Algorithmic Eta-reduction in Type-theory of Acyclic Recursion

Roussanka Loukanova

1,2

Stockholm University, Stockholm, Sweden

Institute of Mathematics and Informatics, Bulgarian Academy of Sciences,

Acad. G. Bonchev Str., Block 8, 1113 Soﬁa, Bulgaria

Keywords:

Algorithms, Acyclic Recursion, Type Theory, Reduction Calculi, Canonical Forms, Eta-reduction.

Abstract:

The paper extends the standard reduction calculus of the higher-order Type-theory of Acyclic Recursion to η-

reduction. This is achieved by adding a restricted η-rule, which is applicable only to terms in canonical forms

satisfying certain conditions. Canonical forms of terms determine the iterative algorithms for computing the

semantic denotations of the terms. Unnecessary λ-abstractions and corresponding functional applications in

canonical forms contribute to algorithmic complexity. The η-rule provides a simple way to reduce complexity

by maintenance of essential algorithmic structure of computations.

1 INTRODUCTION

This paper is part of theoretical development and ap-

plications of a new approach to the mathematical no-

tion of algorithm, originally introduced by the for-

mal languages of recursion (Moschovakis, 1989). The

simply-typed theory of recursion L

introduced in

(Moschovakis, 2006) is a higher-order type theory,

which is a proper extension of Gallin type theory TY

(Gallin, 1975). The book (Moschovakis, 2019) inves-

tigates complexity in its untyped version.

The formal languages FLR introduced in a se-

quence of papers (Moschovakis, 1989; Moschovakis,

1993; Moschovakis, 1997), while untyped systems

formalising untyped domains of recursive functions,

allow full recursion with cyclicity, and are equivalent

to any of the classic theories of the mathematical no-

tion of algorithm. The formal syntax of L

, presented

originally in (Moschovakis, 2006), is typed and al-

lows only recursion terms with acyclic systems of as-

signments, thus, formalising algorithms that end after

ﬁnite iterations of computations. Typed languages of

full recursion L

have the typed syntax of L

, with-

out the acyclicity requirement. The classes of lan-

guages of recursion (FLR, L

, and L

) have two se-

mantic layers: denotational semantics and algorith-

mic semantics. The recursion terms of L

are essen-

tial for representing algorithmic computations of se-

mantic information.

This paper concerns the typed theory of algoritms

, which is already well developed, and provides

new approaches to intelligent foundations, with ver-

satile applications, and especially to the areas of Ar-

tiﬁcial Intelligence (AI). In this paper, we introduce

a technique, which, by adding a simple additional re-

duction rule to the reduction calculi of L

, is impor-

tant for applications of L

to the areas of AI.

The purpose of the reduction calculi of the for-

mal language of L

is to reduce every L

term A to

a term in a canonical form that represents the com-

putational steps for the computation of the denotation

of A. The reduction calculi of L

reﬂect fundamen-

tal algorithmic patterns in computations. Among the

potential applications of L

are intelligent software

systems, e.g., in robotics and in AI, that perform al-

gorithmic procedures, which determine reliable per-

formance. The prominent applications of L

are to

computational semantics of formal and natural lan-

guages, and, in particular, semantics of programming

and other speciﬁcation languages in Computer Sci-

ence and AI, as well as Natural Language Process-

ings (NLP), and computational grammars that cover

semantics.

The reduction calculus of L

reduces every L

term to its canonical form. The canonical forms not

only preserve the denotations of the L

-terms, they

reveal the algorithmic steps encoded by the terms

for computing their denotations. The canonical form

cf(A) of every L

-term A determines the algorithm

that computes the denotation of A in the semantics

domain of every given semantic structure. The λ-rule

of L

is one of the most important reduction rules

Loukanova, R.

Algorithmic Eta-reduction in Type-theory of Acyclic Recursion.

DOI: 10.5220/0009182410031010

In Proceedings of the 12th International Conference on Agents and Artiﬁcial Intelligence (ICAART 2020) - Volume 2, pages 1003-1010

ISBN: 978-989-758-395-7; ISSN: 2184-433X

1003

for the recursion and abstraction operators. Never-

theless, in many cases, the λ-rule, creates redundant

λ-abstractions over variables that do not occur freely

in the recursion terms (e.g., see Example 3.1). For ex-

ample, it is important to simplify terms having such

extra abstractions, when translating, i.e., rendering,

natura language (NL) expressions to semantic repre-

sentations by L

-terms.

This paper presents a restricted η-reduction that

simpliﬁes terms in canonical forms. The η-rule does

not preserve the strictest referential synonymy of its

input and output terms, which are in canonical forms.

