Stanford University

Games, Full Abstraction and Full Completeness (Stanford Encyclopedia of Philosophy)

1. Introduction

1.1 Interpretations of programming languages

The notion of full abstraction arises from the Scott-Strachey approach
to the semantical analysis of programming languages (Scott
& Strachey 1971; Strachey 1966,
1967
), also known as denotational semantics. One
fundamental aim of a denotational semantics of a programming language
({L}) is to give a compositional interpretation
({mathcal{M}}: {L}to D) of the program phrases of ({L})
as elements of abstract mathematical structures (domains)
(D).

We may choose another way of giving meaning to programs, based on
their execution. This operational interpretation is only
defined on the set Prog of programs of
({L}), and involves the definition of a suitable set of program
values, which are the observables of ({L}). If the
execution of program (e) terminates with value (v), a situation
expressed by the notation (e opDownarrow v), then (v) is the
operational meaning of (e). This defines the operational
interpretation of programs as a partial function ({mathcal{O}})
from programs to values, where ({mathcal{O}}(e) = v) when (e
opDownarrow v).

Both interpretations induce natural equivalence relations on program
phrases. In one of its formulations, full abstraction states the
coincidence of the denotational equivalence on a language with one
induced by the operational semantics. Full abstraction has been first
defined in a paper by Robin Milner (1975),
which also exposes the essential
conceptual ingredients of denotational semantics: compositionality,
and the relations between observational and denotational equivalence
of programs. For this reason, full abstraction can be taken as a
vantage point into the vast landscape of programming language
semantics, and is therefore quite relevant to the core problems of the
philosophy of programming languages (White
2004
) and of computer science (Turner
2016
).

1.2 Compositionality

Compositionality (Szabó 2013) is
a desirable feature of a semantical analysis of a programming
language, because it allows one to calculate the meaning of a program
as a function of the meanings of its constituents. Actually, in
Milner’s account (see especially 1975:
sec. 1, 4
), compositionality applies even more generally to
computing agents assembled from smaller ones by means of
appropriate composition operations. These agents may include, beside
programs, hardware systems like a computer composed of a memory,
composed in turn of two memory registers, and a processing unit, where
all components are computing agents. This allows one to include in one
framework systems composed of hardware, of software and even of both.
Now, the syntactic rules that define inductively the various
categories of phrases of a programming language allow us to regard
({L}) as an algebra of program phrases, whose signature is
determined by these rules. One account of compositionality that is
especially suitable to the present setting (Szabó
2013: sec. 2
) identifies a
compositional interpretation of programs with a homomorphism from this
algebra to the domain of denotations associating with every operation
of the algebra of programs a corresponding semantical operation on
denotations.

As an example, consider a simple imperative language whose programs
(mathtt{c}) denote state transformations
({mathcal{M}}(mathtt{c}) : Sigma to Sigma). Among the
operations on programs of this language there is sequential
composition
, building a program (mathtt{c}_1 ; mathtt{c}_2)
from programs (mathtt{c}_1) and (mathtt{c}_2). The intended
operational meaning of this program is that, if (mathtt{c}_1 ;
mathtt{c}_2) is executed starting from a state (sigma in
Sigma), we first execute (mathtt{c}_1) starting form state
(sigma). If the execution terminates we obtain a state (sigma’),
from which we start the execution of (mathtt{c}_2) reaching, if the
execution terminates, a state (sigma”). The latter state is the
state reached by the execution of (mathtt{c}_1 ; mathtt{c}_2) from
state (sigma). From a denotational point of view, we have an
operation of composition on states as functions (Sigma to Sigma),
and the compositional interpretation of our program is given by the
following identity, to be read as a clause of a definition of
({mathcal{M}}) by induction on the structure of programs:

[{mathcal{M}}(mathtt{c}_1 ; mathtt{c}_2) = {mathcal{M}}(mathtt{c}_2) circ {mathcal{M}}(mathtt{c}_1)]

or, more explicitly, for any state (sigma):

[{mathcal{M}}(mathtt{c}_1 ; mathtt{c}_2) (sigma) = {mathcal{M}}(mathtt{c}_2) ({mathcal{M}}(mathtt{c}_1) (sigma)).]

As most programming languages have several categories of phrases (for
instance expressions, declarations, instructions) the algebras of
programs will generally be multi-sorted, with one sort for each
category of phrase. Denotational semantics pursues systematically the
idea of associating compositionally to each program phrase a
denotation of the matching sort (see Stoy
1977
for an early account).

1.3 Program equivalence and full abstraction

The existence of an interpretation of a programming language ({L})
induces in a standard way an equivalence of program
phrases:

Definition 1.1 (Denotational equivalence). Given any
two program phrases (e,e’), they are denotationally
equivalent
, written (e simeq_{mathcal{M}}e’), when
({mathcal{M}}(e) = {mathcal{M}}(e’)).

If ({mathcal{M}}) is compositional, then ({simeq_{mathcal{M}}})
is a congruence over the algebra of programs, whose derived operation,
those obtained by composition of operations of the signature, are
called contexts. A context (Cblbr) represents a program
phrase with a “hole” that can be filled by program phrases
(e) of appropriate type to yield the program phrase (C[e]). By
means of contexts we can characterize easily the compositionality of a
semantic mapping:

Proposition 1.1. If ({mathcal{M}}) is
compositional, then for all phrases (e,e’) and all contexts
(Cblbr):

[tag{1}label{compositionality}{e simeq_{mathcal{M}} e’} Rightarrow {C[e] simeq_{mathcal{M}} C[e’]}.]

This formulation highlights another valuable aspect of
compositionality, namely the referentially transparency of
all contexts, equivalently their extensionality:
denotationally equivalent phrases can be substituted in any context
leaving unchanged the denotation of the resulting phrase. The
implication ((ref{compositionality})) states, in particular, that
({simeq_{mathcal{M}}}) is a congruence. In order to compare
denotational and operational congruence, therefore, we must carve a
congruence out of the naive operational equivalence defined by setting
(e sim e’) if and only if ({mathcal{O}}(e) = {mathcal{O}}(e’)).
This can be done by exploiting program contexts (Cblbr),
representing a program with a “hole” that can be filled by
program phrases (e) of suitable type to yield a complete program
(C[e]).

Definition 1.2 (Observational equivalence) Given any
two program phrases (e,e’), they are observational
equivalent
, written (e simeq_{mathcal{O}}e’), when, for all
program contexts (Cblbr) and all program values (v):

[C[e] opDownarrow v text{ if and only if } C[e’] opDownarrow v.]

Observational equivalence is then a congruence over the algebra of
program phrases, and in fact it is the largest congruence contained in
(sim). From the general point of view of the account of Milner
(1975), that we are following closely,
the context of a computing agent represents one of its possible
environments. If we adopt the principle that “the overt behavior
constitutes the whole meaning of a computing agent”
(Milner 1975: 160), then the contexts
represents intuitively the observations that we can make on the
behavior of the computing agent. In the case of programs, the
observables are the values, so observational equivalence identifies
phrases that cannot be distinguished by means of observations whose
outcomes are distinct values. One consequence of Milner’s
methodological principle is that a computing agent becomes a

transducer, whose input sequence consists of enquiries by, or
responses from, its environment, and whose output sequence consists of
enquiries of, or responses to, its environment. (Milner
1975: 160
)

A behavior of a computing agent takes then the form of a
dialogue between the agent and its environment, a metaphor
that will be at the heart of the game theoretic approaches to
semantics to be discussed in
Section 3.
This behavioral stance, which has its roots in the work of engineers
on finite state devices has also been extended by Milner to a
methodology of modeling concurrent systems, with the aim

to describe a concurrent system fully enough to determine exactly what
behaviour will be seen or experienced by an external observer. Thus
the approach is thoroughly extensional; two systems are
indistinguishable if we cannot tell them apart without pulling them
apart. (Milner 1980: 2)

In addition, the roles of system and observer are symmetric, to such
an extent that

we would like to represent the observer as a machine, then to
represent the composite observer/machine as a machine, then to
understand how this machine behaves for a new observer. (Milner
1980: 19
)

While observational equivalence is blind to the inner details of a
computing agent but only observes the possible interactions
with its environment in which it takes part, denotational equivalence
takes as given the internal structure of a computing agent and, in a
compositional way, synthesizes its description from those of its
internal parts. The notion of full abstraction is precisely intended
to capture the coincidence of these dual perspectives:

Definition 1.3 (Full abstraction). A denotational
semantics ({mathcal{M}}) is fully abstract with respect to
an operational semantics ({mathcal{O}}) if the induced equivalences
({simeq_{mathcal{M}}}) and ({simeq_{mathcal{O}}})
coincide.

As a tool for investigating program properties, full abstraction can
be seen as a completeness property of denotational semantics:
every equivalence of programs that can be proved operationally, can
also be proved by denotational means. Equivalently, a denotational
proof that two terms are not equivalent will be enough to show that
they are not interchangeable in every program context.

