A graph easy class of mute terms A. Bucciarelli, A.Carraro, G.Favro, - PowerPoint PPT Presentation

A graph easy class of mute terms A. Bucciarelli, A.Carraro, G.Favro, A.Salibra ICTCS 2014, Perugia

Terms representing undefinedness. A natural problem arising in λ -calculus is what terms should be considered as representative of undefined programs.

Terms representing undefinedness. A natural problem arising in λ -calculus is what terms should be considered as representative of undefined programs. Ω ≡ ( λ x . xx )( λ x . xx ) is the simplest term that embodies this intuitive idea.

Unsolvable terms.

Unsolvable terms. Every λ -term has one of the following form: ◮ λ x 1 . . . x m . yM 1 . . . M n

Unsolvable terms. Every λ -term has one of the following form: ◮ λ x 1 . . . x m . yM 1 . . . M n ◮ λ x 1 . . . x m . ( λ z . M ) M 1 . . . M n

Unsolvable terms. Every λ -term has one of the following form: ◮ λ x 1 . . . x m . yM 1 . . . M n ◮ λ x 1 . . . x m . ( λ z . M ) M 1 . . . M n If a term β -reduces to a term of the first kind, we say it has a head normal form .

Unsolvable terms. Every λ -term has one of the following form: ◮ λ x 1 . . . x m . yM 1 . . . M n ◮ λ x 1 . . . x m . ( λ z . M ) M 1 . . . M n If a term β -reduces to a term of the first kind, we say it has a head normal form . Definition A term is called unsolvable if it does not have an head normal form.

Unsolvable terms. Every λ -term has one of the following form: ◮ λ x 1 . . . x m . yM 1 . . . M n ◮ λ x 1 . . . x m . ( λ z . M ) M 1 . . . M n If a term β -reduces to a term of the first kind, we say it has a head normal form . Definition A term is called unsolvable if it does not have an head normal form. Unsolvables can be considered as the terms representing the undefined (Barendregt, Wadsworth).

λ -theories and unsolvable terms. Definition A λ -theory is a theory of equations between λ -terms that contains λβ .

λ -theories and unsolvable terms. Definition A λ -theory is a theory of equations between λ -terms that contains λβ . Theorem (Berarducci-Intrigila) There exists a closed unsolvable t such that ∀ M s.t. M � = β I , λβ + { t = M } is a consistent theory, while ∀ M s.t. M = β I , λβ + { t = M } is not consistent.

Easy terms. A closed unsolvable term t is called easy if for any closed term M the theory λβ + { t = M } is consistent.

Easy terms. A closed unsolvable term t is called easy if for any closed term M the theory λβ + { t = M } is consistent. Example ◮ Ω

Easy terms. A closed unsolvable term t is called easy if for any closed term M the theory λβ + { t = M } is consistent. Example ◮ Ω ◮ Ω 3 I , where Ω 3 ≡ ( λ x . xxx )( λ x . xxx )

Easy terms. A closed unsolvable term t is called easy if for any closed term M the theory λβ + { t = M } is consistent. Example ◮ Ω ◮ Ω 3 I , where Ω 3 ≡ ( λ x . xxx )( λ x . xxx ) Ω 3 is unsolvable but not easy.

Easy sets. Definition A set A of closed unsolvable terms is an easy set if for any closed M the theory λβ + { t = M | t ∈ A } is consistent.

Easy sets. Definition A set A of closed unsolvable terms is an easy set if for any closed M the theory λβ + { t = M | t ∈ A } is consistent. Example { Ω( λ x 1 . . . x k +1 . x k +1 ) | k ∈ ω }

Easy sets. Definition A set A of closed unsolvable terms is an easy set if for any closed M the theory λβ + { t = M | t ∈ A } is consistent. Example { Ω( λ x 1 . . . x k +1 . x k +1 ) | k ∈ ω } Theorem The set of easy terms is not an easy set.

Mute terms. Berarducci, “Infinite λ -calculus and non-sensible models”.

Mute terms. Berarducci, “Infinite λ -calculus and non-sensible models”. Definition A term M is a zero term if it does not reduce to an abstraction.

Mute terms. Berarducci, “Infinite λ -calculus and non-sensible models”. Definition A term M is a zero term if it does not reduce to an abstraction. Definition A zero term is mute if it does not reduce to a variable or to a term of the form (Zero term) · Term

Examples and properties of Mute terms. Example ◮ Ω

Examples and properties of Mute terms. Example ◮ Ω ◮ BB , where B ≡ λ x . x ( λ y . xy )

.

Properties of the mute terms. ◮ The set of mute terms is an easy set.

Properties of the mute terms. ◮ The set of mute terms is an easy set. ◮ The set of mute terms is not recursively enumerable, as well as the set of easy sets.

Properties of the mute terms. ◮ The set of mute terms is an easy set. ◮ The set of mute terms is not recursively enumerable, as well as the set of easy sets. Problem Is Y Ω 3 , where Y ≡ λ f . ( λ x . f ( xx ))( λ x . f ( xx )) , easy?

