SYNTAX AND SEMANTICS FOR CINNAMON PROGRAMMING

International Journal of Computer Science & Information Technology (IJCSIT) Vol 9, No 5, October 2017
DOI:10.5121/ijcsit.2017.9510 127
SYNTAX AND SEMANTICS FOR CINNAMON
PROGRAMMING
Kostadin Kratchanov
Department of Software Engineering, Yaşar University, Izmir, Turkey
ABSTRACT
Cinnamons are a new computation model intended to form a theoretical foundation for Control Network
Programming (CNP). CNP has established itself as a programming approach combining declarative and
imperative features. It supports powerful tools for control of the computation process; in particular, these
tools allow easy, intuitive, visual development of heuristic, nondeterministic, or randomized solutions. The
paper providesrigorous definitions of the syntax and semantics of the new model of computation, at the
same time trying to keep the intuition behind clear. The purposely simplified theoretical model is then
compared to both WHILE-programs (thus demonstrating its Turing completeness), and the “real” CNP.
Finally, future research possibilities are mentioned that would eventually extend the cinnamon
programming and its theoretical foundation into the directions of nondeterminism, randomness and
fuzziness.
KEYWORDS
Control network programming, CNP, Programming languages, Programming paradigms, Computation
models, While programs, Theoretical computer science, Recursive automata, Nondeterminism, Semantics
1. INTRODUCTION
This paper is an extended version of the conference publication [1] where cinnamons were first
introduced. Cinnamons are a model of computation (like Turing machines, recursive functions,
while-programs, RAM machines, etc.) to serve is a theoretical model underlying Control Network
Programming. The next section is a brief introduction to and a summary of Control Network
Programming.
1.1. Control Network Programming
Control Network Programming(or CNP) [2-6] is an unusual style of programming in which the
program can be visualized as a set of graphs. The (main part of the) program is called a Control
Network (CN). Control Network Programming can be deciphered as “programming through
control networks”.
Fig. 1a shows the CN of a CNP application. This CN programsimulates the work of the
nondeterministic finite automaton whose graph is shown in Fig. 1b. The NFA accepts strings over
the alphabet {a, b} that include exactly two a’s or at least two b’s. Fig. 1c shows the output after
the execution of the CN program when the input string abbabbb has been entered. The
screenshots in Fig. 1a and 1c are taken from the CNP integrated development environment
SpiderCNP [6-8]environment for developing and running CN programs.In particular, CNs can be
created, tested and edited using a built-in graphic editor. The IDE also provides error messaging,
tracing facilities, etc. All illustrations of CNs in this paper are screenshots from this IDE.
SpiderCNP was developed by T. Golemanov.

128
Fig. 1a CN of an NFA simulator
Fig. 1b NFA

129
Fig. 1c Output
The CN of a CN program is a finite set of subnets.One of the subnets is called the main
subnet.Each subnet consists of nodes (states) and arrows.A chain of primitives is assigned to each
arrow.The primitives are elementary actions or elementary tests.A subnet can call other subnets or
itself.Both primitives and subnets can have parameters. CNs have a graphical representation
(shown in Fig. 1a) and a corresponding textual representation created automatically by the IDE.
During the execution, the system tries to find one or more paths between the initial node of the
main subnet and a final node, executing the primitives along the path. All the above notions will
be rigorously defined below.
In our example, action primitives are add(state: string),init, and print; test primitives are test(ch:
char) and complete. Primitives can be defined by the programmer, or imported from libraries. The
graphical representation of the CN is displayed in Fig. 1a, its textual representation – in Fig. 1d,
and the definitions of the primitives – in Fig. 1e.
Fig. 1d Textual view of CN

130
Fig. 1e Definitions of primitives
As proclaimed in [9], “Primitives + Control Network = Control Network Programming”.
The CN is in general nondeterministic.An interpreter (inference engine) must implement a
strategy for search (inference, computation) in a recursive network.The search involves traversing
the CN, executing the primitives along the way. Search is implemented by a method called
Extended Backtracking (in a recursive network, with inverse execution of action
primitives).Actually, no interpreter exists.The Spider compiler uses an approach similar to the
recursive decent technique well known in compilation. The resulting ‘intermediate program’
corresponds to the CN, but also embodies the algorithm for interpreting it.
The CN may be of:
• Non-procedural nature: then it is a declarative description of the problem – specifying the
problem not an algorithm how to solve it. Therefore, in that case, CNP is a kind of
declarative programming. It should be noted that as declarative programming CNP is
typically easier to understand than, for example, Prolog, due to its being a visual
programming.
• Procedural nature: the CN corresponds to an algorithm, e.g., to its activity diagram. In
this case CNP is imperative programming where the control structure is explicit and even
graphically represented.
Generally, CNP is a combination of declarative and imperative programming – the CN may be a
high-level non-procedural description of the problem, the primitives used in it are simple
procedures.
CNP can be classified as genuine visual programming. The program – the CN – has an explicit
structure represented as a set of graphs. The CNP programmer can visually “see” the program.
We believe “seeing” graphically a program makes it much easier to design, test and
understand,and this results in a faster process of developing a solution to the problem in hand
[2,6].