Full abstraction also functions as a criterion for assessing a
translation (vartheta) from a language ({L}_1) into a (not
necessarily different) language ({L}_2), provided the two languages
have the same sets of observables, say Obs (Riecke
1993
). Then (vartheta) is fully
abstract
if observational equivalence (defined with respect to
Obs) of (e,e’ in {L}_1) is equivalent to observational
equivalence of (vartheta(e),vartheta(e’)). The existence of fully
abstract translation between languages can be used to compare their
expressive power, following a suggestion of (Mitchell
1993; Riecke 1993
): ({L}_1) is no
more expressive than ({L}_2) if there is a fully abstract
translation of ({L}_1) into ({L}_2).

Before going on in this general introduction to full abstraction and
related notions in the area of programming languages semantics, in
order to show the broad relevance of these notions, it is interesting
to observe that there is a very general setting in which it is
possible to study the full abstraction property, suggested by recent
investigations on compositionality in natural and artificial languages
by Hodges (2001) and others. In this
setting, full abstraction is connected to the problem of finding a
compositional extension of a semantic interpretation of a subset (X)
of a language (Y) to an interpretation of the whole language, via
Frege’s Context Principle (see Janssen
2001
on this), stating that the
meaning of an expression in (Y) is the contribution it makes to the
meaning of the expressions of (X) that contain it. In the original
formulation by Frege (X) was the set of sentences and (Y) the set
of all expressions, while in programming theory (X) is the set of
programs, (Y) the set of all program phrases.

A weakening of the definition of full abstraction represents an
essential adequacy requirement for a denotational interpretation of a
language:

Definition 1.4 (Computational adequacy). A
denotational semantics ({mathcal{M}}) is computationally
adequate
with respect to an operational semantics
({mathcal{O}}) if, for all programs (e) and all values (v)

[{mathcal{O}}(e) = v text{ if and only if } {mathcal{M}}(e) = {mathcal{M}}(v).]

An equivalent formulation of computational adequacy allows to
highlight its relation to full abstraction:

Proposition 1.2. Assume that ({mathcal{M}}) is a
compositional denotational interpretation such that ({mathcal{O}}(e)
= v) implies ({mathcal{M}}(e) = {mathcal{M}}(v)). The following
two statements are equivalent:

  1. ({mathcal{M}}) is computationally adequate with respect to
    ({mathcal{O}});
  2. for any two programs (e,e’ in {texttt{Prog}}), [e simeq_{mathcal{M}} e’ text{ if and only if } e simeq_{mathcal{O}} e’]

While the definition of the full abstraction property is
straightforward, fully abstract models for very natural examples of
programming languages have proved elusive, giving rise to a full
abstraction problem. In our discussion of full abstraction we
shall mainly concentrate on the full abstraction problem for the
language PCF (Programming language for Computable Functions, Plotkin
1977
), a simply typed
(lambda)-calculus with arithmetic primitives and a fixed-point
combinator at all types proposed in Scott
1969b
. This language is important because it includes most of
the programming features semantic analysis has to cope with:
higher-order functions, types and recursion, with reduction rules that
provide an abstract setting for experimenting with several evaluation
strategies. Furthermore, PCF is also a model for other extensions of
simply typed (lambda)-calculus used for experimenting with
programming features, like the Idealized Algol of Reynolds (1981).
The efforts towards a solution of the
full abstraction problem for PCF contributed, as a side effect, to the
systematic development of a set of mathematical techniques for
semantical analysis whose usefulness goes beyond their original
applications. We shall describe some of them in
Section 2,
devoted to the semantic analysis of PCF based on partially ordered
structures, the domains introduced by Dana Scott (1970),
that we examine in
Section 2.3.
Technical developments in the theory of domains and also in the new
research area focussed on Girard’s linear logic (Girard
1987
) have led to game
semantics
(Abramsky, Jagadeesan, &
Malacaria 2000; Hyland & Ong 2000
), which is now regarded
as a viable alternative to standard denotational semantics based on
domains. It is to this approach that we shall dedicate
Section 3
trying to provide enough details to orient the reader in an extensive
and still growing literature documenting the applications of games to
the interpretation of a wide spectrum of programming language
features.

2. Sequential higher-order computation: the full abstraction problem for PCF

The full abstraction problem has proved especially hard for a version
of simply typed (lambda)-calculus with arithmetic primitives called
PCF (Programming with Computable Functions) (Plotkin
1977
), a toy programming language
based on the Logic for Computable Functions of Scott (1969)
and Milner (1973).
In this section we introduce (a version
of) the language with its operational and denotational semantics, and
outline how the full abstraction problem arises for this language. The
problem has been one of the major concerns of the theoretical
investigation of programming languages for about two decades, from its
original formulation in the landmark papers (Milner
1977; Plotkin 1977
) to the first
solutions proposed in 1993 (Abramsky et al.
2000; Hyland & Ong 2000
) using game semantics, for which
see
Section 3.

2.1 Syntax of PCF

PCF is a language based on simply typed (lambda)-calculus extended
with arithmetic and boolean primitives, and its type system is defined
accordingly:

Definition 2.1 (PCF types). The set Types
of types of PCF is defined inductively
as follows

  • the ground types num (for
    terms representing natural numbers), bool
    (for terms representing boolean values)
    are types,
  • if (sigma, tau) are types, also ((sigma to tau)) is a
    type.

Parentheses will be omitted whenever possible, with the convention
that they associate to the right, so that a type (sigma_1 to cdots
sigma_n to tau) is equivalent to ((sigma_1 to (sigma_2 to
(cdots (sigma_n to tau)cdots))))

PCF terms are the terms of simply typed (lambda)-calculus extended
with the following arithmetic constants, of the indicated type:

  • a constant ({0}:texttt{num}), representing the natural number
    0;
  • a constant (texttt{succ}) of type (texttt{num}to
    texttt{num}) representing the successor function over natural
    numbers;
  • a constant (texttt{pred}) of type (texttt{num}to
    texttt{num}) representing the predecessor function over natural
    numbers;
  • constants (texttt{tt}) and (texttt{ff});
  • constants of type (texttt{bool}to texttt{num}to
    texttt{num}to texttt{num}) and (texttt{bool}to texttt{bool}to
    texttt{bool}to texttt{bool}) for conditionals of type num
    and of type bool,
    respectively: these are both written as
    ({texttt{if }cdottexttt{ then }cdottexttt{ else }cdot}), and
    we let context make clear what is the intended type of the
    result;
  • a constant (texttt{zero?}) for the test for zero of type
    (texttt{num}to texttt{bool});
  • a unary function symbol ({mathtt{Y}(cdot)}) for the fixed
    point combinator, where ({mathtt{Y}(e)}:sigma) for any (e:sigma
    to sigma).

Terms are built inductively according to rules that allow to infer
judgements of the form (B vdash e : sigma), stating that
term (e) is of type (sigma) under the assumption that the
variables occurring free in (e) are given unique types in a
basis (B) of the form

[{{ x_1:sigma_1,ldots,x_k:sigma_k }}.]

The rule for building
PCF-terms are therefore inference rules for such judgements. In
particular there are rules for typed constants, for example in any
basis (B) there is a judgement (B vdash texttt{zero?} :
texttt{num}to texttt{bool}), and we have rules for typed
(lambda)-abstractions

[frac{B,x:sigma vdash e:tau}{B vdash {lambda x:sigma{, . ,}e}:sigma to tau}]

and applications

[frac{B vdash e_1 : sigma to tau qquad B vdash e_2 : sigma}{B vdash e_1 e_2 : tau}]

and
a rule for the fixed-point operator:

[frac{B vdash e:sigma to sigma} {B vdash {mathtt{Y}(e)}:sigma}.]

2.2 Operational semantics

A PCF program is a closed term of ground type. We specify how
programs are to be executed by defining an evaluation relation (e
opDownarrow v) between closed terms (e) and values (v),
where the values are the constants and abstractions of the form
({lambda x:sigma{, . ,}e}). In particular, values of ground type
bool are (texttt{tt},texttt{ff}),
and values of the ground type num are
({0}) and all terms of the form

[mathtt{n} = underbrace{texttt{succ}(ldots texttt{succ}}_{n}({0}) ldots ) .]

Evaluation is defined
by cases according to the structure of terms, by means of inference
rules for judgements of the form (e opDownarrow v). These rules
state how the result of the evaluation of a term depends on the result
of the evaluation of other terms, the only axioms having the form (v
opDownarrow v) for every value (v). For example there is a rule

[frac{e opDownarrow v}{{texttt{succ }e} opDownarrow {texttt{succ }v}}]

that states that, if the result of the evaluation of
(e) is (v), then the result of the evaluation of ({texttt{succ
}e}) is ({texttt{succ } v}). Similarly we can describe the
evaluation of the other constants. The evaluation of a term of the
form (e_1 e_2) proceeds as follows: first (e_1) is evaluated; if
the evaluation terminates with value (v’), then the evaluation of
(e_1 e_2) proceeds with the evaluation of (v’ e_2); if this
terminates with value (v), this is the value of (e_1 e_2),
formally

[frac{e_1 opDownarrow v’ qquad v’ e_2 opDownarrow v}{e_1 e_2 opDownarrow v}]

For a value of the form ({lambda x:sigma{, .
,}e_1}), its application to a term (e_2) has the value (if any)
obtained by evaluating the term (e_1[e_2/x]) resulting by
substituting (e_2) to all free occurrences of (x) in (e_1):

[frac{e_1[e_2/x] opDownarrow v}{({lambda x:sigma{, . ,}e_1}) e_2 opDownarrow v}.]

