Model-Theoretic Conservative Extension for Definitional Theories

Gengelbach, Arve; Weber, Tjark

doi:10.1016/j.entcs.2018.10.009

Cited by 4 publications

(10 citation statements)

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“…The set U contains the symbols introduced by a theory extension. In contrast to [4], where U is a singleton set, we allow the introduction of several symbols at once, e. g. via constant specification. The set V is the pre-image of type instances Θ u of elements u from U (with Θ a ground type substitution) under the reflexive-transitive, type-substitutive closure of the dependency relation ctxt .…”

Section: Symbol-independent Fragmentmentioning

confidence: 99%

“…We prove this claim in script, to give a flavour of the reasoning involved in the mechanisation. Thereby we amend the earlier proof [4] for the case where a type substitution ρ and • do not commute on a type ς , i. e. ρ(ς • ) = ρ(ς ) • . (This case had been excluded by a faulty lemma inherited from Kunčar and Popescu.…”

Section: Symbol-independent Fragmentmentioning

confidence: 99%

“…The model construction for a definitional theory extension upd::ctxt is guarded with a check: if the symbol to interpret lies in the independent fragment indep_frag_upd (upd::ctxt) upd (total_fragment (sigof ctxt)) of a definitional update upd, the symbol may be interpreted w. r. t. the model (∆ ,Γ ). Otherwise the symbol's interpretation is constructed as discussed in earlier work [17,12,4].…”

Section: Model Constructionmentioning

confidence: 99%

“…• We use the independent fragment to prove a notion of model-theoretic conservativity [4] in a new setting for the lazy ground semantics [17], which delays type variable instantiation and does not instantiate the type of term variables. To the best of our knowledge, this is the first mechanised conservativity result for a logic with overloaded definitions.…”

Section: Introductionmentioning

confidence: 99%

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Mechanisation of Model-theoretic Conservative Extension for HOL with Ad-hoc Overloading

Gengelbach

Pohjola

Weber

2021

Electron. Proc. Theor. Comput. Sci.

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Definitions of new symbols merely abbreviate expressions in logical frameworks, and no new facts (regarding previously defined symbols) should hold because of a new definition. In Isabelle/HOL, definable symbols are types and constants. The latter may be ad-hoc overloaded, i. e. have different definitions for non-overlapping types. We prove that symbols that are independent of a new definition may keep their interpretation in a model extension. This work revises our earlier notion of modeltheoretic conservative extension and generalises an earlier model construction. We obtain consistency of theories of definitions in higher-order logic (HOL) with ad-hoc overloading as a corollary. Our results are mechanised in the HOL4 theorem prover. Const c ty 1 # Const d ty 2 def = c = d ∨ ty 1 # ty 2. A user may introduce (non-built-in) types and constants by theory extension, as described in Section 2.3. For types and constants we generally say symbols. Built-ins We abbreviate Bool for Tyapp «bool» [ ] x → y for Tyapp «fun» [x; y] Equal ty for Const «=» (ty → ty → Bool) s === t for Comb (Comb (Equal (typeof s)) s) t Any type with any of these type constructors at the top level is built-in, as is the constant Equal. These are the only symbols which are not user-defined. A formula is a term of type Bool.

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Section: Symbol-independent Fragmentmentioning

confidence: 99%

Section: Symbol-independent Fragmentmentioning

confidence: 99%

Section: Model Constructionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Mechanisation of Model-theoretic Conservative Extension for HOL with Ad-hoc Overloading

Gengelbach

Pohjola

Weber

2021

Electron. Proc. Theor. Comput. Sci.

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“…In very recent work, Gengelbach and Weber [2017] prove a form of model-theoretic conservativity for Isabelle/HOL over definitional base theories. However, they do not work with standard models, but employ the ground semantics we had developed for proving Isabelle/HOL's consistency [Kunčar and Popescu 2015].…”

Section: :23mentioning

confidence: 99%

Safety and conservativity of definitions in HOL and Isabelle/HOL

Kunčar

Popescu

2017

Proc. ACM Program. Lang.

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Definitions are traditionally considered to be a safe mechanism for introducing concepts on top of a logic known to be consistent. In contrast to arbitrary axioms, definitions should in principle be treatable as a form of abbreviation, and thus compiled away from the theory without losing provability. In particular, definitions should form a conservative extension of the pure logic. These properties are crucial for modern interactive theorem provers, since they ensure the consistency of the logic, as well as a valid environment for total/certified functional programming. We prove these properties, namely, safety and conservativity, for Higher-Order Logic (HOL), a logic implemented in several mainstream theorem provers and relied upon by thousands of users. Some unique features of HOL, such as the requirement to give non-emptiness proofs when defining new types and the impossibility to unfold type definitions, make the proof of these properties, and also the very formulation of safety, nontrivial. Our study also factors in the essential variation of HOL definitions featured by Isabelle/HOL, a popular member of the HOL-based provers family. The current work improves on recent results which showed a weaker property, consistency of Isabelle/HOL's definitions.

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Proof-Theoretic Conservative Extension of HOL with Ad-hoc Overloading

Gengelbach

Weber

2020

Theoretical Aspects of Computing – ICTAC 2020

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Model-Theoretic Conservative Extension for Definitional Theories

Cited by 4 publications

References 10 publications

Mechanisation of Model-theoretic Conservative Extension for HOL with Ad-hoc Overloading

Mechanisation of Model-theoretic Conservative Extension for HOL with Ad-hoc Overloading

Safety and conservativity of definitions in HOL and Isabelle/HOL

Proof-Theoretic Conservative Extension of HOL with Ad-hoc Overloading

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