
In mathematical logic, realizability is a collection of methods in proof theory used to study constructive proofs and extract additional information from them.Formulas from a formal theory are "realized" by objects, known as "realizers", in a way that knowledge of the realizer gives knowledge about the truth of the formula. There are many variations of realizability; exactly which class of formulas is studied and which objects are realizers differ from one variation to another.
Realizability can be seen as a formalization of the Brouwer–Heyting–Kolmogorov (BHK) interpretation of intuitionistic logic. In realizability the notion of "proof" (which is left undefined in the BHK interpretation) is replaced with a formal notion of "realizer". Most variants of realizability begin with a theorem that any statement that is provable in the formal system being studied is realizable. The realizer, however, usually gives more information about the formula than a formal proof would directly provide.
Beyond giving insight into intuitionistic provability, realizability can be applied to prove the disjunction and existence properties for intuitionistic theories and to extract programs from proofs, as in proof mining. It is also related to topos theory via realizability topoi.
Example: Kleene's 1945-realizability
Kleene's original version of realizability uses natural numbers as realizers for formulas in Heyting arithmetic. A few pieces of notation are required: first, an ordered pair (n,m) is treated as a single number using a fixed primitive recursive pairing function; second, for each natural number n, φn is the computable function with index n. The following clauses are used to define a relation "n realizes A" between natural numbers n and formulas A in the language of Heyting arithmetic, known as Kleene's 1945-realizability relation:
- Any number n realizes an atomic formula s=t if and only if s=t is true. Thus every number realizes a true equation, and no number realizes a false equation.
- A pair (n,m) realizes a formula A∧B if and only if n realizes A and m realizes B. Thus a realizer for a conjunction is a pair of realizers for the conjuncts.
- A pair (n,m) realizes a formula A∨B if and only if the following hold: n is 0 or 1; and if n is 0 then m realizes A; and if n is 1 then m realizes B. Thus a realizer for a disjunction explicitly picks one of the disjuncts (with n) and provides a realizer for it (with m).
- A number n realizes a formula A→B if and only if, for every m that realizes A, φn(m) realizes B. Thus a realizer for an implication corresponds to a computable function that takes any realizer for the hypothesis and produces a realizer for the conclusion.
- A pair (n,m) realizes a formula (∃ x)A(x) if and only if m is a realizer for A(n). Thus a realizer for an existential formula produces an explicit witness for the quantifier along with a realizer for the formula instantiated with that witness.
- A number n realizes a formula (∀ x)A(x) if and only if, for all m, φn(m) is defined and realizes A(m). Thus a realizer for a universal statement is a computable function that produces, for each m, a realizer for the formula instantiated with m.
With this definition, the following theorem is obtained:
- Let A be a sentence of Heyting arithmetic (HA). If HA proves A then there is an n such that n realizes A.
On the other hand, there are classical theorems (even propositional formula schemas) that are realized but which are not provable in HA, a fact first established by Rose. So realizability does not exactly mirror intuitionistic reasoning.
Further analysis of the method can be used to prove that HA has the "disjunction and existence properties":
- If HA proves a sentence (∃ x)A(x), then there is an n such that HA proves A(n)
- If HA proves a sentence A∨B, then HA proves A or HA proves B.
More such properties are obtained involving Harrop formulas.
Later developments
Kreisel introduced modified realizability, which uses typed lambda calculus as the language of realizers. Modified realizability is one way to show that Markov's principle is not derivable in intuitionistic logic. On the contrary, it allows to constructively justify the principle of independence of premise:
.
Relative realizability is an intuitionist analysis of computable or computably enumerable elements of data structures that are not necessarily computable, such as computable operations on all real numbers when reals can be only approximated on digital computer systems.
Classical realizability was introduced by and extends realizability to classical logic. It furthermore realizes axioms of Zermelo–Fraenkel set theory. Understood as a generalization of Cohen’s forcing, it was used to provide new models of set theory.
Linear realizability extends realizability techniques to linear logic. The term was coined by to encompass several constructions, such as geometry of interaction models,ludics, interaction graphs models.
Use in proof mining
Realizability is one of the methods used in proof mining to extract concrete "programs" from seemingly non-constructive mathematical proofs. Program extraction using realizability is implemented in some proof assistants such as Coq.