These implement a call-by-name evaluation
strategy: in an application, the term in function position must be
evaluated completely before the term in argument position, which is
then passed as actual parameter. The fixed point combinator is
essential to the encoding of recursive definitions. Its evaluation is
described by the rule

[frac{e({mathtt{Y}(e)}) opDownarrow v}{{mathtt{Y}(e)} opDownarrow v}]

which is the only rule whose
premiss involves the evaluation of a larger term than the one to be
evaluated: this is why the definition of the evaluation relation
cannot be reduced to structural induction.

We shall be especially interested in situations when the evaluation of
a term (e) does not have a value; in these case we say that (e)
diverges, and write (e opUparrow). It is in the presence
of divergent terms that the causal structure of the evaluation process
is exposed. The initial example is in fact a term that diverges in a
very strong sense:

Definition 2.2 (Undefined). For any ground type
(gamma), define (Omega:gamma) as

[{mathtt{Y}({lambda x:gamma{, . ,}x})}]

By inspecting the evaluation rules we see that the only possible
evaluation process gives rise to an infinite regress, therefore
(Omega opUparrow).

We can define the usual boolean operations by means of the conditional
operator, as in the following examples:

[begin{align}
tag{2} texttt{and} &= {lambda x:texttt{bool}, y:texttt{bool}{, . ,}{texttt{if }xtexttt{ then }ytexttt{ else }texttt{ff}}}.
label{etbivalente} \
tag{3} texttt{or} &= {lambda x:texttt{bool}, y:texttt{bool}{, . ,}{texttt{if }xtexttt{ then }texttt{tt}texttt{ else }y}}
label{velbivalente}end{align}]

with the usual
truth tables. However, we have now to take into account the
possibility of divergence of the evaluation process, for example in a
term like (texttt{or}(Omega,texttt{tt})), therefore we extend the
usual truth tables by adding a new boolean value, representing absence
of information, (bot) (read as “undefined”) to
(texttt{tt}) and (texttt{ff}), as the value of the term
(Omega). Here, the first argument to be evaluated is the one on the
left, and if the evaluation of this diverges then the whole evaluation
process diverges. Consider now an operator por
whose interpretation is given by the
table

[tag{4}label{por}
begin{array}{r|lll}
texttt{por}& textit{tt}& textit{ff}&bot \hline
textit{tt}& textit{tt}& textit{tt}&textit{tt}\
textit{ff}& textit{tt}& textit{ff}&bot\
bot & textit{tt}& bot &bot
end{array}]

In this case (texttt{por}(texttt{tt},Omega) =
texttt{por}(Omega,texttt{tt}) = texttt{tt}): this is the
parallel-or which plays a central role in the full
abstraction problem for PCF. It will turn out that is is not definable
by any PCF term, precisely because of its parallel nature. In order to
carry out a semantical analysis of PCF, we need a theory of data types
with partial elements and of functions over them that support
an abstract form of recursive definition through fixed point
equations: this is what is achieved in Scott’s theory of
domains, the original mathematical foundation for denotational
semantics of programming languages as conceived by Strachey (1966,
1967
).

2.3 Denotational semantics

2.3.1 Types as domains

What are the general structural properties of a space of partial data?
The mathematical theory of computation elaborated by Dana Scott (1970)
is an answer to this question, that
takes partially ordered sets generically called domains as
basic structures. The partial order of a domain describes a
qualitative notion of “information” carried by the
elements. In such a framework it is natural to reify divergence by
introducing a new element (bot) representing absence of
information. When (x sqsubseteq y) in this partial order,

(y) is consistent with (x) and is (possibly) more
accurate than
(x [ldots]) thus (x sqsubseteq y) means that
(x) and (y) want to approximate the same entity, but (y) gives
more information about it. This means we have to allow
“incomplete” entities, like (x), containing only
“partial” information. (Scott 1970:
171
)

The resulting partially ordered sets should also have the property
that sequences of approximations, in particular infinite chains (x_0
sqsubseteq x_1 sqsubseteq ldots) should converge to a
limit containing the information cumulatively provided by the
(x_i). The same structure is also exploited in Kleene’s proof
of the First Recursion Theorem in Kleene 1952
(secs. 66, 348–50)
, and will allow to define a notion of
continuous function in terms of preservation of limits.

Definition 2.3 (Complete partial orders). A complete
partial order (cpo) is a partially ordered set ({langle
D,sqsubseteq rangle}) with a least element (bot), and such that
every increasing chain (x_0 sqsubseteq x_1 sqsubseteq ldots) of
elements of (D) has a least upper bound (bigsqcup_n x_n).

Given any set (X), we write (X_{bot}) for the set (X cup {
bot }) obtained by adding a new element (bot). It is natural to
order the elements of (X_{bot}) according to their amount of
information, by setting for (x,y in X_{bot}),

[begin{aligned}
x sqsubseteq y &Longleftrightarrow (x = bot text{ or } x = y).end{aligned}]

Partially ordered structures of the form (X_bot) are called
flat domains, among which we have (texttt{bool}= {{
{textit{tt}},{textit{ff}}}}_bot) and (texttt{num}=
{mathbb{N}}_bot), that will be used to interpret the ground types
of PCF.

A general requirement on domains is that every element be a limit of
its finite approximations, for a notion of finiteness (or
compactness) that can be formulated entirely in terms of the
partial order structure:

Definition 2.4 (Finite elements of a cpo). If (D)
is a cpo, an element (d in D) is finite if, for every
increasing chain (x_0 sqsubseteq x_1 sqsubseteq ldots)

[d sqsubseteq bigsqcup_n x_n Longrightarrow exists x_i left(d sqsubseteq x_i right).]

For (d in D), the notation (mathcal{A}(d)) denotes
the set of finite elements below (d); (mathcal{A}(D)) is the set
of finite elements of (D). Finite elements are also called
compact.

Observe that finite subsets of a set (X) are exactly the finite
elements of the complete lattice of subsets of (X). It is useful
also to observe that this definition only partially matches our
intuition: consider for example the finite element (infty + 1) in
the cpo

[0 sqsubseteq 1 sqsubseteq 2 sqsubseteq cdots sqsubseteq infty sqsubseteq infty + 1.]

Definition 2.5 (Algebraic cpo). A (D) is
algebraic if, for every (d in D), there is an increasing
chain (x_0 sqsubseteq x_1 sqsubseteq ldots) of finite
approximations of (d) such that

[d = bigsqcup_n x_n.]

If (D) is algebraic,
we say that the finite elements form a basis of (D).

One last completeness assumption on algebraic cpo’s is needed in
order to get a category of domains suitable for the interpretation of
PCF:

Definition 2.6. Given a cpo (D), if (X subseteq
D) has an upper bound we say that (X) is consistent, and
write (opuparrow X), or (x opuparrow y) when (X = {{ x,y
}}). (D) is consistently complete if every (X subseteq
D) such that (opuparrow X) has a least upper bound.

The following notion of domain that has proved extremely convenient as
a framework for the denotational semantics of programming languages
(Scott 1982):

Definition 2.7 (Domain). A domain is a
consistently complete algebraic cpo with a countable basis.

2.3.2 An abstract theory of computable functions of higher-types

How can we use the notion of information implicit in the ordering on
the elements of domains to develop an abstract notion of
computability? Clearly, a computable function should preserve
monotonically any increase of information on its inputs:
(f(x) sqsubseteq f(y)) whenever (x sqsubseteq y). In particular,
strict functions (f : D to E) over flat domains, those for
which (f(bot_D) = bot_E), are monotonic.

Consider the domain ({{ 0,1 }}^infty) whose elements are finite
and infinite sequences of bits (0,1), where (u sqsubseteq v) if
either (u) is infinite and (u = v), or (u) is finite and (u)
is a prefix of (v). What properties should be true of a computable
function taking as arguments an infinite sequence of bits ({langle
b_1,b_2,b_3,ldots rangle})? Take as an example the function
(textit{search}:{{ 0,1 }}^infty to {mathbb{B}}_bot) whose
value is ({textit{tt}}) if, for (u in {{ 0,1 }}^infty), (1)
occurs in (u) at least once, otherwise (bot). Think of the
sequence ({langle b_1,b_2,b_3,ldots rangle}) as given one element
at a time: the initial segments obtained in this process are an
increasing chain of finite elements of ({{ 0,1 }}^infty),

[{langle rangle} sqsubseteq {langle b_1 rangle} sqsubseteq {langle b_1,b_2 rangle} sqsubseteq {langle b_1,b_2,b_3 rangle} sqsubseteq ldots]

having ({langle b_1,b_2,b_3,ldots rangle}) as a
limit (i.e., least upper bound). By monotonicity we have a
corresponding increasing chain of values

[textit{search}({langle rangle}) sqsubseteq textit{search}({langle b_1 rangle}) sqsubseteq textit{search}({langle b_1,b_2 rangle}) sqsubseteq textit{search}({langle b_1,b_2,b_3 rangle}) sqsubseteq ldots]

If
(textit{search}({langle b_1,b_2,b_3,ldots rangle}) =
{textit{tt}}), then there must be a finite initial segment
({langle b_1,b_2,ldots,b_n rangle}) for which
(textit{search}({langle b_1,b_2,ldots,b_n rangle}) =
{textit{tt}}), and this will be the cumulative value of the function
for the infinite sequence ({langle b_1,b_2,b_3,ldots rangle}). In
general, a computable function (f : D to E) should (be monotonic
and) have the property that a finite amount of information on the
output (f(x)) must be already obtainable by giving a finite amount
of information on the input (x). This is equivalent to the notion of
continuity originally introduced by Scott in his theory of computable
functions over domains:

Definition 2.8 (Continuous functions). If ({langle
D,sqsubseteq _D rangle},{langle E,sqsubseteq _E rangle}) are
cpo’s and (f : D to E) is monotonic, (f) is
continuous if

[f(bigsqcup_i x_i) = bigsqcup_i f(x_i)]

for every increasing chain (x_0
sqsubseteq x_1 sqsubseteq ldots subseteq D).