131
Clearly, the problem description in CNP can involve nondeterminism. It is also possible to
implement randomized algorithms. The programmer is provided with powerful tools for user
control [10,11]. Control features are also “visually seen” which results in “declarative”
implementation of control.
In summary, CNP makes programming easier. It allows for an easy, natural, and fast
“programming” of declaratively specified problems, problems involving nondeterminism or
randomness, problems that can be naturally visualized by a set of graphs. CNP is an effective and
efficient approach for programming many typical Artificial Intelligence problems, in particular
for implementing heuristic algorithms.
1.2. Cinnamons
Models of computation are fundamental for Computability theory. Numerous models have been
developed. The equivalence in the computational power of most of these models is one of the
most important and astonishing achievements in Theoretical Computer science. RAM-machines
and register machines are the theoretical foundation of contemporary computer architecture,
machine languages and assembly languages, lambda calculus - for functional programming, first-
order logic - for logic programming, WHILE-programs – for high-level imperative programming
languages. Cinnamons are being introduced to serve as a theoretical foundation for Control
Network Programming. They are a new computation model, and their equivalence to WHILE-
programs (and consequently to the other major computation models) is shown in Section 4.
A cinnamon is a simplest CN program. As cinnamons are intended for formal mathematical
study, they are a strictly minimal version of CN programs–they have the same structure, but with
only the most basic and fundamental features and primitives necessary. Sections 2 and 3 are
devoted to the syntax and semantics of the new computation model, respectively. Section 5
includes a concise description of more advanced features that distinguish the “real” CNP from
cinnamon programming while Section 6 sketches some ideas for future study.
Thename cinnamon programming is introduced as a simplified and easily recognizable
substitute for Core Control Network Programming, or Core CNP.
What in traditional programming corresponds to a program, is called here a cinnamon(the formal
definition of a cinnamon is given in the next section). A traditional program is a string in the
corresponding programming language. In contrast, a cinnamon is not simply a string but rather a
‘control network’ – a set of graphs. Cinnamon programming is a type of graphical (visual)
programming. Of course, at a lower level the CN is transformed into a string in an appropriate
language which is manipulated by the CNP compiler. However, in this presentation we prefer to
stay at the higher and more abstract level.
1.3. Background and Perspective
As Jones explicitly maintains in his Preface to [12], the two major computer science fields –
Theory of Computation (which splits into Computability and Computational complexity), and
Programming languages (including Syntax and Semantics) – have much to offer each other. This
overlap of concepts, approaches, and results can be described as a trend observed in the recent
decades, some examples being [13-15].We would like to follow this trend here and use the
terminology from both areas.
A model of computation is defined as a formal mathematical object, followed by the concept of
computation for that model and the notion of what a given specific model computes. Assume that
Turing machines are our computation model. Turing machines – acceptors are used for solving
decision problems, while Turing machines – transducers are used for solving function problems.

132
A Turing machine – acceptor solves a decision problem by accepting or rejecting the element. A
particular Turing machine – transducer calculates a partial function by producing an output
element for an input element.
Note that, ultimately, every Turing machine (including its program), or WHILE-program, etc. has
a unique description in some properly defined language. Therefore, we can talk about the
language of Turing machines, the language of WHILE-programs, and so on. A particular Turing
machine is an element (a well-formed sentence) of the language of Turing machines.
The description of a language is split into syntax (form) and semantics (meaning). Semantics
reveals the meaning of syntactically valid strings in a language. There is a wide range of
semantics proposed for programming languages (see, e.g., [16]). Two of the major approaches are
outlined next. Operational semanticsdescribes how a computation is performed internally in the
corresponding computation model (often also called an abstract machine) – that is, how the
program is interpreted (executed) in the computation model. In contrast, in denotational
semantics we give meaning to the program by specifying the external behavior achieved by the
computation, e.g., the partial function computed.
The above means that introducing a formal model of computation as a mathematical object (e.g.,
Turing machines) and defining how the computation in its terminology is performed (how a
Turing machine operates and what it computes) is actually equivalent to introducing a language
(the language of Turing machines) and specifying its syntax and semantics.
It is customary in mathematics to discuss objects without specifying their representation (talking
about sets of objects, functions between sets of objects, etc.). In computation theory the
representation of objects often plays a central role [17]. However, for clarity and simplicity, we
will still define, as much as we can, the notions and concepts we need using sets without regard
for the representation of their elements. Actually, computational tasks refer to objects that are
represented in some canonical way. The two such main and best studied alternative
representations are strings of symbols and natural numbers. In our presentation, when that must
be made explicit we have chosen to work with natural numbers, and correspondingly functions on
natural numbers.
2. SYNTAX
The syntax of the “language” of cinnamons is defined below.
2.1.Definitions
Let SV = {FINISH, RETURN} be a set of two distinct elements called system vertices(system
nodes, system states), FINISH≠RETURN. A graph is an ordered 6-tuple G = <S, A, source,
target, L, label> consisting of: a nonempty finite set, S of elements called ordinary states
(ordinarynodes, ordinaryvertices), S∩SV=∅; a finite set, A of arrows; two functions source: A
S and target: A (S ∪ SV) mapping an arrow a∈A into its sources∈S and targets’∈ (S ∪
SV) respectively; a set, L of labels; and a function label: A L∪{λ} assigning a label
label(a)∈L∪{λ} to each arrow a∈A. A state is either an ordinary state, or a system state. For a
given s∈S, we will denote by out(s) the set of all arrows with source s(called also arrows
outgoing from s). An initialized graph is an ordered pair <G, so> where G is a graph and so is a
selected state, so∈S called the initial state. Note that FINISH and RETURN have no outgoing
arrows, and that these system states cannot be initial. We will call a graph, G an ordered graph if
the set out(s) for any given vertex s∈S is linearly ordered. An ordered graph which is initialized,
is called an ordered initialized graph.

