294 research outputs found
On Hoare-McCarthy algebras
We discuss an algebraic approach to propositional logic with side effects. To
this end, we use Hoare's conditional [1985], which is a ternary connective
comparable to if-then-else. Starting from McCarthy's notion of sequential
evaluation [1963] we discuss a number of valuation congruences and we introduce
Hoare-McCarthy algebras as the structures that characterize these congruences.Comment: 29 pages, 1 tabl
An Algebraic Characterisation of Concurrent Composition
We give an algebraic characterization of a form of synchronized parallel
composition allowing for true concurrency, using ideas based on Peter Landin's
"Program-Machine Symmetric Automata Theory".Comment: This is an old technical report from 1981. I submitted it to a
special issue of HOSC in honour of Peter Landin, as explained in the Prelude,
added in 2008. However, at an advanced stage, the handling editor became
unresponsive, and the paper was never published. I am making it available via
the arXiv for the same reasons given in the Prelud
Semigroups with if-then-else and halting programs
The "ifāthenāelse" construction is one of the most elementary programming commands, and its abstract laws have been widely studied, starting with McCarthy. Possibly, the most obvious extension of this is to include the operation of composition of programs, which gives a semigroup of functions (total, partial, or possibly general binary relations) that can be recombined using ifāthenāelse. We show that this particular extension admits no finite complete axiomatization and instead focus on the case where composition of functions with predicates is also allowed (and we argue there is good reason to take this approach). In the case of total functions ā modeling halting programs ā we give a complete axiomatization for the theory in terms of a finite system of equations. We obtain a similar result when an operation of equality test and/or fixed point test is included
A superintegrable finite oscillator in two dimensions with SU(2) symmetry
A superintegrable finite model of the quantum isotropic oscillator in two
dimensions is introduced. It is defined on a uniform lattice of triangular
shape. The constants of the motion for the model form an SU(2) symmetry
algebra. It is found that the dynamical difference eigenvalue equation can be
written in terms of creation and annihilation operators. The wavefunctions of
the Hamiltonian are expressed in terms of two known families of bivariate
Krawtchouk polynomials; those of Rahman and those of Tratnik. These polynomials
form bases for SU(2) irreducible representations. It is further shown that the
pair of eigenvalue equations for each of these families are related to each
other by an SU(2) automorphism. A finite model of the anisotropic oscillator
that has wavefunctions expressed in terms of the same Rahman polynomials is
also introduced. In the continuum limit, when the number of grid points goes to
infinity, standard two-dimensional harmonic oscillators are obtained. The
analysis provides the limit of the bivariate Krawtchouk
polynomials as a product of one-variable Hermite polynomials
Monoids with tests and the algebra of possibly non-halting programs
We study the algebraic theory of computable functions, which can be viewed as arising from possibly non-halting computer programs or algorithms, acting on some state space, equipped with operations of composition, if-then-else and while-do defined in terms of a Boolean algebra of conditions. It has previously been shown that there is no finite axiomatisation of algebras of partial functions under these operations alone, and this holds even if one restricts attention to transformations (representing halting programs) rather than partial functions, and omits while-do from the signature. In the halting case, there is a natural āfixā, which is to allow composition of halting programs with conditions, and then the resulting algebras admit a finite axiomatisation. In the current setting such compositions are not possible, but by extending the notion of if-then-else, we are able to give finite axiomatisations of the resulting algebras of (partial) functions, with while-do in the signature if the state space is assumed finite. The axiomatisations are extended to consider the partial predicate of equality. All algebras considered turn out to be enrichments of the notion of a (one-sided) restriction semigrou
Propositional logic with short-circuit evaluation: a non-commutative and a commutative variant
Short-circuit evaluation denotes the semantics of propositional connectives
in which the second argument is evaluated only if the first argument does not
suffice to determine the value of the expression. Short-circuit evaluation is
widely used in programming, with sequential conjunction and disjunction as
primitive connectives.
We study the question which logical laws axiomatize short-circuit evaluation
under the following assumptions: compound statements are evaluated from left to
right, each atom (propositional variable) evaluates to either true or false,
and atomic evaluations can cause a side effect. The answer to this question
depends on the kind of atomic side effects that can occur and leads to
different "short-circuit logics". The basic case is FSCL (free short-circuit
logic), which characterizes the setting in which each atomic evaluation can
cause a side effect. We recall some main results and then relate FSCL to MSCL
(memorizing short-circuit logic), where in the evaluation of a compound
statement, the first evaluation result of each atom is memorized. MSCL can be
seen as a sequential variant of propositional logic: atomic evaluations cannot
cause a side effect and the sequential connectives are not commutative. Then we
relate MSCL to SSCL (static short-circuit logic), the variant of propositional
logic that prescribes short-circuit evaluation with commutative sequential
connectives.
We present evaluation trees as an intuitive semantics for short-circuit
evaluation, and simple equational axiomatizations for the short-circuit logics
mentioned that use negation and the sequential connectives only.Comment: 34 pages, 6 tables. Considerable parts of the text below stem from
arXiv:1206.1936, arXiv:1010.3674, and arXiv:1707.05718. Together with
arXiv:1707.05718, this paper subsumes most of arXiv:1010.367
Proposition Algebra with Projective Limits
Sequential propositional logic deviates from ordinary propositional logic by
taking into account that during the sequential evaluation of a propositional
statement,atomic propositions may yield different Boolean values at repeated
occurrences. We introduce `free valuations' to capture this dynamics of a
propositional statement's environment. The resulting logic is phrased as an
equationally specified algebra rather than in the form of proof rules, and is
named `proposition algebra'. It is strictly more general than Boolean algebra
to the extent that the classical connectives fail to be expressively complete
in the sequential case. The four axioms for free valuation congruence are then
combined with other axioms in order define a few more valuation congruences
that gradually identify more propositional statements, up to static valuation
congruence (which is the setting of conventional propositional logic).
Proposition algebra is developed in a fashion similar to the process algebra
ACP and the program algebra PGA, via an algebraic specification which has a
meaningful initial algebra for which a range of coarser congruences are
considered important as well. In addition infinite objects (that is
propositional statements, processes and programs respectively) are dealt with
by means of an inverse limit construction which allows the transfer of
knowledge concerning finite objects to facts about infinite ones while reducing
all facts about infinite objects to an infinity of facts about finite ones in
return.Comment: 43 pages, 3 table
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