Importantly, the η-rule takes care to maintain closely

the algorithmic meaning of the terms in the entire

reduction sequence, while reducing computational

complexity caused by excessive, superﬂuous lambda-

abstractions and corresponding functional applica-

tions. It preserves the denotations of the input and re-

duced terms. (Loukanova, 2019c; Loukanova, 2019b)

introduce and investigate the properties of more intri-

cate rules and reduction calculi for removing redun-

dant lambda-abstractions. The η-reduction is also in-

teresting. It is easier to use, by extending the ordinal

reduction calculus of L

, since it is applied directly,

only to terms in canonical forms.

(Loukanova, 2011d) introduces very brieﬂy the η-

rule, for the purpose of applying it to semantic rep-

resentations of human language, without looking into

its properties. We reformulate the rule here, for pre-

senting its properties with respect to its existing and

potential applications to the areas of Artiﬁcial Intelli-

gence (AI). In particular, we point to applications that

need context dependent information and algorithmic

computations that depend on states of information, in-

cluding agents.

We should stress that the η-rule introduced here is

about recursion terms and the algorithms designated

by them. It is different from, while related to, the

denotational η-rule in standard λ-calculi.

In Section 2, we give the formal deﬁnitions of the

syntax of L

. We introduce the denotational and algo-

rithmic semantics of L

with some intuitions and ex-

amples. Then, we introduce the rules of the reduction

calculus of the type theory L

, which is central to the

referential intensions, i.e., to the algorithmic meaning

in a selected speciﬁc semantic domain of applications,

and some key theoretical results. The central part of

the paper is on introducing the new, additional η-rule

for reduction of the L

terms, which are in canoni-

cal forms, to simpler η-canonical forms that formalise

more efﬁcient algorithmic computations.

2 TYPE-THEORY OF ACYCLIC

RECURSION

2.1 Types

The set Types is the smallest set deﬁned recursively

as follows, by :

≡ e | t | s | (τ

→ τ

) (Types)

The type e is for entities, also called individuals; s

is for states consisting of various information, e.g.,

such as possible worlds, context, time and/or space

locations, some agents, e.g., a speaker; t is for truth

values. For an elaboration of possible choices of con-

text information, which include speaker agents, see

(Loukanova, 2011b). The type (s → τ) is for context

dependent objects.

2.2 Syntax of L

Vocabulary of L

• Constants: K =

τ∈Types

where, for each τ ∈ Types, K

is a ﬁnite set:

= { c

, c

, . . . , c

}

• Pure Variables: PureV =

τ∈Types

PureV

where, for each τ ∈ Types,

PureV

= { v

, v

, . . . } (a denumerable set)

• Recursion Variables (Memory Locations):

RecV =

τ∈Types

RecV

where, for each τ ∈ Types,

RecV

= { p

, p

, . . . } (a denumerable set)

Deﬁnition 1 (The set Terms of L

≡ c

: τ | x

: τ | B

(σ→τ)

) : τ (1)

| λ(v

)(B

) : (σ → τ) (2)

| A

where { p

= A

, . . . ,

= A

} : σ

(3)

for A

: σ

, . . . , A

: σ

are terms (n ≥ 0), and p

: σ

. . . , p

: σ

are pairwise different recursion variables

(memory locations), and the sequence of assignments

{ p

= A

, . . . , p

= A

} satisﬁes the Acyclicity Con-

straint:

Acyclicity Constraint. A sequence of assignments

of the form:

{ p

= A

, . . . , p

= A

}

is acyclic iff there is a ranking function

rank : { p

, . . . , p

} −→ N

such that, for all p

, p

∈ { p

, . . . , p

}, if p

oc-

curs freely in A

, then rank(p

) < rank(p

ICAART 2020 - 12th International Conference on Agents and Artiﬁcial Intelligence

1004

Intuitively, an acyclic sequence of assignments

{ p

= A

, . . . , p

= A

} deﬁnes recursive computa-

tions of the values den(A

), . . . , den(A

) to be as-

signed to the locations p

, . . . , p

, which close-off

after a ﬁnite number of steps; ranking rank(p

) <

rank(p

) means that the value A

assigned to p

, may

depend on the values of the free occurrences of the

location p

in A

, and of all other free occurrences of

locations with lower rank than p

Some Notations and Abbreviations

• The symbol “=” is a predicate constant of the lan-

guage L

for identity, and also used for the iden-

tity relation, which it denotes

• The symbol “≡” is a meta-symbol for literal iden-

tity between expressions

• The symbol “

≡” is a meta-symbol that we use in

deﬁnitions, e.g., of types and terms, and for the

replacement operation

• Often, we skip some “understood” parentheses

• The type σ of a term A may be depicted either as

a superscript, A

, or by a colon, A : σ

We use the following abbreviations for “folding” and

“unfolding” sequences of assignments. For any terms

: σ

, . . . , A

: σ

, A

n+1

: σ

n+1

, C, D : τ, and recur-

sion variables p

: σ

, . . . , p

: σ

, (where n ≥ 0):