See also
- Curry–Howard correspondence
- Dialectica interpretation
- Harrop formula
Notes
- van Oosten 2000
- A. Ščedrov, "Intuitionistic Set Theory" (pp.263--264). From Harvey Friedman's Research on the Foundations of Mathematics (1985), Studies in Logic and the Foundations of Mathematics vol. 117.
- van Oosten 2000, p. 7
- Rose 1953
- van Oosten 2000, p. 6
- Birkedal 2000
- Krivine, Jean-Louis (2001). "Typed lambda-calculus in classical Zermelo-Fraenkel set theory". Archive for Mathematical Logic. 40 (2): 189–205.
- Krivine, Jean-Louis (2011). "Realizability algebras: a program to well order R". Logical Methods in Computer Science. 7.
- Seiller, Thomas (2024). Mathematical Informatics (Habilitation thesis). Université Sorbonne Paris Nord.
- Girard, Jean-Yves (1989). "Geometry of interaction 1: Interpretation of System F". Studies in Logic and the Foundations of Mathematics. 127: 221–260.
- Girard, Jean-Yves (2001). "Locus Solum: from the rules of logic to the logic of rules". Mathematical Structures in Computer Science. 11: 301–506.
- Seiller, Thomas (2016). "Interaction graphs: Full linear logic". Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science.
References
- Birkedal, Lars; Jaap van Oosten (2000). Relative and modified relative realizability.
- Kreisel G. (1959). "Interpretation of Analysis by Means of Constructive Functionals of Finite Types", in: Constructivity in Mathematics, edited by A. Heyting, North-Holland, pp. 101–128.
- Kleene, S. C. (1945). "On the interpretation of intuitionistic number theory". Journal of Symbolic Logic. 10 (4): 109–124. doi:10.2307/2269016. JSTOR 2269016.
- Kleene, S. C. (1973). "Realizability: a retrospective survey" from Mathias, A. R. D.; Hartley Rogers (1973). Cambridge Summer School in Mathematical Logic : held in Cambridge/England, August 1–21, 1971. Berlin: Springer. ISBN 3-540-05569-X., pp. 95–112.
- van Oosten, Jaap (2000). Realizability: An Historical Essay.
- Rose, G. F. (1953). "Propositional calculus and realizability". Transactions of the American Mathematical Society. 75 (1): 1–19. doi:10.2307/1990776. JSTOR 1990776.
External links
- Realizability Collection of links to recent papers on realizability and related topics.
In mathematical logic realizability is a collection of methods in proof theory used to study constructive proofs and extract additional information from them Formulas from a formal theory are realized by objects known as realizers in a way that knowledge of the realizer gives knowledge about the truth of the formula There are many variations of realizability exactly which class of formulas is studied and which objects are realizers differ from one variation to another Realizability can be seen as a formalization of the Brouwer Heyting Kolmogorov BHK interpretation of intuitionistic logic In realizability the notion of proof which is left undefined in the BHK interpretation is replaced with a formal notion of realizer Most variants of realizability begin with a theorem that any statement that is provable in the formal system being studied is realizable The realizer however usually gives more information about the formula than a formal proof would directly provide Beyond giving insight into intuitionistic provability realizability can be applied to prove the disjunction and existence properties for intuitionistic theories and to extract programs from proofs as in proof mining It is also related to topos theory via realizability topoi Example Kleene s 1945 realizabilityKleene s original version of realizability uses natural numbers as realizers for formulas in Heyting arithmetic A few pieces of notation are required first an ordered pair n m is treated as a single number using a fixed primitive recursive pairing function second for each natural number n fn is the computable function with index n The following clauses are used to define a relation n realizes A between natural numbers n and formulas A in the language of Heyting arithmetic known as Kleene s 1945 realizability relation Any number n realizes an atomic formula s t if and only if s t is true Thus every number realizes a true equation and no number realizes a false equation A pair n m realizes a formula A B if and only if n realizes A and m realizes B Thus a realizer for a conjunction is a pair of realizers for the conjuncts A pair n m realizes a formula A B if and only if the following hold n is 0 or 1 and if n is 0 then m realizes A and if n is 1 then m realizes B Thus a realizer for a disjunction explicitly picks one of the disjuncts with n and provides a realizer for it with m A number n realizes a formula A B if and only if for every m that realizes A fn m realizes B Thus a