From the point of view of denotational semantics, a fundamental
property of continuous functions (D to D) is that they admit a
least fixed point, whose construction can be carried out uniformly and
continuously:

Theorem 2.1 (The Fixed Point Theorem for continuous
functions) Let (f : D to D) be a continuous function and (x in
D) be such that (x sqsubseteq f(x)). Then the element

[bigsqcup_{n in {mathbb{N}}} f^{(n)}(x)]

is the least (y sqsupseteq x) such that (f(y) = y).

Definition 2.9. The least fixed point of a continuous
(f : D to D) is the element of (D) defined by

[{texttt{fix}(f)} =_{textrm{def}} bigsqcup_{n in {mathbb{N}}} f^{(n)}(bot).]

The continuous functions from (D) to (E), for cpo’s
({langle D,sqsubseteq _D rangle}) e ({langle E,sqsubseteq _E
rangle}), form a cpo ([D to E]), ordered pointwise by setting,
for (f,g:D to E):

[f sqsubseteq g Longleftrightarrow forall d in D. f(d) sqsubseteq _E g(d).]

([D to E]) is a domain if (D)
and (E) are, and ({texttt{fix}(cdot)}:[D to D] to D) is
continuous. A further construction on cpo’s which also extends
to domains and is very frequently used is cartesian product:
given cpo’s (D,E), their cartesian product is defined as the
set (D times E) of pairs ({langle d,e rangle}) where (d in
D) and (e in E), ordered pointwise: ({langle d,e rangle}
sqsubseteq {langle d’,e’ rangle}) if and only if (d sqsubseteq
_D d’) and (e sqsubseteq _E e’). We can summarize these
constructions in categorical language (Plotkin 1978, Other Internet
Resources), saying that the category whose objects are domains and
whose morphisms are the continuous functions is cartesian
closed
.

2.3.3 Continuous semantics for PCF

The standard interpretation of PCF consists of a family of
cpos (D^sigma), for each type (sigma), where (D^texttt{num}=
{mathbb{N}}_{bot}) and (D^texttt{bool}= {mathbb{B}}_{bot}),
(D^{sigma to tau} = [D^sigma to D^tau]) and the PCF constants
have the natural interpretation as strict continuous functions of the
appropriate type, for example (texttt{cond}: {mathbb{B}}_{bot} to
{mathbb{N}}_{bot} to {mathbb{N}}_{bot} to {mathbb{N}}_{bot})
is interpreted as:

[textit{cond}(b)(x)(y) = left{ begin{array}{ll}
x&text{if (b = texttt{tt})}\
y &text{if (b = texttt{ff})}\
bot &text{if (b = bot),}
end{array}
right.]

Furthermore, the operator
({mathtt{Y}(cdot)}) is interpreted as the continuous functional
({texttt{fix}(cdot)}) at the appropriate type. This is the
interpretation considered in Scott
1969b
) and Milner 1973.

The possibility that (e) may contain free occurrences of variables
(whose types are given by a basis (B)) slightly complicates the
interpretation of terms, making it depend on a further parameter, an
environment (rho) mapping each free variable (x:tau) of
(e) to an element of (D^tau) (if the latter condition is
satisfied, we say that (rho) respects (B)). Of course,
the environment is irrelevant when (e) is closed.

The standard interpretation of PCF terms (e:sigma) (from a basis
(B)) is then an element ({lll erll rho} in D^sigma), for any
environment (rho) such that (rho) respects (B), built by
structural induction on terms, interpreting application as function
application and (lambda)-abstractions by (continuous) functions.
More generally, an interpretation is continuous if every
(D^sigma) is a cpo and (D^{sigma to tau}) consists of
continuous functions (D^sigma to D^tau).

A model of PCF is an interpretation that satisfies the
expected identities between terms of the same type. We shall omit the
details of the general characterization of models of PCF, for which
the reader is referred to Ong (1995: sec.
3.2)
and Berry, Curien, &
Lévy (1985: sec. 4)
, but just to point out an example of
what must be taken into account when such a generality is needed, in
order to admit interpretations where the elements at function types
are not, strictly speaking, functions, we have to assume a family of
application operations

[cdot_{sigmatau}: D^{sigma to tau} times D^sigma to D^tau]

so that, if (B vdash
e_1 : sigma to tau) and (B vdash e_2 ; sigma), ({lll
e_1e_2rll rho} = {lll e_1rll rho} cdot_{sigmatau} {lll
e_2rll rho} in {{{D}^{tau}}}). A model is
order-extensional if, for all elements (f,g in
{{{D}^{sigma to tau}}}), (f sqsubseteq g) if and only if (f
cdot x sqsubseteq g cdot x) for all (x in {{{D}^{sigma}}}). A
model is extensional if, for all elements (f,g in
{{{D}^{sigma to tau}}}), (f = g) if and only if (f cdot x = g
cdot x) for all (x in {{{D}^{sigma}}}). An element (d in
D^sigma) of a model is definable is there is a closed terms
(e:sigma) such that (d = {lll erll }).

2.4 Relating operational and denotational semantics

The general setting for discussing full abstraction requires that we
introduce the following notions:

Definition 2.11 (Observational preorder and
equivalence) Given PCF terms (e) and (e’) of the same type
(sigma), we write (e precsim_{textrm{obs}} e’) (read (e)
is observationally less defined than (e’)
) if, for every
program context (Cblbr) with a hole of type (sigma) and any
value (v),

[C[e] opDownarrow v text{ implies that } C[e’] opDownarrow v.]

We say that (e) and (e’) are
observationally equivalent, and write (e
simeq_{textrm{obs}} e’), if (e precsim_{textrm{obs}} e’) and
(e’ precsim_{textrm{obs}} e).

Observational equivalence is a congruence. Another congruence on PCF
terms is given by equality of denotations in a model:

Definition 2.11 (Denotational preorder and
equivalence). Given PCF terms (e) and (e’) of the same type
(sigma) relative to a basis (B), we write (e
precsim_{textrm{den}} e’) if ({lll erll rho} sqsubseteq {{lll
e’rll} rho}) for all environments (rho) respecting (B). We
write (e simeq_{textrm{den}} e’) if (e precsim_{textrm{den}}
e’) and (e’ precsim_{textrm{den}} e) .

Proposition 2.1 (Computational adequacy for PCF). The
following two statements hold for the standard model of PCF, and are
equivalent:

  1. For any two PCF terms of the same ground type (e,e’), (e
    simeq_{textrm{den}} e’) implies (e simeq_{textrm{obs}}
    e’);
  2. For any closed PCF term (e) of ground type and any value (v)
    of that type, ({lll erll } = {lll vrll }) if and only if (e
    opDownarrow v);

We can now justify our intuitive interpretation of (bot) in the
standard model, where ground types are interpreted as flat
domains:

Corollary 2.1. For any closed PCF term (e) of
ground type, (e opUparrow) if and only if ({lll erll } =
bot).

In
Section 1.3
we have already defined a very general notion of (equational) full
abstraction, based on synonymy, i.e., equality of interpretation of
terms. In the case PCF, whose intended models are partially ordered at
all types, we can define a stronger property:

Definition 2.12 (Inequational full abstraction). A
continuous model ({langle {{ {{{D}^{sigma}}} mid sigma in
texttt{Types}}},{lll cdotrll cdot} rangle}) of PCF is
inequationally fully abstract if, for closed terms (e,e’),
(e precsim_{textrm{obs}} e’) implies ({lll erll } sqsubseteq
{lll e’rll }).

Definability is the key to full abstraction, as shown by the following
important result of Milner and Plotkin:

Theorem 2.2. A continuous, order-extensional model of
PCF is fully abstract if and only if for every type (sigma),
({{{D}^{sigma}}}) is a domain whose finite elements are
definable.

We turn now to the failure of the full abstraction property for the
standard model of PCF, as shown by Plotkin in his classic study (Plotkin
1977
):

Proposition 2.2. The standard model of PCF is not
fully abstract with respect to call-by-name evaluation.

