Gamma-star Reduction in the Type-theory of Acyclic Algorithms

Roussanka Loukanova

Department of Philosophy, Stockholm University, Stockholm, Sweden

Keywords:

Mathematics Of Algorithms, Recursion, Types, Semantics, Algorithmic Semantics, Denotation, Canonical

Computations.

Abstract:

The paper extends a higher-order type theory of acyclic algorithms by adding a reduction rule, which results

in a stronger reduction calculus. The new reduction calculus determines a strong algorithmic equivalence

between formal terms. It is very useful for simplifying terms, by eliminating sub-terms having superﬂuous

lambda abstraction and corresponding spurious functional applications.

1 INTRODUCTION

In a sequence of papers, see (Moschovakis, 1989;

Moschovakis, 1994; Moschovakis, 1997), Yiannis

Moschovakis introduced a new approach to the math-

ematical notion of algorithm, by the concept of recur-

sion and mutually recursive computations. The ap-

proach, which we call Theory of Moschovakis Recur-

sion, uses a formal language of recursion and ﬁne-

grained semantic distinctions between denotations of

formal terms and algorithms for computing the de-

notations, correspondingly by two semantic layers:

denotational and algorithmic semantics. The initial

work on Moschovakis Recursion was on untyped the-

ory of algorithms and is the basis of (Moschovakis,

2006), which introduced typed theory L

of acyclic

recursion, as formalization of the concepts of algo-

rithmic meaning in typed models. The theory L

was

introduced in (Moschovakis, 2006) by considering its

potentials for applications to algorithmic semantics

of human language, analogously to semantics of pro-

gramming languages, where a given denotation can

be computed by different algorithms.

Our ongoing work on extending the expressive-

ness of L

develops a class of formal languages and

theories of typed acyclic recursion, which cover var-

ious computational aspects of the mathematical con-

cept of algorithm. We target development of formal

systems and calculi, which have applications in con-

temporary intelligent systems. In particular, we de-

velop a class of type theories of algorithms, which

have more adequate computational applications in AI,

by covering context dependent algorithms that de-

pend on AI agents and other contextual parameters.

For instance, such work was initiated in (Loukanova,

2011d; Loukanova, 2013a; Loukanova, 2016c).

Collectively, the classes of typed formal languages

and theories of Moschovakis acyclic, and, respec-

tively, full recursion are formal systems, which we

call typed theory of acyclic recursion (TTofAR), and,

respectfully, typed theory of (full) recursion (TTofR).

TTofAR has potentials for applications to algo-

rithmic semantics of formal and natural languages.

Among the formal languages, we consider applica-

tions of typed theory of recursion to semantics of

programming languages, formalisation of compilers,

languages used in database systems, and many ar-

eas of Artiﬁcial Intelligence (AI), including robotics.

The untyped theory of recursion (Moschovakis, 1997;

Moschovakis, 1989) is applied in (Hurkens et al.,

1998) to model reasoning. The potentials of L

, with

typed acyclic recursion, have been demonstrated for

various applications, in particular, to computational

semantics of human language. Application of L

logic programming in linguistics and cognitive sci-

ence is given in (Hamm and van Lambalgen, 2004).

A sequence of papers initiated extending the orig-

inal L

, and provide applications of L

to computa-

tional semantics and computational syntax-semantics

interface of human language, see (Loukanova, 2013c;

Loukanova, 2013b; Loukanova, 2013a; Loukanova,

2012a; Loukanova, 2012b; Loukanova and Jim´enez-

L´opez, 2012; Loukanova, 2011a; Loukanova, 2011c;

Loukanova, 2011d; Loukanova, 2011e; Loukanova,

2011f; Loukanova, 2011g; Loukanova, 2011b),

(Loukanova, 2016b; Loukanova, 2016a; Loukanova,

2016c; Loukanova, 2015b). By adding polymor-

phism, the work in (Loukanova, 2016a) offers po-

tentials for varieties of applications with polymor-

phic, or otherwise parametric types. The work in

(Loukanova, 2014; Loukanova, 2015b; Loukanova,

2015a; Loukanova, 2017b) provides a formal tech-

Loukanova, R.

Gamma-star Reduction in the Type-theory of Acyclic Algorithms.

DOI: 10.5220/0006662802310242

In Proceedings of the 10th International Conference on Agents and Artiﬁcial Intelligence (ICAART 2018) - Volume 2, pages 231-242

ISBN: 978-989-758-275-2

231

nique for applications of L

and its extended ver-

sions to data science, for representation of factual

and situated content of underspeciﬁed and partial in-

formation. The work (Loukanova, 2017a) extends

and provides a basis for applications to compu-

tational neuroscience for mathematical medeling of

neural structures and connections.

In this paper, at ﬁrst, we present the original reduc-

tion calculus of the type theory of acyclic recursion,

which effectively reduces terms to their canonical

forms. This reduction calculus determines a strict al-

gorithmic equivalence between L

-terms. In the rest

of the paper, we present our contribution. We extend

the reduction calculus of L

to a stronger γ

∗

-reduction

calculus. We call, and denote it also gamma-star

reduction calculus. The new γ

∗

-reduction system is

very useful for simplifying terms, by eliminating sub-

terms having superﬂuous lambda abstraction and cor-

responding spurious functional applications. We give

motivation by using abstract examples, because the

theory has broad applications in technologies. For

better understanding, we give supplementary exam-

ples that render expressions of human language to

-terms. In addition, the theory has direct potentials

for applications to computerised processing of human

language, including large-scale, computational gram-

mars of human language and NLP for AI.