realizer for an implication corresponds to a computable function that takes any realizer for the hypothesis and produces a realizer for the conclusion A pair n m realizes a formula x A x if and only if m is a realizer for A n Thus a realizer for an existential formula produces an explicit witness for the quantifier along with a realizer for the formula instantiated with that witness A number n realizes a formula x A x if and only if for all m fn m is defined and realizes A m Thus a realizer for a universal statement is a computable function that produces for each m a realizer for the formula instantiated with m With this definition the following theorem is obtained Let A be a sentence of Heyting arithmetic HA If HA proves A then there is an n such that n realizes A On the other hand there are classical theorems even propositional formula schemas that are realized but which are not provable in HA a fact first established by Rose So realizability does not exactly mirror intuitionistic reasoning Further analysis of the method can be used to prove that HA has the disjunction and existence properties If HA proves a sentence x A x then there is an n such that HA proves A n If HA proves a sentence A B then HA proves A or HA proves B More such properties are obtained involving Harrop formulas Later developmentsKreisel introduced modified realizability which uses typed lambda calculus as the language of realizers Modified realizability is one way to show that Markov s principle is not derivable in intuitionistic logic On the contrary it allows to constructively justify the principle of independence of premise A xP x x A P x displaystyle A rightarrow exists x P x rightarrow exists x A rightarrow P x Relative realizability is an intuitionist analysis of computable or computably enumerable elements of data structures that are not necessarily computable such as computable operations on all real numbers when reals can be only approximated on digital computer systems Classical realizability was introduced by and extends realizability to classical logic It furthermore realizes axioms of Zermelo Fraenkel set theory Understood as a generalization of Cohen s forcing it was used to provide new models of set theory Linear realizability extends realizability techniques to linear logic The term was coined by to encompass several constructions such as geometry of interaction models ludics interaction graphs models Use in proof miningRealizability is one of the methods used in proof mining to extract concrete programs from seemingly non constructive mathematical proofs Program extraction using realizability is implemented in some proof assistants such as Coq See alsoCurry Howard correspondence Dialectica interpretation Harrop formulaNotesvan Oosten 2000 A Scedrov Intuitionistic Set Theory pp 263 264 From Harvey Friedman s Research on the Foundations of Mathematics 1985 Studies in Logic and the Foundations of Mathematics vol 117 van Oosten 2000 p 7 Rose 1953 van Oosten 2000 p 6 Birkedal 2000 Krivine Jean Louis 2001 Typed lambda calculus in classical Zermelo Fraenkel set theory Archive for Mathematical Logic 40 2 189 205 Krivine Jean Louis 2011 Realizability algebras a program to well order R Logical Methods in Computer Science 7 Seiller Thomas 2024 Mathematical Informatics Habilitation thesis Universite Sorbonne Paris Nord Girard Jean Yves 1989 Geometry of interaction 1 Interpretation of System F Studies in Logic and the Foundations of Mathematics 127 221 260 Girard Jean Yves 2001 Locus Solum from the rules of logic to the logic of rules Mathematical Structures in Computer Science 11 301 506 Seiller Thomas 2016 Interaction graphs Full linear logic Proceedings of the 31st Annual ACM IEEE Symposium on Logic in Computer Science ReferencesBirkedal Lars Jaap van Oosten 2000 Relative and modified relative realizability Kreisel G 1959 Interpretation of Analysis by Means of Constructive Functionals of Finite Types in Constructivity in Mathematics edited by A Heyting North Holland pp 101 128 Kleene S C 1945 On the interpretation of intuitionistic number theory Journal of Symbolic Logic 10 4 109 124 doi 10 2307 2269016 JSTOR 2269016 Kleene S C 1973 Realizability a retrospective survey from Mathias A R D Hartley Rogers 1973 Cambridge Summer School in Mathematical Logic held in Cambridge England August 1 21 1971 Berlin Springer ISBN 3 540 05569 X pp 95 112 van Oosten Jaap 2000 Realizability An Historical Essay Rose G F 1953 Propositional calculus and realizability Transactions of the American Mathematical Society 75 1 1 19 doi 10 2307 1990776 JSTOR 1990776 External linksRealizability Collection of links to recent papers on realizability and related topics