−→

A ≡ p

:= A

, . . . , p

:= A

(4a)

−→

A {C

≡ D} ≡ (4b)

= A

≡ D}, . . . , p

= A

≡ D} (4c)

where, for all i = 1, . . . , n, A

≡ D} are the result of

the replacement of all occurrences of C, respectively

in A

without causing variable clashes.

Denotational Semantics of L

The language L

has denotational semantics provided by a denotational

function den

, for any given typed semantic structure

A with typed domain frames and variable assignments

g in A. The deﬁnition of den

is by structural in-

duction on the terms, for details, see (Moschovakis,

2006) and (Loukanova, 2019c; Loukanova, 2019b).

Often, we shall designate the denotation function by

den without the superscript.

2.3 Reduction Calculus of L

The reduction rules deﬁne a reduction relation be-

tween terms.

The congruence relation, ≡

, is the smallest rela-

tion between L

terms (A ≡

B) that is reﬂexive, sym-

metric, transitive, and closed under: term formation

rules (application, λ-abstraction, and acyclic recur-

sion); renaming bound variables (pure and recursion),

without causing variable collisions; and re-ordering

of the assignments within the acyclic sequences of as-

signments of the recursion terms , i.e., for any permu-

tation π: {1, . . . , n} → {1, . . . , n}:

Reduction Rules of L

Congruence If A ≡

B, then A ⇒ B (cong)

Transitivity If A ⇒ B and B ⇒ C, then A ⇒ C (trans)

Compositionality

• If A ⇒ A

and B ⇒ B

, (rep1)

then A(B) ⇒ A

)

• If A ⇒ B, then λu (A) ⇒ λu (B) (rep2)

• If A

⇒ B

, for i = 0, . . . , n, then (rep3)

where { p

= A

, . . . , p

= A

}

⇒ B

where { p

= B

, . . . , p

= B

}

The Head Rule (head)



where {

−→

A }



where {

−→

B }

⇒ A

where {

−→

A ,

−→

B }

given that no p

occurs freely in any B

, for i = 1,

. . . , n, j = 1, . . . , m

The Beki

c-Scott Rule (B-S)

where {p



where {

−→

B }



−→

A }

⇒ A

where {p

= B

−→

B ,

−→

A }

given that no q

occurs freely in any A

, for i = 1,

. . . , n, j = 1, . . . , m

The Recursion-application Rule (recap)

where {

−→

A }



(B)

⇒ A

(B) where {

−→

A }

given that no p

occurs freely in B, for i = 1, . . . ,

The Application Rule (ap)

A(B) ⇒ A(p) where {p

= B}

given that B is a proper term and p is a fresh loca-

tion

The λ-Rule (λ)

λu(A

where { p

= A

, . . . , p

= A

})

⇒ λuA

where { p

= λu A

, . . . , p

= λu A

where for all i = 1, . . . , n, p

is a fresh lo-

cation and A

is the result of the replacement

of the free occurrences of p

, . . . , p

in A

with

(u), . . . , p

(u), respectively, i.e.:

≡ A

≡ p

(u), . . . , p

≡ p

(u)}

Algorithmic Eta-reduction in Type-theory of Acyclic Recursion

1005

2.4 Some Major Properties of L

The following theorems are important for the algo-

rithmic meanings of the terms.

Theorem 1 (Canonical Form Theorem). (For details,

see (Moschovakis, 2006), § 3.1.) For each term A,

there is a unique, up to congruence, irreducible term

C, denoted by cf(A) and called the canonical form of

A, such that:

1. cf(A) ≡ A

where { p

:= A

, . . . , p

:= A

} for

some explicit, irreducible terms A

, . . . , A

(n ≥ 0)

2. A ⇒ cf(A)

3. if A ⇒ B and B is irreducible, then B ≡

cf(A), i.e.,

cf(A) is the unique, up to congruence, irreducible

term to which A can be reduced

Theorem 2. (See (Moschovakis, 2006), § 3.11.) For

any L

terms A and B, if A ⇒ B, then

den(A) = den(B) (5a)

den(A) = den(cf(A)) (5b)

The canonical forms have a distinguished feature that

is part of their computational (algorithmic) role: they

provide algorithmic patterns of semantic computa-

tions. The more general terms provide algorithmic

patterns that consist of sub-terms with components

that are recursion variables; the most basic assign-

ments of recursion variables (of lowest ranks) pro-

vide the speciﬁc basic data that feeds-up the general

computational patterns. The more general terms and

sub-terms classify language expressions with respect

to their semantics and determine the algorithms for

computing the denotations of the expressions.

