117 research outputs found
Towards Correctness of Program Transformations Through Unification and Critical Pair Computation
Correctness of program transformations in extended lambda calculi with a
contextual semantics is usually based on reasoning about the operational
semantics which is a rewrite semantics. A successful approach to proving
correctness is the combination of a context lemma with the computation of
overlaps between program transformations and the reduction rules, and then of
so-called complete sets of diagrams. The method is similar to the computation
of critical pairs for the completion of term rewriting systems. We explore
cases where the computation of these overlaps can be done in a first order way
by variants of critical pair computation that use unification algorithms. As a
case study we apply the method to a lambda calculus with recursive
let-expressions and describe an effective unification algorithm to determine
all overlaps of a set of transformations with all reduction rules. The
unification algorithm employs many-sorted terms, the equational theory of
left-commutativity modelling multi-sets, context variables of different kinds
and a mechanism for compactly representing binding chains in recursive
let-expressions.Comment: In Proceedings UNIF 2010, arXiv:1012.455
A model theoretic study of right-angled buildings
We study the model theory of countable right-angled buildings with infinite
residues. For every Coxeter graph we obtain a complete theory with a natural
axiomatisation, which is -stable and equational. Furthermore, we
provide sharp lower and upper bounds for its degree of ampleness, computed
exclusively in terms of the associated Coxeter graph. This generalises and
provides an alternative treatment of the free pseudospace.Comment: A number of small typos found by typesetter correcte
Comparing axiomatizations of free pseudospaces
Independently and pursuing different aims, Hrushovski and Srour (On stable non-equational theories. Unpublished manuscript, 1989) and Baudisch and Pillay (J Symb Log 65(1):443–460, 2000) have introduced two free pseudospaces that generalize the well know concept of Lachlan’s free pseudoplane. In this paper we investigate the relationship between these free pseudospaces, proving in particular, that the pseudospace of Baudisch and Pillay is a reduct of the pseudospace of Hrushovski and Srour
Probabilistic thread algebra
We add probabilistic features to basic thread algebra and its extensions with
thread-service interaction and strategic interleaving. Here, threads represent
the behaviours produced by instruction sequences under execution and services
represent the behaviours exhibited by the components of execution environments
of instruction sequences. In a paper concerned with probabilistic instruction
sequences, we proposed several kinds of probabilistic instructions and gave an
informal explanation for each of them. The probabilistic features added to the
extension of basic thread algebra with thread-service interaction make it
possible to give a formal explanation in terms of non-probabilistic
instructions and probabilistic services. The probabilistic features added to
the extensions of basic thread algebra with strategic interleaving make it
possible to cover strategies corresponding to probabilistic scheduling
algorithms.Comment: 25 pages (arXiv admin note: text overlap with arXiv:1408.2955,
arXiv:1402.4950); some simplifications made; substantially revise
Computing overlappings by unification in the deterministic lambda calculus LR with letrec, case, constructors, seq and variable chains
Correctness of program transformations in extended lambda calculi with a contextual semantics is usually based on reasoning about the operational semantics which is a rewrite semantics. A successful approach to proving correctness is the combination of a context lemma with the computation of overlaps between program transformations and the reduction rules.The method is similar to the computation of critical pairs for the completion of term rewriting systems. We describe an effective unification algorithm to determine all overlaps of transformations with reduction rules for the lambda calculus LR which comprises a recursive let-expressions, constructor applications, case expressions and a seq construct for strict evaluation. The unification algorithm employs many-sorted terms, the equational theory of left-commutativity modeling multi-sets, context variables of different kinds and a mechanism for compactly representing binding chains in recursive let-expressions. As a result the algorithm computes a finite set of overlappings for the reduction rules of the calculus LR that serve as a starting point to the automatization of the analysis of program transformations
The notion of independence in categories of algebraic structures, part I: Basic properties
AbstractWe define a formula φ(x;t) in a first-order language L, to be an equation in a category of L-structures K if for any H in K, and set p = {φ(x;a1);i ϵI, ai ϵ H} there is a finite set I0⊂I such that for any f:H→ F in K, .We say that an elementary first-order theory T which has the amalgamation property over substructures is equational if every quantifier-free formula is equivalent in T to a boolean combination of (quantifier-free) equations in Mod(T), the category of models of T with embeddings for morphisms.Thus, we develop a theory of independence with respect to equations in general categories of structures, which is similar to the one introduced in stability (and actually identical to it in the case of equational theories) but which, in our context, has an algebraic character
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