The proof is based on the observation that we can build PCF terms of
type ((texttt{bool}to texttt{bool}to texttt{bool}) to
texttt{num}) that recognize the parallel-or function. Specifically,
consider the “test” terms (T_i) defined as follows,
where (i = {0},1):

(lambda f : texttt{bool}to texttt{bool} to texttt{bool.if } (f
texttt{ tt } bot_{texttt{bool}} texttt{ then})

(texttt{if } (f bot_{texttt{bool}} texttt{ tt}) texttt{ then})

(texttt{if } (f texttt{ ff ff}) texttt{ then})

(bot_{texttt{num}})

(texttt{else } i)

(texttt{else } bot_{texttt{num}})

(texttt{else } bot_{texttt{num}})

Then, ({mathcal{D}lll T_{0}rll} texttt{por} = {0}neq 1 =
{mathcal{D}lll T_1 rll } texttt{por} ), where por
is defined by table ((ref{por})), so
(T_{0} {simeq_{textrm{den}}}T_1) does not hold. However, no
program context in PCF can separate (T_{0}) and (T_1) because
por is not definable. This can be shown
by characterizing in a combinatorial way the relations of dependence
induced by the evaluation process of a program among the evaluation
processes of its (sub)terms, as Plotkin does in the Activity Lemma
(Plotkin 1977: Lemma 4.2). As an
alternative, it is possible to build a computationally adequate models
of PCF whose functions enjoy a weak sequentiality property (that we
discuss below, in
Section 2.5.1)
and where, therefore, the function por
is ruled out: a complete formal proof along these lines is given in
Gunter 1992 (sec. 6.1).

One option to solve the full abstraction problem is to extend the
language: a remarkable result of Plotkin (1977)
shows that adding parallel-or is
enough:

Proposition 2.3. The standard model is fully abstract
for the language PCF extended with parallel-or.

Milner (1977) has shown that there is a
fully abstract model of PCF, by taking the set of closed terms at each
type (sigma) identifying observationally equivalent terms and by
completing the resulting partially ordered set turning it into a
cpo.

Corollary 2.2. There is a unique continuous, order
extensional, inequationally fully abstract model of PCF, up to
isomorphism.

The full abstraction problem for PCF consists in finding a direct
description of the class of domains and continuous functions that make
up the fully abstract model. A solution to this problem would require
a precise criterion for assessing the extent to which a proposed
description of the model is satisfactory. If one accepts the
“precise minimal condition that a semantic solution of the full
abstraction problem should satisfy” given by Jung
& Stoughton (1993)
, namely the
possibility of describing in an effective way the domains (D^sigma)
of a finitary version of PCF (whose only ground type is bool),
then the story of failed attempts to
give such a direct description of the fully abstract model is
justified, with hindsight, by a result of Loader (2001):

Theorem 2.3. Observational equivalence for finitary
PCF is not decidable.

It is still possible, however, that one could find a direct
description of an intensionally fully abstract model (Abramsky
et al. 2000: 411
):

Definition 2.13 (Intensional full abstraction). A
model of PCF is intensionally fully abstract if every
(D^sigma) is algebraic and all its compact elements are
definable.

Pursuing this line of development of the full abstraction problem
leads us to game semantics, which will be the topic of the next
Section. Before that, we outline the main attempts to reduce the model
by means of a semantical characterization of higher-order sequential
computation.

2.5 Towards a sequential semantics

The reason for the failure of full abstraction of the continuous
semantics of PCF is the existence of functions whose evaluation
requires parallel computation. We describe now some proposals for
characterizing sequentiality of functions by means of
properties related to the structure of the domains on which they are
defined. This has been an area of intensive research toward the
solution of the full abstraction problem for PCF, and some of the
insights that emerged from it lead very naturally to the game models
discussed in
Section 3.
In addition, the following summary of attempts at a characterization
of sequentiality is also a very interesting demonstration of the
expressive power of the language of partial order in the semantic
analysis of programming concepts.

Intuitively, a sequential function is one whose evaluation proceeds
serially: this means that it is possible to schedule the evaluation of
its arguments so that the evaluation of the function terminates with
the correct value; if the evaluation of one of them diverges, the
whole evaluation diverges. At each stage of this process there is an
argument whose value is needed to obtain more information on the
output of the function. In order to account for this causal structure
of computations at the semantical level, we need to enrich the domain
structure so that the order on the elements reflect the happening of
computational events and their causal order. This suggests
another way of interpreting the abstract notion of information that
motivated the axioms of a cpo in
Section 2.3.1.
Now,

information has to do with (occurrences of) events: namely the
information that those events occurred. For example in the case of
({mathbb{N}}_{bot}), (bot) might mean that no event occurred
and an integer (n), might mean that the event occurred of the
integer (n) being output (or, in another circumstance being input).
(Plotkin 1978, Other Internet Resources)

2.5.1 Stability

One interpretation of events regards them as the production of values
in the evaluation of an expression. This interpretation originates in
the context of bottom-up computation of recursive programs developed
by Berry (1976), where a recursive
definition is translated into a graph displaying the dependence of
results of an expression on results of its subexpressions. This
context naturally suggests the notion of producer of an event
(x), as a set of events that must have happened in order that (x)
may happen. Reformulating this observation in the language of partial
orders, Berry (1976) defined:

Definition 2.14 (Stability). Let (D_1,ldots,D_n,
D) be flat cpo’s and (f: D_1times ldots times D_n to D)
monotonic (hence continuous). Then (f) is stable if for
every (vec{x} = {langle x_1,ldots,x_n rangle} in D_1times
ldots times D_n) there is a unique minimal element (m(f,x)
sqsubseteq vec{x}) such that (f(m(f,vec{x})) = f(vec{x})).

Clearly, the parallel-or function is not stable: the value
(texttt{por}(bot,{textit{tt}}) = {textit{tt}}=
texttt{por}({textit{tt}},bot)) has no minimal producer. A
remarkable property of stable functions is that they allow to build a
new model of PCF, where ({{{D}^{sigma to tau}}}) is the set of
stable functions on the domains that interpret the types (sigma)
and (tau), which are refinements of Scott domains called
dI-domains (Berry 1978). From
our point of view, the important outcome of these definitions is the
following adequacy result (Gunter 1992: chap.
6
):

Proposition 2.4. The interpretation of PCF terms as
elements of dI-domains, where (D^{sigma to tau}) is the dI-domain
of stable functions from (D^sigma) to (D^tau) with the stable
order, is a computationally adequate model of PCF.

This result completes the argument showing the failure of full
abstraction for the continuous model of PCF at the end of
Section 2.4,
if the informal notion of sequentiality used there is formalized as
stability. The stable model of PCF has recently been shown to be fully
abstract for an extension of PCF (Paolini
2006
).

2.5.2 Sequential functions

The first definitions of sequentiality, due to Vuillemin (1974)
and Milner (1977)
stated that an (n)-ary functions (f)
over flat domains is sequential at argument ({langle
x_1,ldots,x_n rangle})
if there is a sequentiality
index
(i) of (f), depending on ({langle x_1,ldots,x_n
rangle}), such that every increase in the output information must
increase the information at argument (i). For example, the function
(texttt{cond} : {mathbb{B}}_{bot} times {mathbb{N}}_{bot}
times {mathbb{N}}_{bot} to {mathbb{N}}_{bot}) is sequential in
this sense at any input tuple. In fact, its sequentiality index at
({langle bot,m,n rangle}) is 1; its sequentiality index at
({langle {textit{tt}},m,n rangle}) is 2, and its sequentiality
index at ({langle {textit{ff}},m,n rangle}) is 3. There is
however no sequentiality index for the function (texttt{por} :
{mathbb{B}}_{bot} times {mathbb{B}}_{bot} to
{mathbb{B}}_{bot}) at the input ({langle bot,bot
rangle}).

While all sequential functions (over flat domains) are stable,
sequentiality is strictly stronger than stability. For example, the
continuous function from ({mathbb{B}}_bot times {mathbb{B}}_bot
times {mathbb{B}}_bot) to ({mathbb{B}}_bot) defined as the
smallest continuous extension of the three assignments

[{langle {textit{tt}},{textit{ff}},bot rangle} mapsto {textit{tt}}, {langle {textit{ff}},bot,{textit{tt}}rangle} mapsto {textit{tt}}, {langle bot,{textit{tt}},{textit{ff}}rangle} mapsto {textit{tt}}.]

has no sequentiality index at the argument ({langle bot,bot,bot
rangle}), but is stable because the arguments ({langle
{textit{tt}},{textit{ff}},bot rangle},{langle
{textit{ff}},bot,{textit{tt}}rangle},{langle
bot,{textit{tt}},{textit{ff}}rangle}) are pairwise
inconsistent.

The following result adds support to the search for a semantical
characterizations of sequentiality:

Proposition 2.5. Let (f : D_1 times cdots times
D_n to D) be a continuous function, where (D_i,D) are either
({mathbb{N}}_bot) or ({mathbb{B}}_bot). Then (f) is
sequential if and only if it is definable in PCF.

2.5.3 Concrete data structures and sequential algorithms

If the domains needed for an adequate definition of sequentiality are
to describe the causality relations among occurrences of computational
events, then it is necessary to enrich our picture by considering
events as located at places, generalizing the notion of
argument place in the definitions of Vuillemin and Milner which
depends on how a function is presented. This led to a notion of
concrete data structure (cds) (Kahn
& Plotkin 1993
) and to an axiomatization of the
order-theoretic properties of domains of first-order data. Kahn and
Plotkin obtained a representation theorem for the domains described by
their axioms, the concrete domains, in terms of the
states of a process of exploration of a concrete data
structure that consists in filling, given a state (x), any cell
enabled by sets of events that have already happened in (x),
starting from initial cells enabled in the initial, empty
state: this is similar to proving theorems in an abstract deductive
system whose rules are the enablings. As a motivating example, think
of a linked list of, say, natural numbers. The initial cell may be
filled at any time with any value (n_1). This event enables the
second cell of the list, which may then (and only then) be filled with
any value (n_2), and so on for all later cells.