2 SYNTAX AND SEMANTICS OF

TYPED-THEORY OF ACYCLIC

RECURSION

2.1 Syntax

Basic Types: BTypes = { e, t, s }

The basic type e is the type of the entities in the se-

mantic domains and the expressions denoting entities;

t of the truth values and corresponding expressions, s

of the states.

Types: the set Types (i.e., the set of type terms)

is the smallest set deﬁned recursively, by using the

Backus-Naur Form (BNF) notation as follows:

≡ e | t | s | σ | (τ → σ) (1)

The type terms (τ → σ) are the types of functions

from objects of type τ to objects of type σ, and of

expressions denoting such functions.

e → t, the type of characteristic functions

of sets of entities

(2a)

e ≡ (s → e), of state dependent entities

(2b)

t ≡ (s → t), of state dependent truths

(2c)

τ ≡ (s → τ), of state dependent objects

of type τ

(2d)

The vocabulary of L

consists of

Constants:

K =

τ∈Types

where, for each τ ∈ Types, K

= {c

, .. . ,c

, .. . }

Pure variables:

PureVars =

τ∈Types

PureVars

where, for τ ∈ Types, PureVars

= {v

, v

, .. .}

Recursion variables:

RecVars =

τ∈Types

RecVars

where, for each τ ∈ Types, RecVars

= {r

, r

, .. .}

The terms of L

: The set Terms of L

is deﬁned by

recursion expressed by using a typed variant of BNF,

with the type assignments given either as superscripts

or with column sign:

≡ c

| x

| (3a)



(σ→τ)

)



| (3b)



λv

)



(σ→τ)

| (3c)



where { p

= A

, .. . ,

= A

}



(3d)

where for n, m ≥ 0, c

∈ K

is a constant; x

∈

PureVars

∪RecVars

is a pure or recursion variable;

∈ PureVars

is a pure variable; A, B, A

∈ Terms

(i = 0, . . . , n) are terms of the respective types; p

∈

RecVars

(i = 1, . . . , n) are pairwise different recur-

sion variables; and, in the expressions of the form

(3d), the subexpression {p

= A

, . . . , p

= A

}

is a sequence of assignments that satisﬁes the acyclic-

ity constraint:

Acyclicity Constraint AC 1. For any given terms

: σ

, .. ., A

: σ

, and recursion variables p

: σ

..., p

: σ

, the sequence {p

= A

, . . . , p

= A

} is

an acyclic system of assignments iff there is a ranking

function rank : {p

, . . . , p

} −→ N such that, for all

, p

∈ {p

, . . . , p

if p

occurs freely in A

then rank(p

) < rank(p

)

(4)

Usually, the type assignments in the term expres-

sions are skipped. The terms of the form (3d), are

called recursion terms:

where {p

= A

, . . . , p

= A

}]

(5a)

≡ (A

where { p

= A

, . . . , p

= A

}) (5b)

≡ A

where { p

= A

, . . . , p

= A

} (5c)

ICAART 2018 - 10th International Conference on Agents and Artiﬁcial Intelligence

232

2.2 Dynamic Speciﬁcations in Context

by Agents

We shall render the pronoun “his” into a simple recur-

sion variable h for the purpose of the demonstration.

More adequate treatment of the pronouns is not in the

subject of this paper. Here we consider general un-

derspeciﬁcation of terms via free recursion variables

occurring in L

-terms. Using recursion variables al-

lows the recursion terms that contain them, like (6c),

to be expended by adding assignments h

= H, for

some appropriate term H. What is the term H de-

pends on context, and can be speciﬁed dynamically,

when more detailed information is provided, e.g., by

discourse or explicitly by users. Here we do not treat

details of such contextual contributions, by focusing

on the possibility to represent underspeciﬁcation by

“underspeciﬁed” L

-terms containing free recursion

variables. Using free recursion variables in terms

gives potentials for its algorithmic resolving of un-

derspeciﬁcation by adding assignments to canonical

terms.

The reductions in this section can be done by us-

ing the reduction rules of L

, which are given in Sec-

tion (3).

Example 2.1 (Underspeciﬁed pronouns).

John greeted his wife. (6a)

render

−−−→ greeted(wife-of(h))(john) (6b)

⇒

greeted(w)( j) where { j

= john,

= wife-of(h)}

(6c)

The recursion variable h in (6c) is left free, which we

use to represents semantic underspeciﬁcation of the

sentence (6a), in absence of context with an agent that

lacks information to interpret the pronoun. If left un-

bound by any assignment, h can be interpreted as a

deictic pronounobtaining its denotational referents by

the agents’ references provided by context informa-

tion, e.g., by adding h

= tom to the assignments in the

term (7c)–(7e). The result is the dynamic speciﬁca-

tion of the underspeciﬁed term (7b), i.e., (6c), and its

canonical form (7c)–(7e), to the sully speciﬁed term

(7f)–(7i).