Algorithmic Semantics of L

. The notion of algo-

rithmic semantics, i.e., algorithmic intension, in the

languages of recursion, (Moschovakis, 2006), covers

the most essential, computational aspect of the con-

cept of meaning. The referential intension, int(A),

of a meaningful term A is the tuple of functions (a

recursor) that is deﬁned by the denotations den(A

)

(i ∈ {0, . . . n}) of the parts (i.e., the head sub-term A

and of the terms A

, . . . , A

in the system of assign-

ments) of its canonical form:

cf(A) ≡ A

where { p

:= A

, . . . , p

:= A

}

Intuitively, for each meaningful term A, the inten-

sion of A, int(A), is the algorithm for computing its

denotation den(A). Two meaningful expressions are

synonymous iff their referential intensions are natu-

rally isomorphic, i.e., they are the same algorithm.

Thus, the algorithmic meaning of a meaningful term

(i.e., its sense) is the information about how to com-

pute its denotation step-by-step: a meaningful term

has sense by carrying instructions within its struc-

ture, which are revealed by its canonical form, for

acquiring what they denote in a model. The canon-

ical form cf(A) of a meaningful term A encodes its

intension, i.e., the algorithm for computing its deno-

tation, via: (1) the basic instructions (facts), which

consist of { p

:= A

, . . . , p

:= A

} and the head

term A

, which are needed for computing the deno-

tation den(A), and (2) a terminating rank order of

the recursive steps that compute each den(A

), for

i ∈ {0, . . . , n}, for incremental computation of the de-

notation den(A) = den(A

The reduction calculus of the type theory L

effective. The calculus of the intensional synonymy,

i.e., algorithmic equivalence, in the type-theory L

has a restricted β-reduction rule, which contributes

to the high expressiveness of the language of L

see (Moschovakis, 2006) and (Loukanova, 2011a;

Loukanova, 2011c).

Theorem 3 (Referential Synonymy Theorem). (See

(Moschovakis, 2006), § 3.4.) Two terms A, B are ref-

erentially synonymous, A ≈ B, i.e., algorithmically

equivalent, with respect to a given semantic structure

A iff there are explicit, irreducible terms (of appropri-

ate types), A

, A

, . . . , A

, B

, . . . , B

, n ≥ 0, such

that:

1. A ⇒

where { p

:= A

, . . . , p

:= A

2. B ⇒

where { p

:= B

, . . . , p

:= B

3.(a) for every x ∈ PureV ∪ RecV,

x ∈ FreeVars(A

) iff x ∈ FreeVars(B

for i ∈ {0, . . . , n}

(6)

(b) for all g ∈ G:

den(A

)(g) = den(B

)(g), i ∈ {0, . . . , n} (7)

3 ETA-REDUCTION

In this section, at ﬁrst, we present an example from

natural language to motivate the usefulness of the ad-

ditional η-rule, and then we introduce the η-rule and

the extended reduction calculus.

Example 3.1. The detailed steps of rendering the sen-

tence (8a) into a term, and its reduction to the canoni-

cal form (8b)–(8e), are given in (Loukanova, 2019b),

as motivation for γ-reduction, which is more general

and complex.

ICAART 2020 - 12th International Conference on Agents and Artiﬁcial Intelligence

1006

Kim hugs some dog

render

−−−→ A. . . (8a)

cf(A) ≡

[λy

(some



d(y

)



(h(y

)))](k) where (8b)

= kim, (8c)

= λy

λx

hugs(x

)(y

), (8d)

= λy

dog} (8e)

Then, by Theorem 3:

cf(A) 6≈ B ≡



λy

some(d

)



h(y

)



(k) where (9a)

= kim, (9b)

= λy

λx

hugs(x

)(y

), (9c)

= dog } (9d)

The term B in (9a)-(9b) is in a canonical form, but it is

not algorithmically equivalent (referentially synony-

mous) to the term (8b)-(8e), by the reduction calculus

of L

. That is, these two terms are not algorithmi-

cally equivalent with respect to the strictest notion of

algorithm introduced in (Moschovakis, 2006).