Observe that the framework of concrete data structures gives the
necessary notions to reconstruct a semantical version of
sequentiality. Roughly, a monotonic function (f) from states of
(M) to states of (M’) is sequential (at state (x)) if,
for any output cell (c’), there is an input cell (c) that must be
filled in any transition from (x) to (y) such that the transition
from (f(x)) to (f(y)) fills (c’) (if such a (c’) does exist)
(Curien 1986: Def. 2.4.5). The cell
(c) is the sequentiality index for (f) at (x) for
(c’).

The category whose objects are the concrete data structures and whose
morphisms are the sequential functions just defined is, however, not
cartesian closed, not unexpectedly. This observation (for a simple
proof, see Amadio & Curien 1998 (theorem
14.1.12)
) prevents the use of this category as a model of PCF.
However, it is possible to define for every two concrete data
structures (M,M’) a new one (M to M’) whose states represent
sequential algorithms and which is the exponential object of
(M) and (M’) in a cartesian closed category whose morphisms are
sequential algorithms (Curien 1986: sec.
2.5
). The generalizations of the model theory of PCF to
categorical models allows us to obtain a model of PCF from this new
category, even though its morphisms are not functions in the usual
set-theoretic sense. It turns out that the sequential algorithm model
is not extensional, because there are distinct PCF terms that denote
the same continuous function yet represent distinct algorithms. As an
example, consider the following two terms, that denote the same
function but different algorithms:

[begin{aligned}
texttt{lror}(x,y) &= texttt{if }xtexttt{ then }({texttt{if }ytexttt{ then }texttt{tt} texttt{ else }x}) \
&quad texttt{ else }({texttt{if }ytexttt{ then }texttt{tt}texttt{ else }texttt{ff}}) \
texttt{rlor}(x,y) &= texttt{if }ytexttt{ then }({texttt{if }xtexttt{ then }texttt{tt}texttt{ else }y})\
&quad texttt{ else }({texttt{if } xtexttt{ then }texttt{tt}texttt{ else }texttt{ff}}).
end{aligned}]

By suitably
introducing error values (textit{error}_1,textit{error}_2) in the
semantics, and enforcing an error-propagation property of the
interpretations of terms (thus enlarging the observables of the
language), the functions corresponding to the above terms can
then be distinguished: clearly, for the interpreting functions
(textit{lror}) and (textit{rlor}) we have

[begin{aligned}
textit{lror}(textit{error}_1,textit{error}_2) &= textit{error}_1 &textit{rlor}(textit{error}_1,textit{error}_2) &= textit{error}_2end{aligned}]

which
also points to the possibility of proving full abstraction of this
(non-standard) extensional model with respect to an extension of PCF
with control operators (Cartwright, Curien,
& Felleisen 1994
).

Before leaving this overview of the quest for an extensional
characterization of higher-order sequentiality, we should mention
Bucciarelli & Ehrhard (1994) who
introduced a refinement of the dI-domains of Berry supporting a notion
of strongly stable function which allows them to build an
extensional model of PCF, which is not fully abstract. The reason for
the failure of full abstraction in this case depends on the fact that
PCF-definable functionals satisfy extensionality properties that fail
when functions are ordered by the stable order. This was also the
reason that motivated the introduction of bidomains (Berry
1978
), where the stable and extensional
(= pointwise) orderings of functions coexist.

2.6 Historical notes and further readings

The problem of full abstraction has been anticipated in a large amount
of work on the relations between the denotational and operational
interpretations of programming languages. In particular, the
pioneering work on the semantics of recursive programs carried out in
Stanford in the early 1970s by a group of people gathering around
Zohar Manna, and including Jean Marie Cadiou, Robin Milner and Jean
Vuillemin, also interacting with Gilles Kahn.

A related tradition was also quite influential on the background of
the full abstraction problem, namely the characterizations of
semantical notions like continuity and sequentiality inside syntactic
models of the (untyped) (lambda)-calculus based on Böhm trees
(Barendregt 1984), mainly due to
Lévy and Berry (see Berry et al.
1985
and Curien 1992) for
accounts of the search for fully abstract models of PCF along this
line).

The basic papers on full abstraction for PCF are Milner
1977; Plotkin 1977
. They can be read
together as giving a coherent picture of the semantic analysis of this
language. An independent approach to full abstraction came from the
Russian logician Vladimir Sazonov who characterized definability in
PCF in terms of a certain class of sequential computational strategies
(Sazonov 1975, 1976). His work, however,
had no direct influence on the bulk of research on the full
abstraction problem, and only recently there have been attempts to
relate Sazonov’s characterization to the game theoretic
approaches (Sazonov 2007).

Another, completely different approach to full abstraction, exploits
special kinds of logical relations in order to isolate
quotients of the continuous model. The first use of logical relations
in the context of the problem of full abstraction is Mulmuley
1987
, but the resulting construction
of a fully abstract model is obtained by brute force and therefore is
not what the full abstraction problem searches for. Later, Sieber
(1992) and O’Hearn & Riecke
(1995)
have employed refinements of this technique to gain a
better insight into the structure of the fully abstract models,
characterizing the definable elements of the standard continuous model
by means of invariance under special logical relations cutting out the
non-sequential functions.

Detailed accounts of the full abstraction problem for PCF can be found
in Gunter 1992 (chaps 5,6), Streicher
2006
, Ong
1995
, Stoughton 1988 and Amadio
& Curien 1998 (chaps 6, 12, 14)
, in
approximately increasing order of technical complexity. The emphasis
on the recursion-theoretic aspects of PCF and its full abstraction
problem are dealt with in detail in the textbook (Longley
& Normann 2015: chaps 6, 7
); a
shorter account can be found in Longley 2001
(sec. 4)
.

3. Game semantics

3.1 Full completeness

Theorem 2.2
highlights the fundamental role of definability of finite elements in
the fully abstract model of PCF, an aspect that has been stressed
recently in Curien 2007. As a smooth
transition to the formalisms based on games, and partly following the
historical development of the subject, we pause shortly to examine
another aspect of definability that arises at the border between
computation and the proof theory of constructive logical systems. It
has been a remarkable discovery that the structure of natural
deduction proofs for, say, the implicative fragment of intuitionistic
propositional calculus is completely described by terms of the simply
typed (lambda)-calculus, where a provable propositional formula of
the form (sigma to tau) is read as the type of the terms
representing its proofs. This is the propositions-as-types
correspondence
, to be attributed to Curry, de Bruijn, Scott,
Läuchli, Lawvere and Howard, which extends to much richer formal
systems (for a history of this notion see Cardone
& Hindley 2009: sec. 8.1.4
).

The existence of this correspondence makes it possible to speak of a
semantics of proofs, that extends to constructive formal
proofs the denotational interpretations of typed (lambda)-calculi,
and in this context it also makes sense to ask whether an element
(x) of some (D^sigma) in a model of a typed (lambda)-calculus
is the interpretation of some proof of formula (sigma). A further
question asks whether every element of (D^sigma)
satisfying a suitably chosen property is the interpretation of a proof
of formula (sigma). Suitable properties may be for example
invariance under logical relations, suitably defined over each
(D^sigma), like in several results of Plotkin, Statman and others
summarized in Barendregt, Dekkers, &
Statman 2013 (I.3, I.4)
. We can read the latter question as
asking for a strong form of completeness for that system called
full completeness (Abramsky &
Jagadeesan 1994
), whose definition can be better understood in
a categorical semantics of systems of constructive logic. It is common
to interpret formulas (A) of such systems as objects ({lll A rll
}) of suitable categories (mathbb{M}), and proofs (p) of
sequents (A vdash B) as morphisms (lll p rll : lll A rll
longrightarrow lll B rll). While ordinary completeness states that
for every valid sequent (A vdash B) the set (mathbb{M}({lll A
rll },{lll B rll })) of morphisms is not empty, in the present
setting full completeness expresses the stronger requirement that
every morphism (f: lll A rll longrightarrow lll B rll) in a
semantical category (mathbb{M}) arises as the interpretation of
some proof, i.e., (f = {lll p rll }) for some proof (p) of the
sequent (A vdash B). Full completeness results have been proved for
several subsystems of linear logic Girard
(1987)
, see Abramsky (2000) for a
general framework. Furthermore, it has also suggested techniques for
achieving the definition of models of PCF enjoying the strong
definability property required by intensional full abstraction.

3.2 Interaction

In our description of the refinements to the continuous model of PCF
in order to guarantee the definability of finite elements at each
type, we have progressively come closer to an interactive explanation
of computation. For example, the action of a sequential algorithm (M
to M’) (Curien 1986: sec. 3.4)
exploits an external calling agent which triggers a cycle of requests
and responses on input cells leading (possibly) to the emission of an
output value. That interaction should be a central notion in the
analysis of computation, especially in relation to full abstraction,
is perhaps a natural outcome of the observational stance taken in the
definition of operational equivalence. Our short account of game
semantics starts precisely from an analysis of a general notion of
interaction as a motivation to a first formalization of games
which is however rich enough to provide a universe for the
interpretation of a restricted set of types and terms. Later we shall
add to this definition of game and strategies the features needed to
express the constraints that allow strategies to characterize
precisely higher-order, sequential computations, which is the aim set
for denotational semantics by the full abstraction problem. The
present account of the conceptual background of game semantics owes
much to the work of Abramsky and Curien (Abramsky
1994, 1996, 1997; Curien 2003a
).