John greeted his wife. (7a)

render

−−−→ greeted(wife-of(h))(john) (7b)

⇒

greeted(w)( j) where { (7c)

= john, (7d)

= wife-of(h)} (7e)

⇒

context

greeted(w)( j) where { (7f)

= john, (7g)

= wife-of(h), (7h)

= tom} (7i)

An alternative, anaphoric reading of (6a) can be ob-

tained by adding the assignment h

= j to the term

(6b), and thus to the system of assignments in its

canonical form (6c), e.g., dynamically, after the agent

has obtained relevant information. The result is the

speciﬁed term (8c)–(8f):

John greeted his (own) wife. (8a)

render

−−−→ greeted(wife-of(h))(john) (8b)

⇒

context

greeted(w)( j) where { (8c)

= john, (8d)

= wife-of(h), (8e)

= j} (8f)

2.3 Semantics

Denotational Semantics of L

: An L

semantic

structure, also called model, is a tuple A = hT, I i, sat-

isfying the following conditions (S1)–(S4):

(S1) T is a set, called a frame, of sets

T = {T

| σ ∈ Types} (9)

where T

6= ∅ is a nonempty set of entities, T

{0, 1, er} ⊆ T

is the set of the truth values, T

6= ∅

is a nonempty set of objects called states

(S2) T

(τ

→τ

)

= { p | p: T

−→ T

}

(S3) I is a function I : K −→ T, called the inter-

pretation function of A, such that for every c ∈ K

I (c) = c for some c ∈ T

(S4) The set G of the variable assignments for the

semantic structure A is:

G = { g | g: PureVars ∪ RecVars −→ T and g(x) ∈

, for every x: σ}

Deﬁnition 1 (Denotation Function). The denotation

function den, when it exists, of the semantics structure

A, is a function:

den : Terms −→ { f | f : G −→ T } (10)

which is deﬁned, for each g ∈ G, by induction on the

structure of the terms, as follows:

(D1) den(x)(g) = g(x); den(c)(g) = I (c)

(D2) for application terms

den([B

(σ→τ)

)]

)(g) = den(B)(g)(den(C)(g))

(11)

(D3) for λ-terms

den([λv

)



(σ→τ)

)(g) : T

−→ T

where x : τ and B : σ, is the function such that, for

every t ∈ T

[den([λv

)



(σ→τ)

)(g)]





(12)

= den(B)(g{x

= t}) (13)

Gamma-star Reduction in the Type-theory of Acyclic Algorithms

233

(D4) for recursion terms

den([A

where { p

= A

, . . . ,

= A

}]

)

(14a)

= den(A

)(g{p

, . . . , p

}) (14b)

where for all i ∈ {1, . . . , n},

∈ T

are deﬁned by

recursion on rank(p

), so that:

= den(A

)(g{p

, . . . , p

}) (15)

where p

, . . . , p

are all the recursion variables p

∈

, . . . , p

} such that rank(p

) < rank(p

)

Intuitively, a system {p

= A

, . . . , p

= A

} de-

ﬁnes recursive computations of the values to be as-

signed to the locations p

, . . . , p

. When p

occurs

freely in A

, the denotational value of A

, which is as-

signed to p

, may depend on the values of the variable

, as well as on the values of the variables p

hav-

ing lower rank than p

. Requiring a ranking function

rank, such that rank(p

) < rank(p

), i.e., an acyclic

system guarantees that computations end after ﬁnite

number of steps. Omitting the acyclicity condition

gives an extended type system L

, which admits full

recursion. This is not in the subject of this paper.

Algorithmic Semantics: The notion of algorithmic

meaning (algorithmic semantics) in the languages of

recursion covers the most essential, computational as-

pect of the concept of meaning. The algorithmic

meaning, Int(A), of a meaningful term A is the tuple

of functions, a recursor, that is deﬁned by the denota-

tions 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 assignments of its canonical form (see the next sec-

tions) cf(A) ≡ A

where {p

= A

, . . . , p

= A

}. In-

tuitively, for each meaningful term A, the algorithmic

meaning Int(A) of A, is the mathematical algorithm

for computing the denotation den(A).

Two meaningful expressions A and B are algorith-

mically equivalent, A ≈ B i.e., algorithmically syn-

onymous iff their recursors Int(A) and Int(B) are nat-

urally isomorphic, i.e., they are the same algorithms.

Thus, the formal languages of recursion offer a for-

malisation of central computational aspects: denota-

tion, with at least two semantic “levels”: algorithmic

meanings and denotations. The terms in canonical

form represent the algorithmic steps for computing

semantic denotations.

3 REDUCTION CALCULUS

Deﬁnition 2 (Congruence Relation). For any terms

A, B ∈ Te rms, A and B are congruent, A ≡

B, if and

only if one of them can be obtained from the other by

renaming bound variables and reordering assignments

in recursion terms.

3.1 Reduction Rules

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

′

, then

A(B) ⇒ A

′

)

(c-ap)

If A ⇒ B, then

λ(u)(A) ⇒ λ(u)(B)

(c-λ)

If A

⇒ B

, for i = 0, ..., n, then

where { p

= A

, . . . , p

= A

}

⇒ B

where { p

= B

, . . . , p

= B

}

(c-rec)

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

Beki

c-Scott rule: (B-S)

where { p



where {

−→

B }



−→

A }

⇒ A

where { p

= B

−→

B ,

−→

A }

given that no q

occurs free in any A

, for i = 1,

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

Recursion-application rule: (recap)

where {

−→

A }



(B)

⇒ A

(B) where {

−→

A }

given that no p

occurs free in B for i = 1, ..., n

Application rule: (ap)

A(B) ⇒ A(p) where {p

= B}

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

tion

λ-rule: (λ)

λ(u)(A

where { p

= A

, . . . , p

= A

})

⇒ λ(u)A

′

where { p

′

= λ(u)A

′

, . . . ,

′

= λ(u)A

′

}

ICAART 2018 - 10th International Conference on Agents and Artiﬁcial Intelligence

234

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)}

for all i ∈ {1, . . . , n}

(20)

Deﬁnition 3. The reduction relation is the smallest

relation between terms that is closed under the reduc-

tion rules.