Deﬁnition 2 (η-rule). Let A be a term in a canonical

form:

A ≡ A

where {

−→

A ,

n+1

= λvA

n+1

−→

B }

(10a)

≡ A

where { p

:= A

, . . . , p

:= A

n+1

= λvA

n+1

= B

, . . . , q

= B

}

(10b)

with n ≥ 0, k ≥ 0, such that

1. v : σ is a pure variable and p

n+1

: (σ → τ) is a

recursion variable (location).

2. The explicit, irreducible term A

n+1

: τ does not

have any (free) occurrences of v (and p

n+1

3. All the occurrences of p

n+1

in A

−→

A , and

−→

B are

occurrences of the term p

n+1

(v), which are in the

scope of λv (modulo appropriate renaming of v).

Then, for any fresh recursion variable p

n+1

: τ

where {

−→

A ,

n+1

= λvA

n+1

−→

B }

(11a)

⇒

n+1

(v)

≡ p

n+1

} where {

−→

A {p

n+1

(v)

≡ p

n+1

= A

n+1

−→

B {p

n+1

(v)

≡ p

n+1

}}

(11b)

where,

−→

A {p

n+1

(v)

≡ p

n+1

} is the result of the re-

placement A

n+1

(v)

≡ p

n+1

} of all occurrences

of p

n+1

(v) with p

n+1

, in all terms A

−→

A , and

−→

B {p

n+1

(v)

≡ p

n+1

} is the is the result of the re-

placement B

n+1

(v)

≡ p

n+1

} of all occurrences of

n+1

(v) with p

n+1

, in all terms B

−→

B .

Note: In the η-rule and corresponding applica-

tions, for all i ∈ {0, . . . , n} and j ∈ {0, . . . , k}, the re-

placements A

n+1

(v)

≡ p

n+1

} and B

n+1

(v)

≡

n+1

} are such that the occurrences of the term

n+1

(v) in A

and B

are in the scope of λv.

Theorem 4 (Denotational Equivalence by η-rule).

Let A be a term in a canonical form:

A ≡ A

where {

−→

A ,

n+1

= λvA

n+1

−→

B }

(12a)

(n ≥ 0) such that:

1. v : σ is a pure variable and p

n+1

: (σ → τ) is a

recursion variable.

2. The explicit, irreducible term A

n+1

: τ does not

have any (free) occurrences of v (and p

n+1

3. All the occurrences of p

n+1

, in A

−→

A , and

−→

B , are

occurrences of the term p

n+1

(v), which are in the

scope of λv (modulo appropriate renaming of v).

Let p

n+1

: τ be a fresh recursion variable, and A

be a

term as in (13b) (by the η-rule):

A ≡ [A

where {

−→

A ,

n+1

= λvA

n+1

−→

B }]

(13a)

⇒

≡ [A

n+1

(v)

≡ p

n+1

} where

{

−→

A {p

n+1

(v)

≡ p

n+1

= A

n+1

−→

B {p

n+1

(v)

≡ p

n+1

}}]

(13b)

Then,

A 6≈ A

(14a)

cf(A

) ≡

(14b)

and, for all g ∈ G, i ∈ {0, . . . , n}, and j ∈ {1, . . . , k},

the following denotational equalities hold:

den(A)(g) = den(A

)(g) (15)

and:

den(A

)(g{

−→

p , p

n+1

= p

n+1

−→

q })

(16a)

=den



n+1

(v)

≡ p

n+1

}



(g{

−→

n+1

= p

n+1

−→

})

(16b)

Algorithmic Eta-reduction in Type-theory of Acyclic Recursion

1007

and:

den(B

)(g{

−→

p , p

n+1

= p

n+1

−→

})

(17a)

=den



n+1

(v)

≡ p

n+1

}



(g{

−→

n+1

= p

n+1

−→

})

(17b)

where, for all i ∈ {1, . . . , n+1} and j ∈ {1, . . . , k}, the

values p

∈ T

, p

∈ T

, q

∈ T

, and q

∈ T

are

calculated by recursion on the rank of

−→

p , p

n+1

−→

n+1

−→

q ,

−→

Proof. The proof is long, by induction on the rank

values, and is not in the subject of this paper.