133
We will represent initialized graphs graphically, using the following graphical symbols:
Graphical
symbol
l
Element
Arrow with
label l
(Ordinary)
vertex
Initial vertex
FINISH
vertex
RETURN
vertex
In the SpiderCNPIDE, the graphical symbol for an initial node is an oval in green:
A subnet is an ordered pair <G,P> where G is an ordered initialized graph, and P is a list
(possibly empty) with elements called formal subnet parameters.
Let Vars be a countable set. Its elements will be called variables. We will denote the elements of
Var as x0, x1, x2, … Of course, Vars could be the set N of natural numbers itself.
A cinnamon (a substitution for Simple CN) is comprised of a nonempty finite set of subnets one
of which is identified as the main subnet. Formally, a cinnamon, Σ is an ordered pair <Ν, m>
where N is a set of subnets satisfying conditions 1) - 2) below, and m∈N is its main subnet.
This set of subnets must satisfy the following two properties:
1) The sets of vertices of the (graphs of the) subnets are mutually nonintersecting.
2) All subnets share the same label set, L. This label set consists of two types of elements:
primitives, and invocations (the latter also called subnet calls) described below.
Each primitive in the cinnamon may be from one of the following four types:
a) clear(x) (also denoted x:=0)
b) copy(x,y) (also denoted y:=x)
c) inc(x) (also denoted x:=x+1)
d) if nonEq(x,y) (also denoted if x≠y)
Primitives clear, inc, and copy are called elementary action primitives. Primitive if nonEq is
called the elementary test primitive.
Above, xand y are variables. Note that Σ consists of a finite set of subnets, each of which has a
finite set of labeled arrows. Therefore, the number of variables used in it is also finite. We will
denote by VarsΣ
the set of all variables used in the primitives of the cinnamon Σ. If ηis a subnet of
a cinnamon Σ, than Varsη
denotes the set of variables of η. The sets of variables of two distinct
subnets are nonintersecting. Clearly, VarsΣ
is the union of Varsη
for all subnets ηof Σ, VarsΣ
is
finite, and VarsΣ
⊂ Vars.
An invocation has the form:
3) CALLη (a0, a1, …, an-1) where η is a subnet of the cinnamon Σ, and a0 – an-1 are called
actual subnet parameters. Each ai is either a variable from VarsΣ
, or a constant (a
natural number).
The list of actual subnet parameters may be empty, in which case we simply write CALLη. The
number of actual parameters in the subnet call equals the number of formal parameters in the
subnet definition.
s s

134
Both direct and indirect recursion are allowed. For example, it is possible that the invocation
CALL(η) is located in subnet η, that is, subnet η is called from itself.
According to our definition above each arrow has a single label which is an elementary primitive
or a subnet call. This assumption is imposed for simplicity. In practice, it is more convenient to
allow an arrow’s label to be a finite, possibly empty, string of primitives and/or invocations. If
using the new notation, actually denotes
and (empty arrow) denotes .
An example of a cinnamon is shown in Figure 2.
Figure 2 Cinnamon example 1
Here, the cinnamon consists of two subnets, N1 and N2. N1 is the main subnet. It includes the
ordinary states {1, 2, 3}, as well as a RETURN system state. The initial state is 1. Primitives used
in N1 are {p1, p2, p3, p4}; the invocation CALL N2(a,5) is also used. In the invocation, a is an
actual subnet parameter; the second parameter in this call is the constant 5. Subnet N2 uses the
primitives {p1, p3, p4, p5, p6}, and the invocation CALL N1. Its set of states is {4, 5, 6, 7. 8}. N2
also includes two RETUN states and one FINISH state. Each one of the primitives p1 – p6 must
be of the forms discussed above, i.e., either an elementary action primitive or elementary test
primitive. Subnet N1 has no formal subnet parameters while subnet N2 has two formal subnet
parameters, x and y.This is an example of indirect recursion: subnet N2 is called from within
subnet N1, and N1 is called from within N2. Clearly, invocating the main subnet form the same or
other subnet is not a hindrance.
As emphasized, the arrows going out of a given node, are linearly ordered. For convenience we
accept that on the graphical representations the order of the arrows is from the left to the right and
up – down. For example, the arrow 1 2 in N1 is before arrow 1 3. If using this default rule on
the graphical representation is difficult, we may also show the order explicitly, as in Fig. 3. Here,
arrow 1 3 precedes arrow 1 2.

135
Figure 3 Indicating order of arrows
In our definition of a cinnamon above we have chosen a minimal set of allowed elementary
primitives consisting of the specified three elementary action primitives (x:=0, y:=x, and x:=x+1)
and one test primitive (if x≠y). As we will see further, this minimal set is functionally complete in
the sense that these primitives are enough for implementing any computation. It would have been
possible to choose alternative sets, e.g., action primitives {x:=0, y:=x+1, y:=x÷1}. As far as the
test primitive is concerned, its condition could have been replaced by any one of conditions =, <,
>, ≤, ≥ (similarly to [13,18] in the case of WHILE-programs). For our considerations, however,
our choice of elementary primitives proves to be the most convenient.
It is also possible to choose a different set of elementary primitives if working with a based-on-
strings representation of data. Let A be a fixed alphabet with at least two different symbols. We
could, for example, choose the following set of elementary primitives: clear(l) (creating an empty
string l=λ), sep(l,h,t) (separating a string l into its head h and tail t), cons(h,t,l) (constructing a
new string l with head h and tail t = adding a new symbol to the beginning of a string), ifEq(a,a1)
(comparing symbols), ifEmpty(l) (if l = λ).
It is worth mentioning that the concept of a cinnamon is an elaborated, filled with flesh version of
an introduced much earlier but not so widely used skeletal notion – the notion of a recursive
automaton / recursive finite-state automaton / recursive transition network / recursive control
graph [3,19-27]. It is a computation model equivalent in computational power to nondeterministic
push-down automata. Basically, the difference between an automaton and a recursive automaton
is that the latter is a set of nondeterministic automata in which transitions are labeled either by an
element of the input alphabet (a primitive in the case of cinnamons) or by the name of a
nondeterministic automaton from the set (a subnet in the case of cinnamons). Cinnamons are a
more complicated concept than recursive automata – for example, primitives and subnets have
parameters, the set of outgoing arrows is ordered, the behavior in the basic definition is
deterministic, there are concepts like failure and backward execution (the latter differences are of
semantic nature).
As is customary in computation theory literature, we can allow for convenience the usage of
macro primitives. A macro definition is a sequence of primitives. For example, the sequence
<clear(z), nonEq(x,z)>can be considered as a macro definition for the macro statement
nonZero(x) which would be a new test primitive. Such an abbreviation will be called a macro
statement (plural: macros).