The reduction relation is denoted by ⇒. That is,

for any two terms A and B, A reduces to B, denoted by

A ⇒ B, iff B can be obtained from A by ﬁnite number

of applications of reduction rules.

Deﬁnition 4 (Term Irreducibility). We say that a term

A ∈ Terms is irreducible if and only if

for all B ∈ Te rms, if A ⇒ B, then A ≡

B (21)

The following theorems are major results that are

essential for algorithmic semantics.

Theorem 1 (Canonical Form Theorem: existence and

uniqueness of the canonical forms). (Moschovakis,

2006) For each term A, there is a unique, up to con-

gruence, 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 (Referential Synonymy Theorem). (See

(Moschovakis, 2006)) Two terms A, B are algorithmi-

cally equivalent, i.e., synonymous, A ≈ B, if and only

if there are explicit, irreducible terms of correspond-

ing types, A

: σ

, ..., A

: σ

, B

: σ

, ..., B

: σ

(n ≥ 0), such that:

⇒

where { p

= A

, . . . , (22a)

= A

} (22b)

⇒

where { p

= B

, . . . , (22c)

= B

} (22d)

and for all i = 0, ..., n,

den(A

)(g) = de n(B

)(g), for all g ∈ G (23)

4 ALGORITHMIC PATTERNS

AND λ-ABSTRACTIONS

In this section we demonstrate the technique of under-

speciﬁed, parametric algorithm, i.e., algorithmic pat-

terns that represent classes of speciﬁed algorithm in

reduction steps . We use the technique with some ex-

amples to motivate the γ

∗

-reduction introduced in the

second part of the paper.

A Parametric Algorithm: Now, we can use a more

general term of an algorithmic pattern, as paramet-

ric algorithm. For any proper terms W,J, G

, G

that

to not contain free occurrences of the pure variables

, x

, e.g., constants, the following reductions can

be done by using the reduction rules of L

, which are

given in Section (3).

≡ q(W(h))(J) where { (24a)

= (G

) + G

))} (24b)

⇒

q(w)( j) where {

= (q

+ q

= J,

= W(h),

= G

), q

= G

)}

(24c)

The term P

in (24a)– (24c) can be preceded by a

sequence of λ-abstractions, as in the term P

in (25a)–

(25b). By using the reduction rules given in Sec-

tion (3), P

can be reduced to the term (25d)–(25i).

≡ λ(x

)λ(x

)



q(W(h))(J) where { (25a)

= (G

) + G

))}



(25b)

⇒ λ(x

)λ(x

)



q(w)( j) where {

= (q

+ q

= J,

= W(h),

= G

), q

= G

)}



(25c)

⇒ λ(x

)λ(x

)



′

)(x

)



′

)(x

)



( j

′

)(x

))

where {

(25d)

′

= λ(x

)λ(x

)

′

)(x

′

)(x

))

(25e)

′

= λ(x

)λ(x

)

, (25f)

′

= λ(x

)λ(x

)

W(h)

, (25g)

′

= λ(x

)λ(x

)

(25h)

Gamma-star Reduction in the Type-theory of Acyclic Algorithms

235

′

= λ(x

)λ(x

)

}

(25i)

The term (25d)–(25i) has vacuous λ-abstractions,

e.g., (25f), (25g), (25h), (25i), which denote con-

stant functions, and corresponding applications, e.g.,

in (25d), that give the same values.

The γ

∗

-reduction, introduced in this paper,reduces

such spurious sub-terms.

5 GAMMA-STAR REDUCTION

5.1 The γ

∗

-Rule

In the following sections, we give the deﬁnition of

the γ

∗

-rule, see Table 1, and its major properties. Ex-

panding the reduction calculus of L

with the γ

∗

-rule

simpliﬁes some terms, by reducing sub-terms with

vacuous λ-abstractions, while maintaining closely the

original algorithmic structure. By using the γ

∗

-rule,

the canonical forms determine more efﬁcient versions

of algorithms, by maintaining the essence of the com-

putational steps.

Deﬁnition 5 (Strong γ

∗

-condition). A recursion term

A ∈ Terms satisﬁes the strong γ

∗

-condition for an

assignment p

= λ(

−→

)λ(v

: (

−→

ϑ → (ϑ → τ)),

with respect to λ(v), if and only if A is of the form:

(26a)–(26c):

A ≡ A

where {

−→

A , (26a)

= λ(

−→

u )λ(v)P, (26b)

−→

B } (26c)

with the sub-terms of correspondingly appropriate

types, and which is such that the following holds:

1. The term P ∈ Terms

does not have any (free) oc-

currences of v in it, i.e., v 6∈ FreeV(P)

2. All the occurrences of p in A

−→

A , and

−→

B are oc-

currences in sub-terms p(

−→

u )(v), modulo renam-

ing the variables

−→

u , v

In such a case, we also say that the assignment

= λ(

−→

u )λ(v)P satisﬁes the γ

∗

-condition in the re-

cursion term A in (26a)–(26c).

6 THE γ

∗

-REDUCTION

Adding the ⇒

∗

to the reduction rules of L

deter-

mines an extended reduction relation between terms

as follows.