Deﬁnition 3 (η-reduction). The η-reduction relation

in L

is the smallest relation ⇒

∗

between L

-terms,

such that:

(1) For any L

-terms A and B,

if cf(A) ⇒

B, then A ⇒

∗

B (η-red)

(2) ⇒

∗

is closed under transitivity, congruence, and

is compositional with respect to term formation

rules, i.e.:

Transitivity If A ⇒

∗

B and B ⇒

∗

C, then A ⇒

∗

(η-tr)

Congruence If A ≡

B, then A ⇒

∗

B (η-cong)

Compositionality

• If A ⇒

∗

and B ⇒

∗

, then A(B) ⇒

∗

)

(η-rep1)

• If A ⇒

∗

B, then λu(A) ⇒

∗

λu(B) (η-rep2)

• If A

⇒

∗

, for i = 0, . . . , n, then (η-rep3)

where { p

= A

, . . . , p

= A

}

⇒

∗

where { p

= B

, . . . , p

= B

}

Deﬁnition 4 (η-equivalence relation ≈

). For any

-terms A and B

A ≈

B ⇐⇒ for some C,

cf(A) ⇒

∗

C and C ≈ B

(18)

Corollary 1. For any L

-terms A and B,

(1) if A ⇒

∗

B, then den(A) = den(B)

(2) if A ≈

B, then den(A) = den(B)

Proof. ((1)) is proved by induction on the deﬁnition

of the ⇒

∗

. Then, ((2)) follows by the Deﬁnition 4 of

≈

Corollary 2. There exist (many) L

-terms A, B, and

C such that

A ⇒ B ⇒

∗

C =⇒ A ≈ B =⇒ A ≈

B ≈

C (19a)

while C 6≈ B and C 6≈ A (19b)

4 USEFULNESS OF THE

ETA-RULE

In this section, we present some arguments for the use

of η-rule and η-reduction, for existing and potential

applications.

Maintenance of Essential Algorithmic Computa-

tions. While the equalities (15), (16a), (17a) are

about denotations, they do not use the denotational

traditional β-conversion or any other syntactical ma-

nipulations over the terms A and A

, except the η-rule

for A ⇒

∗

Preservance of Algorithmic Structure. The η-rule

preserves very closely the computational structure of

the term A in a canonical form. The term A

, which is

such that A ⇒

∗

, is also in a canonical form, with

parts that are almostly the same as the correspond-

ing parts of A, with the exception of the replacements

n+1

(v)

≡ p

n+1

} and skipped λv from the term part

of p

n+1

= λvA

n+1

to the term part of p

n+1

= A

n+1

The where assignments formalise some essentials

of object declarations in object oriented programming

languages, and in general, of function declarations in

programming languages.

The equalities of the denotations of the corre-

sponding parts, given in (16a)–(16b) and (17a)–(17b),

are proved by recursion on rank. The denotations of

the parts may, in general, strictly depend on the values

of the recursion variables that have lesser rank. The

denotational equality of the corresponding parts, e.g.,

of A

and A

n+1

(v)

≡ p

n+1

}, in (20a)–(20b), holds

for the variable assignment g due to its update for the

local variables within the scope of where, which are

constrained by the recursion assignments.

den(A

)(g{

−→

p , p

n+1

= p

n+1

−→

q })

(20a)

=den



n+1

(v)

≡ p

n+1

}



(g{

−→

n+1

= p

n+1

−→

})

(20b)

The recursion variables

−→

p , p

n+1

−→

, p

n+1

−→

q ,

−→

are bound, i.e., constrained to the values determined

by the assignments in the scope of where. In case that

rank(p

) > rank(p

n+1

), the value den(A

)(g) of the

part A

can depend on the value of p

n+1

, and respec-

tively, den



n+1

(v)

≡ p

n+1

}



(g) on the value of

n+1

. Without the constraints p

n+1

= λvA

n+1

and

n+1

= A

n+1

, a variable assignment g can be such

that for some a ∈ T

, g(p

n+1

)(a) 6= g(p

n+1

). Then,

without the speciﬁed update of g, outside the scope of

ICAART 2020 - 12th International Conference on Agents and Artiﬁcial Intelligence

1008

where, i.e. outside the local system of assignments,

it is possible that den(A

)(g) 6= den



n+1

(v)

≡

n+1

}



(g).

Local Scope of Recursion Operator. We explain

the role of the local scope of the recursion operator

designated by the constant where.

Let us consider again the terms in Example 3.1.

The corresponding parts, which are subject of the η-

rule, λy

(some



d(y

)



(h(y

))), of the term (8b)–(8e),

and λy

(some





(h(y

))), of (9a)–(9b), depend on

the values of d and d

, respectively. The denota-

tions of these terms are equal only within the local

scopes of where, with the variable assignment up-

dated, i.e., constrained by the corresponding assign-

ments d

= λy

dog and d

= dog. Without these con-

straints, it is possible that, for some g ∈ G, we have

the denotational inequality in (21a)–(21b):

den



λy

(some



d(y

)



(h(y

)))



(g) (21a)

6= den



λy

(some





(h(y

)))



(g) (21b)

because g(d) ∈ T

(ee→(ee→

t))

and g(d

) ∈ T

(ee→

can be

any objects in these domains. For example,



g(d)



(k)

can depend on a, and it is possible that there is some

k ∈ T

, such that



g(d)



(k) 6= g(d

) so that, the fol-

lowing holds:

den



λy

(some



d(y

)



(h(y

)))



(k) (22a)

6= den



λy

(some





(h(y

)))



(k) (22b)

Then, from (22a) by the denotation function, den, we

have (23a).