The relevant notion of interaction has been isolated as the result of
contributions that come from widely different research areas
intensively investigated only in relatively recent years, notably
linear logic (Girard 1987) and the
theory of concurrent processes. It is in these areas that a notion of
composition as interaction of modules takes shape.
We give here just a simple example where the composition of modules in
the form of “parallel composition + hiding” is found in
nature, in order to connect it with the origin of this idea in the
semantics of concurrent processes developed by Hoare (1985),
and also to afford a first glimpse into
a simplified game formalism.

Consider a module (S) with four channels labeled
(a_textrm{in},a_textrm{out},r_textrm{in},r_textrm{out}). The
module is intended to return on channel (a_textrm{out}) the
successor of the number (n) incoming through channel
(a_textrm{in}), therefore its behavior can be specified as
follows:

  • (S) receives an input signal (mathbf{?}_textrm{in}) on
    channel (r_textrm{in}), then
  • emits a signal (mathbf{?}_textrm{out}) on channel
    (r_textrm{out}), and
  • waits for a value (n) on channel (a_textrm{in}) and then,
    after receiving it,
  • emits a value (n+1) on channel (a_textrm{out}).

(This pattern of interaction is formally identical to the
handshake protocol which is used in hardware design to
synchronize components in order to avoid hazards caused by
interference of signals.) This behavior can be mapped on the channels
as follows:

[a rectangular box with the letter S inside, on the upper left a line ending in a white circle is labeled '?_in', on the lower left a line ending in a black circle is labeled 'n+1_out', on the upper right a lined ending in a black circle is labeled '?_out', and on the lower right a line ending in a white circle is labeled 'n_in']

Figure 1: A module for the successor
function.

where (circ) means input or, more generally, a passive
involvement of the module in the corresponding action, whereas
(bullet) means output, or active involvement in the
action. We can describe the behavior of (S) using traces
(Hoare 1985), i.e., finite sequences of
symbols from the infinite alphabet ( alpha S = {{
mathbf{?}_textrm{in},mathbf{?}_textrm{out},n_textrm{in},m_textrm{out}
}}: )

[tau S = {{ varepsilon,mathbf{?}_textrm{in},mathbf{?}_textrm{in} mathbf{?}_textrm{out}, mathbf{?}_textrm{in} mathbf{?}_textrm{out} n_textrm{in},mathbf{?}_textrm{in} mathbf{?}_textrm{out} n_textrm{in} n+1_textrm{out},ldots }}]

If we consider another instance (S’) of (S)
with alphabet ( alpha S’ = {{
mathbf{?}_textrm{in}’,mathbf{?}_textrm{out}’,n_textrm{in}’,m_textrm{out}’
}} ) we can compose (S) and (S’) by identifying (= connecting)
channels (r_textrm{out},r_textrm{in}’), and (a_textrm{in} ,
a_textrm{out}’), and the signals passing through them, as shown:

[begin{aligned}
mathbf{?}_textrm{out} , mathbf{?}_textrm{in}’ &leadsto x\
n+1_textrm{in} , n+1_textrm{out}’ &leadsto yend{aligned}]

This represents the parallel composition of the modules,
(S | S’):

[two rectangles, the one on the left containing 'S' and the one on the right 'Sprime'. The upper left of the left one has a line ending in a white circle labeled '?_in', on the lower left a line ending in a black circle labeled 'n+2_out'. The upper right of the right rectangle has a line ending in a black circle labeled '?prime_out' and the lower right a line labeled 'nprime_in'. The two rectangles are connected by two lines. The upper with a black circle on the left and a white circle on the right labeled 'x'. The lower with a white circle on the left and at black circle on the right labeled 'y'.]

Figure 2

The behavior of the compound module is described by the set of traces

[{{ varepsilon,
mathbf{?}_textrm{in},
mathbf{?}_textrm{in} x,
mathbf{?}_textrm{in} x mathbf{?}_textrm{out}’,
mathbf{?}_textrm{in} x mathbf{?}_textrm{out}’ n_textrm{in}’,
mathbf{?}_textrm{in} x mathbf{?}_textrm{out}’ n_textrm{in}’ y,
mathbf{?}_textrm{in} x mathbf{?}_textrm{out}’ n_textrm{in}’ y n+2_textrm{out}, ldots }}]

The symbols (x,y) can now be hidden, representing the
behavior of the final system

[two rectangles with dashed lines for sides, the one on the left containing 'S' and the one on the right 'Sprime'. The upper left of the left one has a line ending in a white circle labeled '?_in', on the lower left a line ending in a black circle labeled 'n+2_out'. The upper right of the right rectangle has a line ending in a black circle labeled '?prime_out' and the lower right a line labeled 'nprime_in'. The two rectangles are connected by two dashed lines. The upper with a black circle on the left and a white circle on the right and not labeled. The lower with a white circle on the left and at black circle on the right and not labeled. A larger rectangle with solid sides encloses the two rectangles so that the left side of the left rectangle and the right side of the right rectangle are on the left and right sides of the enclosing rectangle]

Figure 3

whose traces have the required form

[{{
varepsilon,
mathbf{?}_textrm{in},
mathbf{?}_textrm{in} mathbf{?}_textrm{out}’,
mathbf{?}_textrm{in} mathbf{?}_textrm{out}’ n_textrm{in}’,
mathbf{?}_textrm{in} mathbf{?}_textrm{out}’ n_textrm{in}’ n+2_textrm{out}, ldots }}.]

This example contains
many of the ingredients on which game semantics is based. There is the
idea of a System, whose behavior is triggered by an incoming request
from its Environment: in a game formalism these are the roles of
Proponent and Opponent in a two-person game. The behavior of each
module is described as the trace of its possible interactions with
other agents, and the behaviors can be composed by a peculiar change
of role whereby the module who plays as System (in the above example,
(S) emitting a request signal on channel (r_textrm{out})) is made
to behave as Environment with respect to (S’) when this signal is
received in input on channel (r_textrm{in}’). Let us see how this
example can be generalized.

3.3 Games and strategies

We only give the definitions needed to understand the basic
constructions on games and to see how these form a category, following
Abramsky 1997 and Hyland 1997 that
contain more formal details and proofs.

3.3.1 Games

Definition 3.1 A game (G) is specified by
giving a set of moves (M_G), a labeling (ell_G)
of the moves as either moves of the Proponent ((P)) or as
moves of the Opponent ((O)). Furthermore, there is a set of
positions (P_G) which is a set of sequences of moves where:
(1) the two players alternate, starting with (O); (2) if (s in
P_G) then every prefix (s’) of (s) is also in (P_G).

As an example, consider a game associated with the data-type of
Boolean values, (G_texttt{bool}). There are three possible
moves,

  • an (O)-move (?_texttt{bool}) and
  • two (P)-moves (textit{tt}, textit{ff})

(i.e., (ell_texttt{bool}(?_texttt{bool}) = O,
ell_texttt{bool}({textit{tt}}) = ell_texttt{bool}({textit{ff}})
= P)). The positions in this game are

[?_{texttt{bool}}, ?_{texttt{bool}} texttt{tt}, ?_{texttt{bool}} texttt{ff}:]

think of
(?_texttt{bool}) as a cell (as in a concrete data structure) which
can be filled by one of the two values ({textit{tt}}) and
({textit{ff}}), or as a question by the Opponent that admits as
answers by the Proponent either ({textit{tt}}) or
({textit{ff}}). Similarly we can describe a game (G_texttt{num})
with an (O)-move (?_texttt{num}) and (P)-moves (n in
{mathbb{N}}).

3.3.2 Strategies and their composition

The players move in a game (G) alternately, at each move reaching a
legal position in (P_G). Their behavior is best thought of as
describing a strategy that prescribes deterministically what is
(P)’s response to (O) in a position where it is its turn to
move.

Definition 3.2. A strategy (sigma) on a
game (G) is a prefix-closed set of even-length positions of (G)
such that, each time (sab,sac in sigma), we have (b=c).

For example, the strategies on (G_texttt{num}) are (varepsilon)
and all sequences (?_texttt{num}n), corresponding respectively to
the elements (bot) and (n) of the domain ({mathbb{N}}_bot).