Table 1: The γ

∗

-rule

(γ

∗

)

A ≡ A

where {

−→

A , (27a)

= λ(

−→

u )λ(v)P, (27b)

−→

B } (27c)

⇒

∗

′

where {

−→

′

, (27d)

′

= λ(

−→

u )P

′

, (27e)

−→

′

} (27f)

where

• the term A ∈ Terms satisﬁes the (strong) γ

∗

condition (in Deﬁnition 5) for p

= λ(

−→

u )λ(v)P

• p

′

∈ RecVars

(

−→

ϑ →τ)

is a fresh recursion variable

•

−→

′

≡

−→

X {p(

−→

u )(v)

≡ p

′

(

−→

u )} is the result of the

replacements X

{p(

−→

u )(v)

≡ p

′

(

−→

u )}, i.e., of all

occurrences of p(

−→

u )(v) by p

′

(

−→

u ), in all parts

in (27d)–(27f), modulo renaming the variables

−→

u , v

Deﬁnition 6 (γ

∗

-reduction). The γ

∗

-reduction relation

is the smallest relation, ⇒

∗

⊆ Terms×Terms (also

denoted by ⇒

∗

), between terms that is closed under

the L

-reduction rules, given in Section 3.1, and the

∗

-rule, given in Table 1.

We refer to the set of all L

reduction rules ex-

tended with the γ

∗

-rule as the set of γ

∗

-reduction rules.

In addition to the notations ⇒

∗

, and ⇒

∗

, for the

∗

-reduction, we also use the usual notation for re-

ﬂexive and transitive closure of a relation, given in

(28a). To specify that the γ

∗

-rule has been applied

certain number of times (including zero times), pos-

sibly intervened by applications of some of the other

reduction rules, we use the notation (28b)–(28c).

A ⇒

∗

B ⇐⇒ A ⇒

∗

by n applications of reduction rules,

possibly γ

∗

(n ≥ 0)

(28a)

A ⇒

∗

B ⇐⇒ A ⇒

∗

[n]

B, for n ≥ 0

by using ⇒-rules and

n applications of the γ

∗

-rule

(28b)

A ⇒

∗

B ⇐⇒ A ⇒

∗

[n]

B for n ≥ 1

by using ⇒-rules and

n applications of the γ

∗

-rule

(28c)

Deﬁnition 7 (γ

∗

-irreducible terms). We say that a

ICAART 2018 - 10th International Conference on Agents and Artiﬁcial Intelligence

236

term A ∈ Te rms is γ

∗

-irreducible if and only if

for all B ∈ Terms, A ⇒

∗

B =⇒ A ≡

B (29)

Deﬁnition 8 (γ

∗

-irreducible recursion terms for a spe-

ciﬁc assignment c

= λ(

−→

u )λ(v)C). We say that a re-

cursion term

A ≡ A

where {

−→

A , c

= λ(

−→

u )λ(v)C,

−→

B }

is γ

∗

-irreducible for the assignment c

= λ(

−→

u )λ(v)C,

with respect to λ(v), if and only if the conditions for

the γ

∗

-rule are not satisﬁed for it, i.e., either

(1) v ∈ FreeV(C), or

(2) v 6∈ FreeV(C), and not all of the occurrences of c

in A

−→

A , and

−→

B are sub-occurrences in a term

−→

u )(y), modulo congruence by renaming the

variables

−→

u , y ∈ PureVars.

Theorem 3 (Criteria for γ

∗

-irreducibility). By struc-

tural induction:

1. If A ∈ Const ∪Vars, then A is γ

∗

-irreducible.

2. An application term A(B) is γ

∗

-irreducible if and

only if A is explicit and irreducible and B is imme-

diate.

3. A λ-term λ(x)A is γ

∗

-irreducible if and only if A is

explicit and irreducible.

4. A recursion term A

A ≡ [A

where { p

= A

, . . . , p

= A

}] (n ≥ 0)

is γ

∗

-irreducible if and only if

(a) all of the parts A

, .. ., A

are explicit and

irreducible, and

(b) A does not satisfy the γ

∗

-condition

Proof. By structural induction on terms and inspec-

tion of the γ

∗

-reduction rules.

7 CANONICAL FORMS AND

∗

-REDUCTION

Theorem 4 (Extended γ

∗

-Canonical Form Theorem).

For every A ∈ Terms, the following holds:

1. (Existence of a γ

∗

-canonical form of A) There ex-

ist explicit, irreducible A

, . . . , A

∈ Terms (n ≥ 0)

such that the term A

where { p

= A

, . . . , p

} is γ

∗

-irreducible, i.e., irreducible and does

not satisfy the γ-condition, and

(A) ≡ A

where { p

= A

, . . . ,

= A

(30)

Thus, cf

(A) is γ

∗

-irreducible.

2. A constant c ∈ K or a recursion variable p ∈

RecVars occurs freely in cf

(A) if and only if it

occurs freely in A.

3. A ⇒

∗

(A)

4. If A is γ

∗

-irreducible, then cf

(A) ≡

5. If A ⇒

∗

B, then cf

(A) ≡

(B)

6. (Uniqueness of cf

(A) up to congruence)if A ⇒

∗

B and B is γ

∗

-irreducible, then B ≡

(A),

i.e., cf

(A) is unique, up to congruence, γ

∗

irreducible term. We write

A ⇒

gscf

B ⇐⇒ B ≡

(A) (31a)

A ⇒

gscf

(A) (31b)

Proof. The statement (1) is proved by induction on

term structure, using the deﬁnition of the cf

(A). The

statements (2) and (3) are provedby induction on term

structure, using the criteria for γ

∗

-irreducibility 3. (4)

is proved by induction on the deﬁnition of the γ

∗

reduction relation. (5) follows from (3) and (4).