I (some)





g(d)



(k)





g(h)



(k)



(23a)

6= I (some)



g(d

)





g(h)



(k)



(23b)

For all variable assignments g in a given semantic

structure A, we get den(A)(g) = den(A

)(g), because

the recursion variables

−→

p , p

n+1

−→

, p

n+1

−→

q ,

−→

, are

bound, i.e., constrained to the values determined by

the assignments in the scope of where. The term A

carries the binding constraints together with the scope

of where.

Theorem 4 is not just about denotational seman-

tics of terms related by the η-rule. Many different

terms (as in many formal languages) have the same

denotations. Some manipulations and operators over

terms, do not preserve denotations, i.e., they may pro-

duce new terms that have different denotations from

the original terms over which they operate. While the

η-rule does not preserve the original, strict algorith-

mic steps, i.e., the referential intension, Theorem 4

proves that the η-rule, applied on a canonical form

cf(A), preserves its denotational semantics, by mini-

mal divergence from the algorithmic steps determined

by the original canonical form cf(A) on which it is

applied. This additional reduction is useful for var-

ious tasks, e.g., in translations between NL expres-

sions and generation of NL from semantic representa-

tions. Theorem 4 extended to the ≈

equivalence by

Corollary1 shows that the ≈

equivalence is one of

the many equivalence relations between terms, which

is stronger than denotational equality and weaker than

referential synonymy.

5 EXISTING AND POTENTIAL

APPLICATIONS

In this section, we point to existing and potential ap-

plications of the Type-Theory of Acyclic Recursion

, in areas that are within the subareas of Artiﬁ-

cial Intelligence (AI). The reduction calculs of L

ex-

tended to η-reduction is useful in such applications,

for reducing algorithmic complexity.

• programming languages: for algorithmic (proce-

dural) semantics of programming languages

• compiler programming languages: for automatic

conversion of recursive programs into iterative

programs, where the reduction calculus is build

into parts of compilation processing

• algorithm speciﬁcations with higher-order type

theory of algorithms

• data science / database

• Computational Semantics (Loukanova, 2016)

• Syntax-Semantics Interface in NLP, computa-

tional grammar, and lexicon of human language

(Loukanova, 2019d; Loukanova, 2019a)

• Language Processing / Technology

• Computational Neuroscience (Loukanova, 2017)

6 OUTLOOK AND FUTURE

WORK

We have formulated the η-rule to simplify the canon-

ical forms in some cases with occurrences of vacuous

λ-abstractions and corresponding functional applica-

tions, by preserving all other structural components

Algorithmic Eta-reduction in Type-theory of Acyclic Recursion

1009

of the canonical terms. The proof of Theorem 4 in-

volves details of general canonical forms and induc-

tion steps. It demonstrates what part of a canonical

form is reduced, by inessential divergence from the

strictest referential synonymy of the entire term. It

demonstrates that the replacements preserve the de-

notation of the entire term.

The presented η-rule has applications to computa-

tional semantics of human languages and to semantics

of programs and compilers. Replacements based on

the η-rule are not always possible because there are

intervening (algorithmic) structures. The effect of a

general β-replacement would have been like reversing

iteration to recursive program interpreter. Something

like a compiler that translates a program with recur-

sion (i.e. a L

term) to a program with induction,

and then ﬁnds functional components that compute

constant functions (not depending on inputs), and in

attempts to provide the constant value, without using

the “vacuous” function applications, is reversing the

tail recursion into recursion. The presented η-rule

avoids this.

In the analyses of certain classes of human lan-

guage expressions, at least those that we have covered

in recent work, the η-rule provides simpliﬁcation of

canonical terms that are otherwise irreducible.

REFERENCES

Gallin, D. (1975). Intensional and Higher-Order Modal

Logic: With Applications to Montague Semantics.

North-Holland Publishing Company, Amsterdam and

Oxford, and American Elsevier Publishing Company.