136
Figure 4 Cinnamon for subtraction
2.2 Examples
A second example of a cinnamon is shown in Fig. 4. It consists of a single (main) subnet
Subtract. After understanding the semantics (computation) of cinnamons, one can convince
himself that this cinnamon computes the function subtraction of natural numbers; more precisely,
=
− , ≥ ;
0, < .
Macros are one method for separating parts of a cinnamon into a named module of some sort and
re-use them. Another, more powerful method is using subnets. In the following example (Figure
4), we use a modified version of the previous example – the cinnamon for subtraction, but now it
is not a complete cinnamon but rather a subnet within a cinnamon. The cinnamon has four
subnets. One is subnet subtr already discussed. Subnet dec ‘defines’ a new ‘operation’ –
decrement, using thesubnet for subtraction. Using dec, subnet sum ‘defines’ another new
‘operation’ – summation. The overall control structure is determined by the main subnet. This
example actually implements the function
if x≠y then z:=x+y else z:=0
Note that ordering of outgoing arrows from a certain node plays an important role. For example,
according to the default ordering of arrows outgoing from node s0 in subnet main arrow
s0 FINISH labeled if nonEq(x,y), call sum(x,y,z) precedesthe arrow labeled clear(z), and the
loop s2 s2 in subnet sum precedes arrow s2 RETURN.
Our next example (Fig. 6) illustrates a cinnamon based on the definition with data representation
using strings instead of natural numbers. Here, copy(t,l) is a macro statement with macro
definition <sep(l,h,t, cons(h,t,l)> (l := t).
Assuming that the alphabet A includes symbols a, b, c, +, and *, this cinnamon plays the role of a
recognizer for arithmetic expressions defined by the (ambiguous) grammar
E E + E / E * E / a / b / c
Equivalently, this grammar can be defined by the syntax diagram in Fig. 7. The structure of the
graph of the cinnamon duplicates that of the syntax diagram. The left recursion has been avoided.

137
Above, a cinnamon was defined abstractly using purely mathematical notions (such as sets and
graphs) and notations. Alternatively, we could’ve had defined a formal language, or a
programming language in which a cinnamon is specified. For our purposes in this paper the more
Figure 5 A larger example
abstract approach is preferable as it presents the concepts in a much more clear and neat manner.
In reality, for developing and running CN programs, one usually uses an integrated development
environment such as SpiderCNPor Bouquet [6].

138
3. SEMANTICS
In the previous section we presented the syntax of the ‘language’ of cinnamon programming, in
other words, what syntactically a cinnamon is. As already discussed this is technically not a
language as a cinnamon, unlike a program, is not a string but rather a set of graphs.
Figure 6 Recognizer of arithmetic expressions
Figure 7 Syntax diagram of the grammar of arithmetic expressions
Now we turn our attention to the semantics of a cinnamon – in other words, what a cinnamon
‘computes’. This specification of the computation process can be considered as an operational
semantics. One can also consider a partial function which is defined by this computation thus

139
saying that a cinnamon as a syntactic construct specifies this function – and that can be
considered as a denotational semantics.
3.1Informally On The Computation In A Cinnamon
We will describe first the ‘computation’ performed in a cinnamon, and then will address the
semantics of the model more rigorously. Our exemplary cinnamon (which is a simplified version
of the ‘technical example’ from [3]) is shown below.
Figure 8 Technical example
The computation starts from the initial state of the main subnet by ‘forward’ traversal along the
arrows, and executing each primitive on the way. An elementary test primitive can be executed
successfully, or unsuccessfully. After unsuccessful execution, the direction of the traversal is
switched to ‘backwards’. How a primitive is executed forwards and backwards is given in its
definition. The traversal strategy is an extended version of backtracking. When the control is
returned to a state, the next not attempted yet outgoing arrow is tried. If no non-attempted
outgoing arrows exist, the control goes backwards through the arrow along which the state was
reached. A very interesting point of cinnamon programming is that the backward traversal can
enter backwards a subnet, and continue backwards along the arrow used before. This is not
possible in traditional programming because when completing a procedure all information from
inside the procedure is lost. In contrast to traditional procedure/function call in programming
languages, in cinnamon programming the data are not restored from a data stack but are restored
by backwards execution. If the traversal reaches a FINISH node than the computation finishes
successfully. If the traversal gets stuck in the initial node of the main subnet then the computation
finishes unsuccessfully – no solution has been found.
One possible traversal is shown in Fig. 9.The execution starts from the initial state, 1 of the main
subnet, Alpha. Action primitive A is executed forwards, then test primitive if C is executed
successfully. Next invocation Call Beta is executed, after which the control is in the initial state,
10 of subnet Beta. Primitive I is now executed, and control moves to state 11. However, state 11

140
has no outgoing arrows, and mode of traversal changes to ‘Backwards’. Primitive I is now
executed backwards and control is back in state 10. The next in order not-attempted outgoing
arrow is the one with primitive P which is now executed. Then test primitive if U is successfully
executed, RETURN state is reached, and control jumps back to subnet Alpha, more specifically to
state 4. Test primitive if D follows, but its execution is unsuccessful. Therefore, backwards
execution is triggered. Primitive if D is executed backwards, and control returns to state 4. There
are no other outgoing arrows from 4. Therefore, subnet Beta is entered backwards. The system
remembers that the last executed arrow before returning from subnet Beta was the one with label
if U. Now, if U is executed backwards. State 12 has no remaining not-attempted arrows, so
control continues moving backwards and primitive P is executed backwards. Now the control is
in state 10. Outgoing arrow S has not been attempted, so the mode changes back to ‘Forward’ and
primitive S, and then T and if U are executed forwards. Now, return from subnet Beta is
performed and control reaches state 4. Assume that test primitive if D is now successful. Thus, the
control reaches state FINISH, and the computation completes successfully.
Figure 9 One possible execution in the technical example
Now we can proceed to a formal definition of the cinnamon semantics.
3.2 Formal Semantics
Let Vars be the countable set of variables defined earlier in the Section on Syntax. Let Σ be a
given cinnamon, and VarsΣ
⊂Vars be its finite set of variables. As a matter of fact, we never use
values of Vars outside of VarsΣ
but the definitions remain simpler for Var. We will also need two
special ‘system variables’ FORW and FAILURE. We assume that SV = {FORW, FAILURE}, SV
∩ Vars = ∅, and Varss = SV ∪ Vars.
An environment, ε is a partial function Varss N. Terms with similar meaning used in
literature are state of computation, store, single-assignment store. If for v ∈ Varssthe partial
function σ is defined then the natural number ε(v) is called the value of variable v. If ε(v) is not
defined, we will also say that the value of variable v is not defined. The values ε(FORW) and
ε(FAILURE) can only be 0 or 1, therefore the two system variables will more often be called
system flags. The set of all environments is denoted Env.
Informally, VarsΣ
imitates the data store (operational unit) in our model. We also need a control
unit which in cinnamon programming is much more complex than in most other computation
models as cinnamon programming is intended for declarative solution of problems of
nondeterministic nature. Following our general strategy in this paper, we will employ
mathematical definitions rather than explicitly specifying the abstract machine or use
programming style terminology. However, in order to simplify the presentation, we will still use