Deﬁnition 9 (γ

∗

-equivalence (γ

∗

-synonymy) relation

≈

∗

). For any A,B ∈ Terms:

A ≈

∗

B ⇐⇒ cf

(A) ≈ cf

(B)

(32)

When A ≈

∗

B, we say that A and B are γ

∗

-equivalent,

alternatively, γ

∗

-synonymous.

Note 1. If we have added an additional restriction in

the γ

∗

-condition of the γ

∗

-rule that all the occurrences

of the sub-terms p(

−→

u )(v) have to be in the scope

of λ(v) (modulo renaming congruence), the ⇒

∗

-rule

would have preserved all the free variables of A in

(A), including the pure variables, not only the re-

cursion variables, so that FreeV(cf

(A)) = FreeV(A)

(see the γ

∗

-Canonical Form Theorem 4). In this

strong γ

∗

-reduction, we refrain from adding such an

extra restriction. Note also that the replacements

{p(

−→

u )(v)

≡ p

′

(

−→

u )}, B

{p(

−→

u )(v)

≡ p

′

(

−→

u )} in

the γ

∗

-rule (Table 1) are not necessarily “free”, in the

inverse sense that the ⇒

∗

-rule may remove occur-

rences of v which are in the scope of λ(v), in some

parts, due to the clause (2) in the γ

∗

-condition (5).

8 SOME PROPERTIES OF THE

∗

-EQUIVALENCE

Theorem 5 (γ

∗

-Equivalence Theorem). Two terms

A, B are algorithmically γ

∗

-synonymous, A ≈

∗

B, if

and only if there are explicit, irreducible terms of cor-

responding types, A

: σ

, B

: σ

(i = 0, . . . , n), (n ≥ 0),

such that:

A ⇒

gscf

where { p

= A

, . . . , p

= A

}

≡ cf

(A) (i.e., γ

∗

-irreducible)

(33a)

Gamma-star Reduction in the Type-theory of Acyclic Algorithms

237

B ⇒

gscf

where { p

= B

, . . . , p

= B

}

≡ cf

(B) (i.e., γ

∗

-irreducible)

(33b)

and for all i = 0, ..., n,

den(A

)(g) = de n(B

)(g), for all g ∈ G (34)

Proof. The theorem follows from Deﬁnition 9 of γ

∗

equivalence and Theorem 2.

Deﬁnition 10 (Syntactic Synonymy (Equivalence)

≈

). For any A,B ∈ Terms,

A ≈

B ⇐⇒ cf(A) ≡

cf(B) (35)

For more details about syntactic synonymy, see

Moschovakis (Moschovakis, 2006). The difference

between syntactic and algorithmic synonymies is that

syntactic synonymy does not apply to denotationally

equivalent constants and syntactic constructs such as

λ-terms. For instance, assuming that dog and canine

are constants, such that den(dog) = den(canine), it

holds that dog ≈ canine (by the Referential Syn-

onymy Theorem 2), because both terms are in canon-

ical forms, with the same denotations, i.e., they de-

note the same function obtainable by the same al-

gorithm, determined by the interpretation function I

of the semantics structure A = hT, I i. On the other

hand, dog 6≈

canine, since dog 6≡

canine. Also,

den(dog) = den(λ(x)dog(x)) (by the clauses (D1),

(D3) of the Deﬁnition 1 of the denotation function).

Therefore, dog ≈ λ(x)dog(x) (by the Referential Syn-

onymy Theorem 2), because both terms are in canoni-

cal forms. These two terms are syntactically different,

dog 6≈

λ(x)dog(x), because dog 6≡

λ(x)dog(x).

Theorem 6. For any A, B ∈ Terms,

A ⇒ B =⇒ A ≈

B (36a)

=⇒ A ≈ B (36b)

=⇒ A ≈

∗

B =⇒ A |=| B (36c)

Proof. By using the deﬁnitions.

Theorem 7. For any A, B ∈ Terms,

cf(A) ≈

∗

cf(B) ⇐⇒

cf(A) ⇒

∗

′

, cf(B) ⇒

∗

′

, and A

′

≈

∗

′

for some A

′

, B

′

∈ Terms

(37a)

cf(A) ≈

∗

cf(B) ⇐⇒ A ≈

∗

B (37b)

Proof. The directions ⇐= are proved by using Deﬁ-

nition 9, Referential Synonymy Theorem 2, and Ex-

tended γ

∗

-Canonical Form Theorem 4.

Corollary 1. For all A, B,C ∈ Terms,

A ⇒ B ⇒

∗

C =⇒ A ≈ B =⇒ A ≈

∗

B ≈

∗

(38)

while there exist (many) terms A, B,C ∈ Terms such

that

A ⇒ B ⇒

∗

C, C 6≈ B, and C 6≈ A (39)

Proof. (38) follows from Deﬁnition 9, the Canonical

Form Theorems 1, and 4.

By Deﬁnition 9 of γ

∗

-equivalence between two

terms A, B as algorithmic synonymy between their

∗

-canonical forms, various properties of algorithmic

synonymy are inherited by γ

∗

-equivalence, reﬂected,

e.g., by the γ

∗

-EquivalenceTheorem 5 and the compo-

sitionality of γ

∗

-equivalence, with the very restricted

form of β-reduction.