Loukanova, R. (2011a). From Montague’s Rules of Quan-

tiﬁcation to Minimal Recursion Semantics and the

Language of Acyclic Recursion. In Bel-Enguix, G.,

Dahl, V., and Jim

enez-L

opez, M. D., editors, Biology,

Computation and Linguistics, volume 228 of Fron-

tiers in Artiﬁcial Intelligence and Applications, pages

200–214. IOS Press, Amsterdam; Berlin; Tokyo;

Washington, DC.

Loukanova, R. (2011b). Modeling Context Information

for Computational Semantics with the Language of

Acyclic Recursion. In P

erez, J. B., Corchado, J. M.,

Moreno, M., Juli

an, V., Mathieu, P., Canada-Bago,

J., Ortega, A., and Fern

andez-Caballero, A., editors,

Highlights in Practical Applications of Agents and

Multiagent Systems, volume 89 of Advances in Intel-

ligent and Soft Computing, pages 265–274. Springer

Berlin Heidelberg.

Loukanova, R. (2011c). Reference, Co-reference and

Antecedent-anaphora in the Type Theory of Acyclic

Recursion. In Bel-Enguix, G. and Jim

enez-L

opez,

M. D., editors, Bio-Inspired Models for Natural and

Formal Languages, pages 81–102. Cambridge Schol-

ars Publishing.

Loukanova, R. (2011d). Semantics with the Language of

Acyclic Recursion in Constraint-Based Grammar. In

Bel-Enguix, G. and Jim

enez-L

opez, M. D., editors,

Bio-Inspired Models for Natural and Formal Lan-

guages, pages 103–134. Cambridge Scholars Publish-

ing.

Loukanova, R. (2016). Relationships between Speci-

ﬁed and Underspeciﬁed Quantiﬁcation by the Theory

of Acyclic Recursion. ADCAIJ: Advances in Dis-

tributed Computing and Artiﬁcial Intelligence Jour-

nal, 5(4):19–42.

Loukanova, R. (2017). Binding Operators in Type-Theory

of Algorithms for Algorithmic Binding of Functional

Neuro-Receptors. In Ganzha, M., Maciaszek, L., and

Paprzycki, M., editors, Proceedings of the 2017 Fed-

erated Conference on Computer Science and Informa-

tion Systems, volume 11 of Annals of Computer Sci-

ence and Information Systems, pages 57–66. Polish

Information Processing Society.

Loukanova, R. (2019a). Computational Syntax-Semantics

Interface with Type-Theory of Acyclic Recursion for

Underspeciﬁed Semantics. In Osswald, R., Retor

C., and Sutton, P., editors, IWCS 2019 Workshop on

Computing Semantics with Types, Frames and Re-

lated Structures. Proceedings of the Workshop, pages

37–48. The Association for Computational Linguis-

tics (ACL).

Loukanova, R. (2019b). Gamma-Reduction in Type The-

ory of Acyclic Recursion. Fundamenta Informaticae,

170(4):367–411.

Loukanova, R. (2019c). Gamma-star canonical forms in the

type-theory of acyclic algorithms. In van den Herik, J.

and Rocha, A. P., editors, Agents and Artiﬁcial Intel-

ligence, pages 383–407, Cham. Springer International

Publishing.

Loukanova, R. (2019d). Syntax-semantics interfaces of

modiﬁers. In Rodr

ıguez, S., Prieto, J., Faria, P.,

Kłos, S., Fern

andez, A., Mazuelas, S., Jim

enez-

opez, M. D., Moreno, M. N., and Navarro, E. M.,

editors, Distributed Computing and Artiﬁcial Intelli-

gence, Special Sessions, 15th International Confer-

ence, pages 231–239, Cham. Springer International

Publishing.

Moschovakis, Y. N. (1989). The formal language of recur-

sion. Journal of Symbolic Logic, 54(04):1216–1252.

Moschovakis, Y. N. (1993). Sense and denotation as algo-

rithm and value. In Oikkonen, J. and V

anen, J.,

editors, Logic Colloquium ’90: ASL Summer Meet-

ing in Helsinki, volume Volume 2 of Lecture Notes in

Logic, pages 210–249. Springer-Verlag, Berlin.

Moschovakis, Y. N. (1997). The logic of functional recur-

sion. In Dalla Chiara, M. L., Doets, K., Mundici,

D., and van Benthem, J., editors, Logic and Scientiﬁc

Methods, volume 259, pages 179–207. Springer, Dor-

drecht.

Moschovakis, Y. N. (2006). A Logical Calculus of Mean-

ing and Synonymy. Linguistics and Philosophy,

29(1):27–89.

Moschovakis, Y. N. (2019). Abstract Recursion and Intrin-

sic Complexity, volume 45 of Lecture Notes in Logic.

Cambridge University Press.

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