Regular mute terms

Hereditarily n -ary terms. Definition Let n > 0 and ¯ x ≡ x 1 , . . . x k be distinct variables. The set of hereditarily n-ary λ -terms over ¯ x , H n [¯ x ], is the smallest set of terms such that:

Hereditarily n -ary terms. Definition Let n > 0 and ¯ x ≡ x 1 , . . . x k be distinct variables. The set of hereditarily n-ary λ -terms over ¯ x , H n [¯ x ], is the smallest set of terms such that: ◮ For all i = 1 , . . . , k x i ∈ H n [¯ x ]

Hereditarily n -ary terms. Definition Let n > 0 and ¯ x ≡ x 1 , . . . x k be distinct variables. The set of hereditarily n-ary λ -terms over ¯ x , H n [¯ x ], is the smallest set of terms such that: ◮ For all i = 1 , . . . , k x i ∈ H n [¯ x ] ◮ For all fresh distinct variables ¯ y ≡ y 1 , . . . , y n , t 1 ∈ H n [¯ x , ¯ y ] , . . . , t n ∈ H n [¯ x , ¯ y ] λ y 1 . . . λ y n . y i t 1 . . . t n ∈ H n [¯ x ]

Examples of hereditarily terms. ◮ λ x . xx ∈ H 1 = H 1 []

Examples of hereditarily terms. ◮ λ x . xx ∈ H 1 = H 1 [] ◮ λ y . yx ∈ H 1 [ x ]

Examples of hereditarily terms. ◮ λ x . xx ∈ H 1 = H 1 [] ◮ λ y . yx ∈ H 1 [ x ] λ x . xxx is not an hereditarily n -ary term.

A hierarchy of sets based on hereditarily terms. Definition Let ¯ x ≡ x 1 , . . . x k and ¯ y ≡ y 1 , . . . , y n be distinct variables. ◮ H 0 n [¯ x ] = H n [¯ x ]

A hierarchy of sets based on hereditarily terms. Definition Let ¯ x ≡ x 1 , . . . x k and ¯ y ≡ y 1 , . . . , y n be distinct variables. ◮ H 0 n [¯ x ] = H n [¯ x ] ◮ H m +1 x ] = { s [ u / y ] : s ∈ H m u ≡ u 1 , . . . , u n ∈ H m [¯ n [¯ x , ¯ y ] , ¯ n [¯ x ] } n

A hierarchy of sets based on hereditarily terms. Definition Let ¯ x ≡ x 1 , . . . x k and ¯ y ≡ y 1 , . . . , y n be distinct variables. ◮ H 0 n [¯ x ] = H n [¯ x ] ◮ H m +1 x ] = { s [ u / y ] : s ∈ H m u ≡ u 1 , . . . , u n ∈ H m [¯ n [¯ x , ¯ y ] , ¯ n [¯ x ] } n x ] = � ◮ S n [¯ m H m n [¯ x ].

A new class of mute terms. Theorem Given s 0 , . . . , s n ∈ S n , the term s 0 . . . s n is mute.

A new class of mute terms. Theorem Given s 0 , . . . , s n ∈ S n , the term s 0 . . . s n is mute. Proof. Sketch: the key point of the proof is that every reduction path can be seen as starting from a term of this form: ( λ y 1 . . . λ y n . y i t 1 . . . t n ) M 1 . . . M n � �� � � �� � � �� � n terms n terms n abstractions with t j , M j ∈ S n

A new class of mute terms. Theorem Given s 0 , . . . , s n ∈ S n , the term s 0 . . . s n is mute. Proof. Sketch: the key point of the proof is that every reduction path can be seen as starting from a term of this form: ( λ y 1 . . . λ y n . y i t 1 . . . t n ) M 1 . . . M n � �� � � �� � � �� � n terms n terms n abstractions with t j , M j ∈ S n This means that at each step the whole term has a shape among those who are allowed for mute terms.

Definition Terms of the form s 0 . . . s n ∈ S n where s i belongs to S n , are called Regular mute terms.

Definition Terms of the form s 0 . . . s n ∈ S n where s i belongs to S n , are called Regular mute terms. M n is the set of regular mute terms of the form s 0 . . . s n .

Definition Terms of the form s 0 . . . s n ∈ S n where s i belongs to S n , are called Regular mute terms. M n is the set of regular mute terms of the form s 0 . . . s n . M is the set of all regular mute.

Definition Terms of the form s 0 . . . s n ∈ S n where s i belongs to S n , are called Regular mute terms. M n is the set of regular mute terms of the form s 0 . . . s n . M is the set of all regular mute. Example ◮ Ω ∈ M 1

Definition Terms of the form s 0 . . . s n ∈ S n where s i belongs to S n , are called Regular mute terms. M n is the set of regular mute terms of the form s 0 . . . s n . M is the set of all regular mute. Example ◮ Ω ∈ M 1

Definition Terms of the form s 0 . . . s n ∈ S n where s i belongs to S n , are called Regular mute terms. M n is the set of regular mute terms of the form s 0 . . . s n . M is the set of all regular mute. Example ◮ Ω ∈ M 1 ◮ ( λ x . x ( λ y . yx ))( λ x . xx ) ∈ M 1

Definition Terms of the form s 0 . . . s n ∈ S n where s i belongs to S n , are called Regular mute terms. M n is the set of regular mute terms of the form s 0 . . . s n . M is the set of all regular mute. Example ◮ Ω ∈ M 1 ◮ ( λ x . x ( λ y . yx ))( λ x . xx ) ∈ M 1