141
popular notions such as a stack of a particular type of elements for example, instead of describing
it in formal mathematical language as a function N Elements.
We will define the operational semantics of cinnamons by presenting formally an interpreter. The
algorithm of this interpreter is specified by its UML activity diagram shown in Figure 10.
The corresponding virtual machine includes as components two stacks and a number of global
variables. Stack arr_ST stores the current path of arrows and its elements are ordered pairs of
theform <ARR, Remaining>. ARR can take as values the name of any of the arrows in
thecinnamon,as well as the special value separator which signals that jump to a subnet has been
performed. Stack subnet_ST keeps information about the nested subnets invoked. Its elements
have the format <state after the subnet call, last value of Remaining when the call was made,
values of parameters

142
Figure 10a Activity diagram of the interpreter I
Figure 10b Activity diagram of the interpreter II

143
of the calling subnet, values of local variables of the calling subnet>. The global variables are: the
system flags FORW and FAILURE, the current state STATE of the CN, the list (stack) Remaining
of the outgoing arrows for the current state that have not been attempted yet, current arrow ARR,
list PARAM of values of the parameters of the current subnet, list LOCALS of the values of the
local variables of the current subnet, current primitive PRIM to be executed. The following
notations have been also used in the activity diagram: init(SN) is the initial node of subnet SN,
out(STATE) is the list of arrows outgoing from STATE, and target(ARR) is the state which is the
target of arrow ARR.
The interpreter presented above specifies the operational semantics of the language of cinnamons,
and is a theoretical model only. Instead of an interpreter, the SpiderCNP IDE contains a
recursive-decent-type compiler [3,6-8].
Note that the activity diagrams Process_Arr_Forw and Process_Arr_Backwcontain actions called
‘execute’. This stands for ‘execute current primitive PRIM’. In the former activity diagram, the
execution is forwards, while in the latter it is performed backwards.
The backwards execution involves restoring the data. That means that normally the action
primitives must have two types of action – forward action (when moving forwards) and backward
action (when moving backwards and the data are being restored). Recall that the primitives can be
of two types: elementary action primitives and elementary test primitives. Only execution of
action primitives (clear, copy, and inc) changes the environment. Execution of test primitives
does not affect the environment. We can easily define a backward action for inc. However, there
is no natural way to define a backward action to clear and copy. Technically, their backward
action is empty; in reality, a programmer never uses these two action primitives in positions in the
CN where a backward action might be necessary. Values of system flags may be changed by an
action primitive, or directly by the control unit.

144
Figure 10c Activity diagram of the interpreter III
If a cinnamon is started in an initial environment ε, then in the course of the ‘computation’ the
control moves in accordance with the activity diagram, and from time to time action primitives
will be executed thus changing the values of variables. If and when the cinnamon halts, the final
environment will in general be different from ε. We thus interpret a cinnamon Σ as a partial
function [[Σ]] : Env Env. The value [[Σ]] (ε) is the final environment after executing the
cinnamon Σ with initial environment ε, provided Σ halts. If Σ does not halt when started in initial
environment ε, then [[Σ]] (ε) is undefined. Instead of saying ‘Executing the cinnamon Σ’ one can
equivalently say ‘The cinnamon Σ computes’ the function [[Σ]]. This function is the denotational
semantics of Σ. Above, we use emphatic brackets [[ ]] following the tradition in studies on
denotational semantics.
The followingmore formal definition of the execution of each of the allowed primitives follows
elements from [18] for the case of while-programs. For ε∈ Env, x ∈ Varss, and a ∈ N, let ε[x 
a] denote the environment that is identical to ε except for the value of x, which is a. Formally,
ε [x  a](y) = ε (y) if y≠x, and ε [x  a](x) = a.