Assume that the (γ∗)-rule, see Table 1, is applied

to a term A in canonical form, i.e., A ≡

cf(A). By ap-

plication of the (γ∗)-rule until we obtain the γ

∗

canon-

ical form cf

(A) of A. The corresponding parts in

the assignments (27b)–(27e) are not denotationally

equivalent, since they are not of the same type. By

the γ

∗

-Equivalence Theorem 5, A ≈ cf(A) 6≈ cf

(A).

The γ

∗

-reduction calculus does not preserve per se

the algorithmic synonymy between terms. That is, in

general, it is possible that A ≈ B, while A 6≈

∗

Nevertheless, the γ

∗

-reduction relation ⇒

∗

be-

tween terms is very useful. For any terms A and B,

a γ

∗

-reduction A ⇒

∗

(A) preserves the most es-

sential algorithmic components of the canonical form

cf(A) in cf

(A). It reduces vacuous λ-abstractions,

which denote constant functions, and corresponding

applications that give the constant values.

9 APPLICATIONS OF THE

∗

-RULE

In this section, we give pattern examples for possi-

ble renderenings of expressions in human language

to L

-terms that can represent their algorithmic se-

mantics. A deﬁnition of a rendering relation be-

tween human language expressions and their seman-

tic representations by L

-terms is not in the subject

of this paper. Rendering can be deﬁned in a computa-

tional mode, via syntax-semantics interfaces, within a

computational grammar, e.g., see (Loukanova, 2011f;

Loukanova, 2017b). Typically, L

offers alternative

terms for representing algorithmic semantics of hu-

man language expressions. The choice would depend

on applications.

Developments of new, hybrid machine learning

techniques and statistical approaches for extraction of

semantic information from text can provide more pos-

sibilities for rendering human language expressionsto

-terms.

ICAART 2018 - 10th International Conference on Agents and Artiﬁcial Intelligence

238

Example 9.1.

Kim hugs some dog

render

−−−→ A (40a)

A ≡

λ(y

)



some



dog





λ(x

)hug(x

)(y

)





(kim)

(40b)

Proposition 1. Given that A is the term in (40b), its

canonical and γ

∗

-canonical forms, cf(A) and cf

(A),

are as in (41) and (42), correspondingly:

cf(A) ≡

λ(y

)



some



′

)



(h(y

))

i

(k) where

= λ(y

)λ(x

)hugs(x

)(y

′

= λ(y

)dog, k

= kim}

(41)

(A) ≡



λ(y

)some(d)



h(y

)



(k) where

= λ(y

)λ(x

)hugs(x

)(y

= dog, k

= kim}

(42)

cf(A) 6≈ cf

(A) (43a)

cf(A) ≈

∗

(A) (43b)

Proof. The following reductions hold for the term A

in (40b).

A ⇒ . . . (44a)

⇒

λ(y

)



some



′

)



h(y

)





where

′

= λ(y

)dog,

= λ(y

)λ(x

)hugs(x

)(y

)}

(kim)

(44b)

⇒

λ(y

)



some



′

)



(h(y

))

i

(k)

where { h

= λ(y

)λ(x

)hugs(x

)(y

′

= λ(y

)dog, k

= kim}

(44c)

⇒

∗



λ(y

)some(d)



h(y

)



(k) where

= λ(y

)λ(x

)hugs(x

)(y

= dog, k

= kim}

(44d)

(43a) follows from Theorem 2 and (43b) from Theo-

rem 5.

The term in (44d), and thus, the term cf

(A) in

(41) too, is in a canonical form, but it is not algorith-

mically equivalent to the term (44c), i.e., to cf(A) in

(42) too, by the original reduction calculus of L

Moschovakis (Moschovakis, 2006). The term (44d)

is simpler than (44c), which has an extraneous, vac-

uous λ-abstraction over y

in the assignment d

′

λ(y

)dog, while the term dog does not have any (free)

occurrences of y

, i.e., the values of λ(y

)dog stored

in d

′

are constant and do not depend on λ(y

). The

terms (44c) and (44d) denote very similar algorithms

that are γ

∗

-equivalent, by applying the γ

∗

-rule. The

term (44d) is in γ

∗

-canonical form.

Example 9.2. Assume that the sentence (45) is ren-

dered to a L

-term B ≡ B

that is given in (46a)–

(46h).

[Jim]

sent Mia the article about

the [discovery of Protein353 by [him]

]

render

−−−→ B

(45)

Alternatively, depending on speciﬁc applications, B

may be a term that is reduced to the term in (46a)–

(46h).

B ⇒ B

≡ (46a)

λ(z)

λ(x)



send(m

)(a

)(z) where (46b)

= the(r

), (46c)

= article-about(b

), (46d)

= the(d

), (46e)

= discovery-of-by(p

)(z), (46f)

= protein353, (46g)

= mia}



(jim) (46h)

The λ(x) abstractions inside the assignments in (47a)–

(47h) are the typical result of the (λ)-rule of the re-

duction calculus of L

, in this case, to the term B

⇒

(λ)

≡ (47a)

λ(z)

λ(x)send(m

(x))(a

(x))(z) whe re (47b)

= λ(x)the(r

(x)), (47c)

= λ(x)article-about(b

(x)), (47d)

= λ(x)the(d

(x)), (47e)

= λ(x)discovery-of-by(p

(x))(z), (47f)

= λ(x)protein353, (47g)

= λ(x)mia}

(jim) (47h)

Another application of the (λ)-rule reduces the term

to B

in (48a)–(48h).