We would like to consider the behavior of the successor module
described above as an element of a set (G_texttt{num}multimap
G_texttt{num}) of strategies that compute functions over the natural
numbers. If we consider only the sequences of interactions of (S)
taking place either on the left or on the right side of the module of
Figure 1,
we see that they describe positions of (G_texttt{num}), with an
inversion of polarity (active/passive) depending on which side the
interactions take place: the module is initially passive, and becomes
active upon receiving a request from the environment. Such inversion,
represented by the complementary labeling of the moves
(overline{lambda_G}), assigning to Proponent the moves of the
Opponent in (G) and conversely, is essential to the definition of a
game (G multimap H):

Definition 3.3. Given any pair of games (G,H), the
game (G multimap H) has moves (M_{G multimap H}) the disjoint
union (M_G + M_H) of the games (G) and (H), where

[lambda_{G multimap H} (m) =
begin{cases}
overline{lambda_{G}} (m) = &text{if (m in M_G),}\
lambda_{H} (m) = &text{if (m in M_H).}
end{cases}]

and a position in (P_{G multimap H}) is any alternating sequence
(s) of moves (of (M_{G multimap H})) whose restrictions (s
upharpoonright M_G,s upharpoonright M_H) to the moves in (G) and
(H), respectively, are positions of (G) and (H).

The strategy that interprets (texttt{succ}:texttt{num}to
texttt{num}) corresponds to the behavior of the module (S) used
above as a guiding example. The parallel composition + hiding approach
used to compose two instances of the successor module can now be
reinterpreted as composition of strategies, suggesting a general
pattern:

[begin{array}{ccc|c|cccc|c}
G_texttt{num}& multimap & G_texttt{num}&& & G_texttt{num}& multimap & G_texttt{num}& \ hline
& & && & & & mathbf{?}_textrm{in} & O\
& &mathbf{?}_textrm{in}’ &O& &mathbf{?}_textrm{out} && &P\
mathbf{?}_textrm{out}’& & &P& & & & &O\
vdots& & & vdots && & & &vdots \
n_textrm{in}’& & &O& & & & & P\
& & n+1_textrm{out}’ &P& &n+1_textrm{in}& & &O \
& & &O& & & &n+2_textrm{out} &P
end{array}]

Definition 3.4. The composition (tau circ sigma)
on (G multimap K) of strategies (sigma) on (G multimap H) and
(tau) on (H multimap K) consists of the sequences of moves of
(M_G + M_K) obtained by hiding the moves of (M_H) from the
sequences (s) of moves in (M_G + M_H + M_K) such that (s
upharpoonright G,H) is in (P_{G multimap H}) and (s
upharpoonright H,K) is in (P_{H multimap K}).

There is one strategy that deserves a special name, because it is the
identity morphism in the category whose objects are games and whose
morphisms from (G) to (H) are the strategies on (G multimap H).
The copy-cat strategy (textsf{id}) on (G multimap G) is
defined as the set of sequences of moves (s) such that the
restriction of (s) to the left instance of (G) coincides with its
restriction to the right instance.

3.4 Special kinds of strategies

The game formalism just introduced is not detailed enough to
characterize the kind of sequential computation at higher types needed
to achieve definability. For this purpose, a richer structure on games
is needed, making them closer to dialogue games between
Proponent and Opponent exchanging questions and
answers. This allows to formulate restrictions on plays by
matching answers with the corresponding questions in an appropriate
manner. The strategies for this refined game notion, that we study
next essentially through examples, will yield a richer notion of
morphism between games, allowing to make finer distinctions of a
computational nature needed for intensionally fully abstract model of
PCF, following essentially the approach of Hyland
& Ong (2000)
drawing also material
from Abramsky & McCusker (1999) and
Curien (2006).

The moves of the refined game notion will be either questions
or answers played by Proponent or by the Opponent. We have
then four classes of moves each represented by a kind of (round or
square) bracket: Proponent’s questions ‘(’;
Opponent’s answers ‘)’; Opponent’s questions
‘[’; and Proponent’s answer ‘]’. This
labeling of the moves subsumes under the usual well-formedness
criterion for bracket sequences, at one time: the alternation between
Proponent and Opponent, the fact that Opponent is the first to move
and that each answer of a player answers a unique question of the
partner. This is not enough, however: a further justification
structure on questions and answers is needed to discipline the nesting
of (sub)dialogues in the evaluation of higher-order functions,
allowing to characterize the well-bracketed strategies.
Consider now the strategy in ((G^{11}_texttt{bool}to
G^{12}_texttt{bool}to G^1_texttt{bool}) to G_texttt{bool}),
described informally using a labeling of the copies of
(G_texttt{bool}) as shown:

  • (1)Opponent asks
    question (?) in (G_texttt{bool});
  • (2)Proponent asks
    question (?_1) in (G^1_texttt{bool}), justified by (?), in
    order to know about the output of the input value (f);
  • (3.1)
    if Opponent asks
    question (?_{11}), Proponent answers ({textit{tt}}) in
    (G_texttt{bool}): the computation examines first the first argument
    of (f);
  • (3.2)
    if Opponent asks
    question (?_{12}), Proponent answers ({textit{ff}}) in
    (G_texttt{bool}): the computation examines first the second
    argument of (f);

Here, the Proponent’s moves at steps (3.(i)) answer the
question asked by Opponent at step (1), not the questions asked by the
Opponent at steps (3.1), (3.2) that are still pending. This violates a
“no dangling question mark” condition on dialogues
introduced under this name by Robin Gandy in his unpublished work on
higher-type computability (and well-known in the tradition of game
semantics for intuitionistic logic initiated by Lorenzen (1961)).
Strategies such as these interpret
control operators that do not exist in the fully abstract game model
of PCF, but do exist, for example, in the model based on sequential
algorithms (Curien 1986: sec. 3.2.7,
3.2.8
). A different phenomenon occurs in a variation of the
previous example:

  • (1)Opponent asks
    question (?) in (G_texttt{bool});
  • (2)Proponent asks
    question (?_1) in (G^1_texttt{bool});
  •   (3.1)  if
    Opponent asks question (?_{11}),
    Proponent answers ({textit{tt}}) in
    (G^{11}_texttt{bool});
  •    (3.1.1)   if
    Opponent answers ({textit{tt}}) in
    (G^1_texttt{bool}), Proponent answers ({textit{tt}}) in
    (G_texttt{bool});
  •   (3.2)  if
    Opponent answers ({textit{tt}}) in
    (G^1_texttt{bool}), Proponent answers ({textit{ff}}) in
    (G_texttt{bool})

Here the strategy prescribes a response to the moves by Opponent
depending on the internal detail of the latter’s behavior. The
response prescribed to Proponent by the strategy to the initial
question should not depend on what happens between the
Proponent’s question (?_1) and the Opponent’s answer
({textit{tt}}). This is the property of innocence, that
limits the amount of detail that a strategy for (P) can access. For
this reason, failure of innocence allows strategies to model storage
phenomena.

This gives us the necessary terminology to understand the statement of
the intensional full abstraction theorem proved in Hyland
& Ong 2000 (th. 7.1)
, where
the types of PCF are interpreted as games and terms as innocent and
well-bracketed strategies, see also Abramsky et
al. 2000 (th. 3.2)
, Curien 2006
(th. 5.1)
:

Theorem 3.1. For every PCF type (sigma = sigma_1
to cdots to sigma_n to kappa) with (kappa = texttt{num}) or
(kappa = texttt{bool}), every (compact) innocent and
well-bracketed strategy corresponds to the denotation of a closed
term.

This closes our quick overview of game semantics applied to the full
abstraction problem for PCF, but opens a broad research area in the
classification of programming disciplines according to the possible
combinations of restrictions (innocence, well-bracketing) on general
strategies for games as defined above. An introductory picture (the
“semantic square” by Abramsky and his students) of this
landscape, that we leave to the contemplation of the reader, can be
found in Abramsky & McCusker
1999
.

3.5 Historical notes and further readings

Games as a semantic framework have a longstanding tradition, from
ancient logic onwards. Here we list of the main sources and further
readings pertaining to game semantics applied to programming
languages.

The use of game semantic for dealing with the full abstraction problem
for PCF originates from Abramsky et al.
2000
and Hyland & Ong 2000.
Hanno Nickau (1994) proposed
independently a game model similar to that of Hyland and Ong: their
games are sometimes called collectively “H2O
games”.

As a background for game semantics, from intuitionistic logic we have
the very early Lorenzen (1961) on
dialogue games, then from linear logic Lafont and Streicher (1991)
and Blass (1992)
and from Coquand’s game
theoretical analysis of classical provability (Coquand
1995
). From combinatorial game theory
the categorical account by Joyal (1977),
“the first person to make a category of games and winning
strategies” according to Abramsky &
Jagadeesan (1994)
. A readable historical account of the first
uses of games in the interpretation of constructive logical
formalisms, especially linear logic, is included in Abramsky
& Jagadeesan 1994
. It should be
observed that games for logic require winning strategies in order to
capture validity, an issue that we have not dealt with at all in this
entry.

Connections with concrete data structures were first noticed by Lamarche
(1992)
and Curien
(1994)
, see Curien
2003b
. Antonio
Bucciarelli (1994)
explains the connections between
Kleene’s unimonotone functions and concrete data structures: the
use of dialogues in the former is mentioned in Hyland
& Ong 2000 (sec. 1.4)
.

Finally, among the introductions to game semantics for PCF and other
languages, we suggest Abramsky 1997; Abramsky
& McCusker 1999
. The latter also contains a description of
the applications of game semantics to imperative languages, notably
Idealized Algol. Other excellent introductions to game semantics are
Hyland 1997 and Curien
2006
. A broad account of the use of
games in the semantics of programming languages with many pointers to
Lorenzen games, and intended for a philosophical audience, is Jürjens
2002
.

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