145
The execution of the action primitive has the following semantics:
The execution of the elementary test primitive nonEq(x,y) changes the value of the system
variable FAILURE only. Formally, its semantics is as follows:
ε, if ε(x)≠ε(y) & FORW = 1
[[noneq(x,y]](ε) = ε[FAILURE1], if ε(x)=ε(y) & FORW = 1
ε , if FORW = 0
As we discussed, the interpretation of a cinnamon Σ as a partial function [[Σ]] : Env Env. At
the same time, for any given natural number j, we might want to consider a cinnamon Σ as an
agent computing a j-ary partial function fΣ: Nj
N. If we want to emphasize the arity of the
function, we will write fΣ
(j)
: Nj
N.
Let a cinnamon Σ and a natural number j be given. Let n be the number of variables of the
cinnamon Σ. A partial function fΣ: Nj
N defined in the following way, is called the partial j-
ary function computed by Σ
Σ
Σ
Σ :
a) Suppose j < n. Let <a1, a2, …, aj>∈Nj
. Let ε∈ Env be an environment such that ε(xi) =
aifor any i, 1 ≤ i ≤ j, ε(x0) = 0, and ε(xi) = 0 for any i > j. Then fΣ(a1, a2, …, aj) is defined
iff [[Σ]](ε)(x0) is defined, and if both are defined then
fΣ(a1, a2, …, aj) = [[Σ]](ε)(x0).
b) Suppose j ≥ n. Let ε∈ Env be an environment such that ε(xi) = aifor any i ≤ n-1, and ε(x0)
= 0. Then fΣ(a1, a2, …, aj) is defined iff [[Σ]](ε)(x0) is defined, and if both are defined
then
fΣ(a1, a2, …, aj) = [[Σ]](ε)(x0).
In other words, if j < n where n is the number of variables of Σ, and ε(x1), ε(x2), …. ε(xj) are the
values of the j variables x1, ..., xj of Σin the starting environment while the values of the other
variables are 0, then f(ε(x1), ε(x2), …. ε(xj)) takes the value [[Σ]](ε(x0)) of the variable x0 in the
environment after the termination of the computation of Σ if it terminates. If the cinnamon has
less variables than the parameters of the function then we simply use the first n variables and
ignore the remaining ones.
Clearly, a given cinnamon Σ computes a nullary (constant) function fΣ
(0)
: {0} N, a unary
function fΣ
(1)
: N N, a binary function fΣ
(2)
: N2
N, and so on up to N(n-1)
N, and further to
infinity.
As an example, let us consider the cinnamon from Fig. 4, and let Var = {z,x,y,…}. Let us first
consider the function f(2)
: N2
N. It is easy to check that f(2)
(7,3) = 4, f(2)
(7,7) = 0, f(2)
(7,0) = 7,
and f(2)
(3,7) = 0. f(1)
(7) = f(2)
(7,0) = 7. f(0)
() = f(2)
(0,0) = 0. f(3)
(7,3,8) = f(2)
(7,3) = 4. f(4)
(7,3,8,2) =
f(2)
(7,3) = 4.

146
4. TURING COMPLETENESS
In this section, we show the Turing-completeness of cinnamon programming. Of course, the
power of cinnamon programming cannot be seen and appreciated when computing simple
deterministic functions but when dealing with nondeterministic problems. Actually, for
deterministic algorithms, cinnamon programming is a departure from the idea of structured
programming. However, as cinnamon programming is another computation model, it is still
worthy to show its equivalence in computational power to the other, well-established computation
models which are all equivalent in power according to the famous Church-Turing thesis.
Computing models are typically considered as transducers (solvers of function problems) or
acceptors (solvers of decision problems). A computation model is said to be Turing-complete if
its computational power is equivalent to that of the Turing machine, and consequently to all other
equivalent models. The computation model which is most similar to cinnamons, is the while-
programs. Therefore, we discuss below the equivalence between cinnamons and while-programs.
The result is formulated in the next theorem.
Theorem:
{i) For every partial function f: Nj
N computable by a while-program P, there
is a cinnamon Σ such that fΣ
(j)
= f.
{ii) For every partial function fΣ
(j)
: Nj
N computable by a cinnamon Σ, there is
a while-program P that computes fΣ
(j)
.
As usual for this type of results, for the proof of part (i) we need to simulate the while-program P
by a suitable cinnamon, and to prove part (ii) we need to simulate the cinnamon Σ by a suitable
while-program.
Slightly different (but equivalent) definitions of the concept of a while-program can be found in
the literature. We will use here the definition given in [18]. This definition includes the simple
assignments x := 0, x := y+1 and x := y, and four statement constructs: sequential composition
{p;q}, conditional if x<y then p else q, for-loop for y do p, and while-loop while x<y do p.
Programs built inductively from these constructs are called while-programs. The function
computed by a while-program is defined similarly to that of a cinnamon.
The simulation of all the constructs from the definition of a while-program by cinnamons is
shown below:
Sequential composition {P, Q} from a while program can be simulated in the corresponding
cinnamon by simulating separately P and Q, and then combining them by unifying the FINISH
node of the part simulating P with the initial node of the part simulating Q. The simulation of the
conditional statement if x < y then P else Q is illustrated below:

147
where SimP is the cinnamon fragment simulating P, and simQ is the simulation of Q. A for-loop
for y do P can be simulated by:
The simulation of while x<y do P is illustrated by:
where subnet P simulates segment P of the while-program.
We will not provide a formal proof of Part (ii) of the Theorem. We can notice, however, that the
cinnamon interpreter presented earlier is an algorithm, and as such can be coded in a
programming language or by a while-program.
5. FULL CNP VS. CINNAMON PROGRAMMING
The cinnamon, or core CNP, is a minimal theoretical computation model underlying the ‘real’
Control Network Programming. We include below a short list of some extensions and extra
features supported by the real CNP.
• Subnets can have local variables.
• In a subnet invocation the programmer can indicate which state of the subnet will be
considered as its initial state for the particular subnet call.
• The label of an arrow is a (possibly empty) sequence of primitives and/or subnet calls,
not just a single primitive or a single subnet invocation.
• The user defines their own primitives which are arbitrary procedures of the underlying
programming language (not only the given minimal set of three elementary action
primitives and one test primitive). In addition to forward action, any user-defined
primitive can also have backward action. The definitions of the primitives (plus
eventually definitions of constants and helping functions) form the operational unit (in

