction theorem)

A logic L has the standard deduction theorem (DT) if it holds that φ1, ..., φn ⊢ ψ if and only if φ1, ..., φn−1 ⊢ φn → ψ.

Definition (general deduction theorem)

A logic L has the general DT whenever φ1, ..., φn ⊢ ψ iff φ1, ..., φn−1 ⊢ α→(φn, ψ), where α→ is an L-formula, with one or more occurrences of φn and ψ.

Capturing adequacy criterion 2

◮ the clauses on triviality and decidability are clear enough ◮ the one on the deduction theorem needs elucidation ◮ Definition (standard deduction theorem)

A logic L has the standard deduction theorem (DT) if it holds that φ1, ..., φn ⊢ ψ if and only if φ1, ..., φn−1 ⊢ φn → ψ.

Definition (general deduction theorem)

A logic L has the general DT whenever φ1, ..., φn ⊢ ψ iff φ1, ..., φn−1 ⊢ α→(φn, ψ), where α→ is an L-formula, with one or more occurrences of φn and ψ.

◮ Definition (preservation of the general DT)

A translation T : L1 − → L2 is said to preserve the general deduction theorem whenever L1 has the standard deduction theorem and T (L1) has the general deduction theorem.

Capturing the criteria: general recursive translations

The following is adapted from French’s recursive translations [Fre10]:

Definition (General-Recursive)

Let T1, ..., Tw be auxiliary mappings defined inductively on L1-formulas. A translation T : L1 − → L2 is general-recursive if for every n-ary connective # and L1- formulas φ1, ..., φn, there is an L2-formula #T (p1, ..., pm) such that T (#(φ1, ..., φn)) = #T (T1(φi)/p1, ..., Tw(φk)/pm) where {φi, ..., φk} ⊆ {φ1, ..., φn}.

Claim: Conservative GR-translations satisfy criteria 1-3

Adequacy Criterion 1: L2 is at least as expressive as L1 only if everything that can be said in terms of the connectives of L1 can also be said in terms of the connectives of L2. (α) for each n-ary (composite) connective ⊗ in L1 and L1-formulas φ1, ..., φn, there must be L2-formulas δ⊗(p1, ..., pm) (possibly m = n) and ψ1, ..., ψm such that ⊗(φ1, ..., φn) has a similar deductive behaviour with δ⊗(ψ1/p1, ..., ψm/pm).

◮ α is satisfied in conservative general-recursive translations

Claim: Conservative GR-translations satisfy criteria 1-3

Adequacy Criterion 1: L2 is at least as expressive as L1 only if everything that can be said in terms of the connectives of L1 can also be said in terms of the connectives of L2. (α) for each n-ary (composite) connective ⊗ in L1 and L1-formulas φ1, ..., φn, there must be L2-formulas δ⊗(p1, ..., pm) (possibly m = n) and ψ1, ..., ψm such that ⊗(φ1, ..., φn) has a similar deductive behaviour with δ⊗(ψ1/p1, ..., ψm/pm).

◮ α is satisfied in conservative general-recursive translations

◮ conservativeness forces a similar deductive behaviour of

⊗(φ1, ..., φn) and δ⊗(ψ1, ..., ψm)

◮ and ψi is obtained from some φi through the translation

Claim: Conservative GR-translations satisfy criteria 1-3

[ Adequacy Criterion 2] It cannot hold that L2 is more expressive than L1 when

◮ L2 is trivial and L1 is non trivial; ◮ L2 is decidable and L1 is not decidable; ◮ L1 satisfies the standard DT and the language

fragment of L2 purportedly as expressive as L1 does not satisfy (not even) a general formulation of DT;

Claim: Conservative GR-translations satisfy criteria 1-3

[ Adequacy Criterion 2] It cannot hold that L2 is more expressive than L1 when

◮ L2 is trivial and L1 is non trivial; ◮ L2 is decidable and L1 is not decidable; ◮ L1 satisfies the standard DT and the language

fragment of L2 purportedly as expressive as L1 does not satisfy (not even) a general formulation of DT;

Proposition (non-triviality)

All conservative translations preserve non-triviality

Proposition (undecidability [FD01])

If L1 is undecidable, then there is no computable compositional translation T : L1 − → L2, where L2 is decidable.

Claim: Conservative GR-translations satisfy criteria 1-3

◮ even the general version of DT requires compositionality ◮ T is general-recursiveC if T is compositional for →:

T (φ → ψ) = δ→(T (φ), ..., T (ψ)).

Proposition

Let L1 satisfy the standard deduction theorem. If T : L1 − → L2 is a conservative general-recursiveC translation, then T (L1) has the general deduction theorem.

Proposition

Conservative general-recursive translations form a non-trivial pre-order on logics.

Criterion for translational expressiveness

L2 is at least as expressive as L1 if there is a conservative general-recursive translation T : L1 − → L2.

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• jcicki.

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