Monoids with tests and the algebra of possibly non-halting programs

Jackson, Marcel; Stokes, Tim

doi:10.1016/j.jlamp.2014.08.007

Cited by 8 publications

(15 citation statements)

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“…It is desirable to achieve such a representation for the general class of C-monoids with no restriction on the C-algebra, which can be considered as future work. Note that the term f • T in the standard functional model of a C-monoid represents the aspect of the domain of the function, as used in [4,11]. It is interesting to study the relation between these two concepts in the current set up.…”

Section: Resultsmentioning

confidence: 99%

Axiomatization of `if-then-else` over possibly non-halting programs and tests

Panicker

Krishna

Bhaduri

2017

Int. J. Algebra Comput.

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In order to study the axiomatization of the if-then-else construct over possibly non-halting programs and tests, the notion of C-sets was introduced in the literature by considering the tests from an abstract C-algebra. This paper extends the notion of C-sets to C-monoids which include the composition of programs as well as composition of programs with tests. For the class of C-monoids where the C-algebras are adas a canonical representation in terms of functional C-monoids is obtained.2010 Mathematics Subject Classification. 08A70, 03G25 and 68N15.There are multiple studies (e.g., see [3,7,13,14]) on extending two-valued Boolean logic to three-valued logic. However McCarthy's logic (cf. [17]) is distinct in that it models the short-circuit evaluation exhibited by programming languages that evaluate expressions in sequential order, from left to right. In [6] Guzmán and Squier gave a complete axiomatization of McCarthy's three-valued logic and called the corresponding algebra a C-algebra, or the algebra of conditional logic. While studying if-then-else algebras in [15], Manes defined an ada (Algebra of Disjoint Alternatives) which is essentially a C-algebra equipped with an oracle for the halting problem.Jackson and Stokes in [11] studied the algebraic theory of computable functions, which can be viewed as possibly non-halting programs, together with composition, if-then-else and while-do. In this work they assumed that the tests form a Boolean algebra. Further, they demonstrated how an algebra of non-halting tests could be constructed from Boolean tests in their setting. Jackson and Stokes proposed an alternative approach by considering an abstract collection of non-halting tests and posed the following problem:Characterize the algebras of computable functions associated with an abstract C-algebra of non-halting tests.The authors in [19] have approached the problem by adopting the approach of Jackson and Stokes in [10]. The notion of a C-set was introduced through which a complete axiomatization for if-then-else over a class of possibly non-halting programs and tests, where tests are drawn from an ada, was provided.In this paper, following the approach of Jackson and Stokes in [10], we extend the notion of C-sets to include composition of possibly non-halting programs and of these programs with possibly non-halting tests. This object is termed a Cmonoid and we show that every C-monoid where the tests are drawn from an ada is embeddable in a canonical model of C-monoids, viz., functional C-monoids.

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Section: Resultsmentioning

confidence: 99%

Axiomatization of `if-then-else` over possibly non-halting programs and tests

Panicker

Krishna

Bhaduri

2017

Int. J. Algebra Comput.

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“…The difference between the two evaluation strategies can be seen in the cases of U ∧ F and U ∨ T . The connectives are not commutative when evaluated sequentially as in [8] and [11], but are commutative under the parallel paradigm used here.…”

Section: Testsmentioning

confidence: 99%

“…In [11], Jackson and Stokes attempted to model only computable operations (with no oracle for the halting problem). The signature again facilitates the expression of domain as well as program-test composition.…”

Section: Introductionmentioning

confidence: 99%

“…As noted there, this induces an algebra of non-halting tests in which tests are evaluated sequentially. Panicker et al [12] generalized [11] in the concatenationfree case to allow arbitrary non-halting tests within if-then-else statements, but under sequential evaluation. Recently, they extended this signature by including composition as well, in [13].…”

Section: Introductionmentioning

confidence: 99%

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Algebraic properties of `if-then-else` and commutative three-valued tests

Soo

Stokes

2019

Int. J. Algebra Comput.

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This paper establishes a finite axiomatization of possibly non-halting computer programs and tests, with the if-then-else operation. The model is a two-sorted algebra, with one sort being the programs and the other being the tests. The main operation on programs is composition, and 1 and 0 represent the programs skip and loop (i.e. never halts) respectively. Programs are modeled as partial functions on some state space [Formula: see text], with tests modeled as partial predicates on [Formula: see text]. The operations on the tests are the usual logical connectives ∧, ∨, [Formula: see text], [Formula: see text] and [Formula: see text]. In addition, there is the hybrid operation of if-then-else, and the test-valued operation [Formula: see text] on programs which is true when a program halts, and undefined otherwise. The halting operation [Formula: see text] implies that operations of domain [Formula: see text] and domain join ∨ may also be expressed. When tests are assumed to be possibly non-halting, the evaluation strategy of the logical connectives affects the result. Here we model parallel evaluation, as opposed to the common sequential (or short-circuit) evaluation strategy. For example, we view [Formula: see text] as false if either [Formula: see text] or [Formula: see text] is false, even if the other does not halt.

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“…In order to address a problem posed by Jackson and Stokes in [8], present authors introduced the notion of C-sets and studied axiomatization of if-then-else over C-algebras in [15,16]. Every C-algebra has an inbuilt if-then-else action using which we introduce a notion of annihilators in a natural manner, which aid in studying various structural properties of C-algebras.…”

Section: Introductionmentioning

confidence: 99%

On the structure of C-algebras through atomicity and if-then-else

2020

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This paper introduces the notions of atoms and atomicity in Calgebras and obtains a characterisation of atoms in the C-algebra of transformations. Further, this work presents some necessary conditions and sufficient conditions for the atomicity of C-algebras and shows that the class of finite atomic C-algebras is precisely that of finite adas. This paper also uses the if-then-else action to study the structure of C-algebras and classify the elements of the C-algebra of transformations.2010 Mathematics Subject Classification. 08A70, 03G25 and 68N15.

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Monoids with tests and the algebra of possibly non-halting programs

Cited by 8 publications

References 38 publications

Axiomatization of `if-then-else` over possibly non-halting programs and tests

Axiomatization of `if-then-else` over possibly non-halting programs and tests

Algebraic properties of `if-then-else` and commutative three-valued tests

On the structure of C-algebras through atomicity and if-then-else

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Monoids with tests and the algebra of possibly non-halting programs

Cited by 8 publications

References 38 publications

Axiomatization of if-then-else over possibly non-halting programs and tests

Axiomatization of if-then-else over possibly non-halting programs and tests

Algebraic properties of if-then-else and commutative three-valued tests

On the structure of C-algebras through atomicity and if-then-else

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Product

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Axiomatization of `if-then-else` over possibly non-halting programs and tests

Axiomatization of `if-then-else` over possibly non-halting programs and tests

Algebraic properties of `if-then-else` and commutative three-valued tests