⇒

(λ)

≡ (48a)

λ(z)λ(x)send(m

(z)(x))(a

(z)(x))(z) w here

(48b)

Gamma-star Reduction in the Type-theory of Acyclic Algorithms

239

= λ(z)λ(x)the(r

(z)(x)), (48c)

= λ(z)λ(x)article-about(b

(z)(x)), (48d)

= λ(z)λ(x)the(d

(z)(x)), (48e)

= λ(z)λ(x)

discovery-of-by(p

(z)(x))(z),

(48f)

= λ(z)λ(x)protein353, (48g)

= λ(z)λ(x)mia}

(jim) (48h)

The term B

in (49a)–(49h) is the result of applying

the Recursion-application rule (recap) to B

in (48a)–

(48h).

⇒

(recap)

≡ (49a)

λ(z)λ(x)send(m

(z)(x))(a

(z)(x))(z)

(jim) (49b)

where

= λ(z)λ(x)the(r

(z)(x)), (49c)

= λ(z)λ(x)article-about(b

(z)(x)), (49d)

= λ(z)λ(x)the(d

(z)(x)), (49e)

= λ(z)λ(x)

discovery-of-by(p

(z)(x))(z),

(49f)

= λ(z)λ(x)protein353, (49g)

= λ(z)λ(x)mia} (49h)

The term B

is reduced to the term B

in (50a)–(50h),

by successive applications of the reduction rule (ap)

to the head part in (49b), the Compositionality rule

(c-rec) for recursion terms, the Head rule (head), and

Congruence of the order of the recursion assignments.

⇒ B

≡ (50a)

λ(z)λ(x)send(m

(z)(x))(a

(z)(x))(z)

( j) (50b)

where

= λ(z)λ(x)the(r

(z)(x)), (50c)

= λ(z)λ(x)article-about(b

(z)(x)), (50d)

= λ(z)λ(x)the(d

(z)(x)), (50e)

= λ(z)λ(x)

discovery-of-by(p

(z)(x))(z),

(50f)

= λ(z)λ(x)protein353, (50g)

= λ(z)λ(x)mia, j

= jim} (50h)

The denotations of the terms λ(z)λ(x)protein353 and

λ(z)λ(x)mia, ‘saved’ respectively in p

and m

, by

(50g) and (50h), are constant functions that do not

depend on the argument roles of the abstractions

λ(z)λ(x).

The term B

is reduced to B

, by four successive

applications of the γ

∗

-rule, for the assignments p

λ(z)λ(x)protein353 and m

= λ(z)λ(x)mia.

⇒

∗

≡ (51a)

λ(z)λ(x)send(m)(a

(z)(x))(z)

( j) (51b)

where

= λ(z)λ(x)the(r

(z)(x)), (51c)

= λ(z)λ(x)article-about(b

(z)(x)), (51d)

= λ(z)λ(x)the(d

(z)(x)), (51e)

= λ(z)λ(x)

discovery-of-by(p)(z),

(51f)

= protein353, (51g)

= mia, j

= jim} (51h)

Now, the term B

satisﬁes the γ

∗

-condition for the as-

signment (51f), with respect to λ(x). Application of

the γ

∗

-rule to B

, reduces B

to B

⇒

∗

≡ (52a)

λ(z)λ(x)send(m)(a

(z)(x))(z)

( j) (52b)

where

= λ(z)λ(x)the(r

(z)(x)), (52c)

= λ(z)λ(x)article-about(b

(z)(x)), (52d)

= λ(z)λ(x)the(d(z)), (52e)

= λ(z)discovery-of-by(p)(z),

(52f)

= protein353, (52g)

= mia, j

= jim} (52h)

Now, the term B

satisﬁes the γ

∗

-condition for the as-

signment (52e), with respect to λ(x). Application of

the γ

∗

-rule to B

, reduces B

to B

⇒

∗

≡ (53a)

λ(z)λ(x)send(m)(a

(z)(x))(z)

( j) (53b)

where

= λ(z)λ(x)the(r

(z)(x)), (53c)

= λ(z)λ(x)article-about(b(z)), (53d)

= λ(z)the(d(z)), (53e)

= λ(z)discovery-of-by(p)(z),

(53f)

= protein353, (53g)

= mia, j

= jim} (53h)

In (54a), the term B

is reduced to B

, i.e., B

⇒

∗

by two successive applications of the γ

∗

-rule, at ﬁrst

for r

= λ(z)λ(x)article-about(b(z)), with respect to

ICAART 2018 - 10th International Conference on Agents and Artiﬁcial Intelligence

240

λ(x), and then for a

= λ(z)λ(x)the(r(z)), with re-

spect to λ(x).

⇒

∗

[2]

≡ (54a)

λ(z)λ(x)send(m)(a(z))(z)

( j) (54b)

where

= λ(z)the(r(z)), (54c)

= λ(z)article-about(b(z)), (54d)

= λ(z)the(d(z)), (54e)

= λ(z)discovery-of-by(p)(z),

(54f)

= protein353, (54g)

= mia, j

= jim} (54h)

10 FUTURE WORK

We work on applications of the type-theory of acyclic

algorithms. For example, most promising results have

been achieved in language processing of formal and

natural languages. Speciﬁc applications are computa-

tional semantics and computational syntax-semantics

interfaces. These lines of work continue.

A new direction of applications is to computa-

tional neuroscience, by algorithmic modelling of pro-

cedural, factual, and declarative memory, and depen-

dencies between those, by mutual recursion.

Along such applications to advanced technologies

and AI, we work on theoretical developments. The

results in this paper are part of such long-term work.

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