148
SpiderCNP it is implemented as a file called SpiderUnit). Technically no difference is
made between action and test primitives.
• User-defined primitives can be defined in separate units which are linked within the
project.
• It should be possible that the source code of these units is written in different
programming languages – interoperability.
• Object-oriented programming can be used.
• There exists a system state STOP which is equivalent to a state without outgoing arrows.
• The programmer may define costs of the arrows of the control network.
• User has very powerful control tools – control states and system options – in order to
direct and control the CN traversal [24,25]. In particular, that allows simple ‘visual’
implementation of
o heuristic algorithms [4,5,28]
o nondeterministic algorithms [29]
o randomized algorithms [30]
• The programmer can define the solution scope, that is, how many solutions will be found
(if they exist) – a single solution, a fixed in advance number of solutions, all solutions,
prompting after each solution if another solution should be sought. For example, the
recognizer of arithmetic expressions (Fig. 6) will recognize as legal the expression a+b*c
with operation + preceding the operation *, but another solution is possible which
corresponds to operation * preceding +.
6. FURTHER STUDY: CINNAMONS AS A THEORETICAL MODEL WITH
NONDETERMINISM AND RANDOMNESS
In this section, we address a few ideas for future study.
As discussed in detail in [6] for the case of CNP, the concept of a cinnamon is inherently
declarative and nondeterministic in nature. This comes as no surprise at all, as its skeleton is
recursive nondeterministic automata – see Section 2 above. At the same time, in Section 3, we
defined the cinnamon as a deterministic computation model, and then in Section 4 pointed at its
Turing-completeness, that is, its equivalence in computational power to the other deterministic
models of computation. It would seem natural to actually define cinnamons as a nondeterministic
model (computing a relation rather than a partial function) and relate them to other
nondeterministic models such as nondeterministic Turing machines, nondeterministic while-
programs [31-35], etc. This direction of study is clearly related to the issue of solution scope.
Further, nondeterministic computation is closely related to the definitions of a search problem,
and the complexity class NP for search problems – therefore, treatment of cinnamons in this
context are possible. This ‘nondeterministic’ avenue of research, however, as the other suggested
topics of this section, lay beyond the scope of the current publication.
As mentioned, the ‘real’ CNP has control tools that make the ‘declarative’ implementation of
randomized algorithms easy and visually clear. It would be, therefore, reasonable to appropriately
extend the concept of a cinnamon and study it as a model of randomized (probabilistic)
computation, in particular in relation to existing randomized (probabilistic) computation models
and complexity classes [36-38].
Historically, CNP has roots in rule-based systems and fuzzy rule-based systems [39]. Considering
fuzzy cinnamons could be another possible direction for study.

149
REFERENCES
[1] K. Kratchanov (2017), Cinnamons: A Computational Model Underlying Control Network
Programming. In: Computer Science and Information Technology, v. 73, 7th Intl Conf. on Computer
Science, Engineering & Applications (ICCSEA 2017), Copenhagen, Denmark (N. Meghanathan & D.
Wyld – eds.), AIRCC Publ. Co., pp. 1-20. (http://airccj.org/CSCP/vol7/csit77301.pdf)
[2] K. Kratchanov, E.Golemanova and T. Golemanov (2008),“Control Network Programming Illustrated:
Solving Problems With Inherent Graph-Like Structure”, In: Proc. 7th IEEE/ACIS Int. Conf. on
Computer and Information Science (ICIS 2008), May 2008, Portland, Oregon, USA, 453-459.
[3] K. Kratchanov, E. Golemanova and T. Golemanov (2009),“Control Network Programs and Their
Execution”, In: Proc. 8thWSEAS Int. Conf. on AI, Knowledge Engineering & Data Bases (AIKED
’09), Feb 2009, Cambridge, UK, 417-422.
[4] K. Kratchanov, E. Golemanova, T. Golemanov and Y. Gökçen (2012),“Implementing Search
Strategies in Winspider II: Declarative, Procedural, and Hybrid Approaches”, In: I. Stanev and K.
Grigorova (eds.): Knowledge-Based Automated Software Engineering, Cambridge Scholars Publ.,
115-135.
[5] E. Golemanova (2013),“Declarative Implementations of Search Strategies for Solving CSPs in
Control Network Programming”, WSEAS Transactions on Computers, 12 (4), 174-183.
[6] K. Kratchanov, T. Golemanov, B. Yüksel and E. Golemanova (2014),“Control network programming
development environments”,WSEAS Transactions on Computers, 13, 645-659.
[7] T. Golemanov (2012), “SpiderSNP: An Integrated Environment for Visual Control Network
Programming”, Annals of Ruse University, 51, ser. 3.2, 123-127 (in Bulgarian).
[8] T. Golemanov (2014), Development and Study of an Integrated Development Environment for
Control Network Programming, Ph.D. Dissertation, Ruse Univ.
[9] K. Kratchanov, B. Yüksel, T. Golemanov, and E. Golemanova (2014), Learning Control Network
Programming withthe Bouquet Cloud Compiler. In: Recent Advances in Educational Technologies
and Education, Proc. 2014 Intl. Conf. on Educational Technologies and Education (ETE 2014),
Interlaken, Switzerland, February 22-24, 2014, 29-36.
Also: http://www.inase.org/library/2014/interlaken/bypaper/EDU/EDU-02.pdf
[10] K.Kratchanov, T.Golemanov and E.Golemanova (2009). “Control Network Programming: Static
Search Control with System Options”, In: Proc. 8thWSEAS Int. Conf. on AI, Knowledge Engineering
& Data Bases (AIKED ’09), Feb 2009, Cambridge, UK, 423-428.
[11] K.Kratchanov, T.GolemanovE.Golemanova and T.Ercan (2010),“Control Network Programming with
SPIDER: Dynamic Search Control”, In: Knowledge-Based and Intelligent Information and
Engineering Systems, Proc. 14thIntl. Conf. (KES 2010), Cardiff, UK, Sep 2010, Part II, Lect. Notes
in Computer Science (Lect. Notes in Artificial Intelligence), v.6277, Springer, 253-262.
[12] N. Jones (1997), Computability and Complexity from a Programming Perspective, MIT Press.
[13] A. Kfoury, R. Moll and M. Arbib (1982, reprints 2011, 2013), A Programming Approach to
Computability, Springer.
[14] M. Fitting (1987), Computability Theory, Semantics, and Logic Programming, Oxford Univ. Press.
[15] C. Moore and S. Martens (2011), The Nature of Computation, Oxford Univ. Press.
[16] K. Slonneger and B. Kurtz (1995), Formal Syntax and Semantics of Programming Languages: A
Laboratory Based Approach, Addison-Wesley.
[17] O. Goldreich (2010), P, NP, and NP-Completeness: The Basics of Computational Complexity,
Cambridge Univ. Press.
[18] D. Kozen (2006), Theory of Computation, Springer.
[19] W. Woods (1970), “Transition Network Grammars of Natural Language Analysis”, Comm. Of the
ACM, 13, 591-606.
[20] E. Popov and G. Firdman (1976), Algorithmic Foundations of Intelligent Robots and Artificial
Intelligence, Nauka (in Russian).
[21] A. Barr and E. Feigenbaum (eds.) (1981), The Handbook of Artificial Intelligence, v. 1, Pitman.
[22] K. Kratchanov (1985), On the Foundations of Rule-Based Systems, Dpt. Comp. Sci. Techn. Report
CSTR 34/85, Brunel Univ., Uxbridge, UK.
[23] K. Gough (1988), Syntax Analysis and Software Tools, Addison-Wesley.
[24] R. Alur, M. Benedikt, K. Etessami, P. Godefroid, T. Reps, and M. Yannakakis (2005), “Analysis of
recursive state machines”,ACM Trans. on Programming Languages and Systems, 27(4):786–818.
[25] I. Tellier (2006), “Learning Recursive Automata from Positive Examples”, RSTI – RIA – 20(2006)
New Methods in Machine Learning, 775-804.

150
[26] S. Chaudhuri (2008), “Subcubic Algorithms for Recursive State Machines”,
https://www.cs.rice.edu/~sc40/pubs/popl08.pdf.
[27] S. LaValle (2009), “Recursive Automata”, https://courses.engr.illinois.edu/cs373/fa2009/recaut.pdf.
[28] K.Kratchanov, E.Golemanova, T.Golemanov and T,Ercan (2010), “Non-Procedural Implementation
of Local Heuristic Search in Control Network Programming”, In: Knowledge-Based and Intelligent
Information and Engineering Systems, Proc. 14thIntl. Conf. (KES 2010), Cardiff, UK, Sep 2010, Part
II, Lect. Notes in Computer Science (Lect. Notes in Artificial Intelligence), v.6277, Springer, 263-
272.
[29] K. Kratchanov, E. Golemanova, T. Golemanov and B. Külahçıoğlu (2012), “Using Control Network
Programming in Teaching Nondeterminism”, In: Proc. 13thInt. Conf. on Computer Systems and
Technologies (CompSysTech’12), Ruse, (B. Rachev, A. Smrikarov – eds.), ACM Press, New York,
391-398.
[30] K. Kratchanov, E. Golemanova, T. Golemanov and B. Külahçıoğlu (2012),“Using Control Network
Programming in Teaching Randomization”, In: Proc. Int. Conf. Electronics, Information and
Communication Engineering, Macau (EICE 2012), ASME, 67-71.
[31] E. Dijkstra (1975),“Guarded Commands, Nondeterminacy and Formal Derivations of Programs”,
Comm. of the ACM, 18, 453-457. Also: E. Dijkstra. “Guarded Commands. Non-Determinacy and a
Calculus for the Derivation of Programs” (EWD418), 1974
https://www.cs.utexas.edu/users/EWD/transcriptions/EWD04xx/EWD418.html
[32] G. Mascari and M. Zilli (1985),“While Programs with Nondeterministic Assignments and the Logic
ALNA”,Theoretical Computer Science, 40, 211-235.
[33] J. van Leeuwen (Ed.) (1990, 1992), Handbook of Theoretical Computer Science, v. B: Formal Models
and Semantics. Elsevier and MIT Press.
[34] K. Apt, F, de Boer, E. Olderog (2010), Verification of Sequential and Concurrent Programs, 3rd ed.,
Springer.
[35] K. Mamouras (2015),“Synthesis of Strategies and the Hoare Logic of Angelic Nondeterminism”, In:
Foundations of Software Science and Computation Structures. 18thInt. Conf. FOSSACS 2015 Held as
Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2015, London,
11-18 Apr 2015, Proc. LNCS 9034, Springer 2015 (A. Pitts – ed.), 25-40.
[36] R. Motwani, P. Raghavan. Randomized Algorithms. Cambridge Univ. Press, 1995.
[37] S. Arora, B. Barak. Computational Complexity: A Modern Approach, Cambridge Uni. Press, 2009.
[38] D. Antonova, D. Kunkle. Theory of Randomized Computation. 2005.
http://www.ccs.neu.edu/home/kunkle/papers/techreports/randAlgo.pdf.
[39] K. Kratchanov (1985), Towards the Fundamentals of Fuzzy Rule-Based Systems, CSTR 35/85, Dept.
Comp. Sci., Brunel Univ., Uxbridge, UK
AUTHOR
Dr Kostadin Kratchanov taught at Ruse Univ. (Bulgaria), Technical Univ. of Sofia
Plovdiv Branch (Bulgaria), Brunel Univ. (UK), European Univ. of Lefke (Northern
Cyprus), Univ. of Bahrain (Bahrain), Grande Prairie Regional Coll. (Canada),
Mount Royal Univ. (Canada), and is currently with Yasar Univ. (Turkey). His
research interests are in Theoretical Computer Science, Theory of Computation,
Discrete Structures, Fuzzy Automata, Applications of Category Theory in
Computer Science, Programming Languages and Paradigms, Artificial Intelligence,
Analysis and Design of Algorithms, Software Engineering.

SYNTAX AND SEMANTICS FOR CINNAMON PROGRAMMING

More Related Content

What's hot (19)

Similar to SYNTAX AND SEMANTICS FOR CINNAMON PROGRAMMING (20)

More from AIRCC Publishing Corporation (20)

Recently uploaded (20)

SYNTAX AND SEMANTICS FOR CINNAMON PROGRAMMING