Guarded Kleene Algebra with Tests: Coequations, Coinduction, and Completeness

Guarded Kleene Algebra with Tests (GKAT) is an efficient fragment of KAT, as it allows for almost linear decidability of equivalence. In this paper, we study the (co)algebraic properties of GKAT. Our initial focus is on the fragment that can distinguish between unsuccessful programs performing different actions, by omitting the so-called early termination axiom. We develop an operational (coalgebraic) and denotational (algebraic) semantics and show that they coincide. We then characterize the behaviors of GKAT expressions in this semantics, leading to a coequation that captures the covariety of automata corresponding to these behaviors. Finally, we prove that the axioms of the reduced fragment are sound and complete w.r.t. the semantics, and then build on this result to recover a semantics that is sound and complete w.r.t. the full set of axioms

Guarded Kleene Algebra with Tests: Coequations, Coinduction, and Completeness

Guarded Kleene Algebra with Tests (GKAT) is an efficient fragment of KAT, as it allows for almost linear decidability of equivalence. In this paper, we study the (co)algebraic properties of GKAT. Our initial focus is on the fragment that can distinguish between unsuccessful programs performing different actions, by omitting the so-called early termination axiom. We develop an operational (coalgebraic) and denotational (algebraic) semantics and show that they coincide. We then characterize the behaviors of GKAT expressions in this semantics, leading to a coequation that captures the covariety of automata corresponding to these behaviors. Finally, we prove that the axioms of the reduced fragment are sound and complete w.r.t. the semantics, and then build on this result to recover a semantics that is sound and complete w.r.t. the full set of axioms

Guarded Kleene algebra with tests: verification of uninterpreted programs in nearly linear time

Guarded Kleene Algebra with Tests (GKAT) is a variation on Kleene Algebra with Tests (KAT) that arises by restricting the union (+) and iteration (*) operations from KAT to predicate-guarded versions. We develop the (co)algebraic theory of GKAT and show how it can be efficiently used to reason about imperative programs. In contrast to KAT, whose equational theory is PSPACE-complete, we show that the equational theory of GKAT is (almost) linear time. We also provide a full Kleene theorem and prove completeness for an analogue of Salomaa’s axiomatization of Kleene Algebra

Synchronous Kleene algebra

AbstractThe work presented here investigates the combination of Kleene algebra with the synchrony model of concurrency from Milner’s SCCS calculus. The resulting algebraic structure is called synchronous Kleene algebra. Models are given in terms of sets of synchronous strings and finite automata accepting synchronous strings. The extension of synchronous Kleene algebra with Boolean tests is presented together with models on sets of guarded synchronous strings and the associated automata on guarded synchronous strings. Completeness w.r.t. the standard interpretations is given for each of the two new formalisms. Decidability follows from completeness. Kleene algebra with synchrony should be included in the class of true concurrency models. In this direction, a comparison with Mazurkiewicz traces is made which yields their incomparability with synchronous Kleene algebras (one cannot simulate the other). On the other hand, we isolate a class of pomsets which captures exactly synchronous Kleene algebras. We present an application to Hoare-like reasoning about parallel programs in the style of synchrony

Position Automata for Kleene Algebra with Tests

Kleene algebra with tests (KAT) is an equational system that combines Kleene and Boolean algebras. One can model basic programming constructs and assertions in KAT, which allowed for its application in compiler optimization, program transformation and dataflow analysis. To provide semantics for KAT expressions, Kozen first introduced emph{automata on guarded strings}, showing that the regular sets of guarded strings plays the same role in KAT as regular languages play in Kleene algebra. Recently, Kozen described an elegant algorithm, based on ``derivatives'', to construct a deterministic automaton that accepts the guarded strings denoted by a KAT expression. This algorithm generalizes Brzozowski's algorithm for regular expressions and inherits its inefficiency arising from the explicit computation of derivatives. In the context of classical regular expressions, many efficient algorithms to compile expressions to automata have been proposed. One of those algorithms was devised by Berry and Sethi in the 80's (we shall refer to it as Berry-Sethi construction/algorithm, but in the literature it is also referred to as position or Glushkov automata algorithm). In this paper, we show how the Berry-Sethi algorithm can be used to compile a $KAT$ expression to an automaton on guarded strings. Moreover, we propose a new automata model for KAT expressions and adapt the construction of Berry and Sethi to this new model

An information-based theory of topics and grammatical relations.

This dissertation proposes a formal semantic characterization of topichood and an account of the relationship between Topic and core Grammatical Relations. The theoretical framework employed is a form of HPSG (Pollard & Sag (1994)). The notion of Topic has been widely invoked in descriptions both of sentence structure and of intersentential discourse relations. Despite this a formal characterization of this notion is lacking in the literature. It is proposed here that Topics should be seen as predication targets at an underlying semantic level, and that the Topic-Comment relation is analogous to that between possible worlds (situations) and the propositional contents which they support. A Topic is interpreted as a point whose location has to be fixed in some conceptual space formed by the Comments, and this metaphor is extended to the overall Topic of a discourse sequence. Formally, it is suggested that Topics and Comments can be treated as the points and open sets respectively of a topological space. It is claimed that this captures well-known semantic restrictions on which NPs can be made Topics of a sentence. The proposed treatment is also extended to intersentential Topic relations. This account of Topics is made the basis of a revision to the relational hierarchy, which underlies many relational theories of grammar. It is proposed that basic predicates in language are maximally binary and sensitive to topichood, their initial Subject being the default predication target or Topic. Predicates of greater valency are treated as composite, and the effects of the relational hierarchy are derived from rules governing the process of composition. A number of cross-linguistic phenomena are examined which bear on the relationship between Topics and core Grammatical Relations, including the double Subject constructions characteristic of Japanese and other East Asian languages, the clitic doubling of Objects which is an areal phenomenon of the Balkans, and the so called "Object agreement" of Amharic. Finally a chapter is devoted to the nature of Indirect Objects, which are argued (against standard views) to rank above Direct Objects. It is claimed that with this approach an important part of the relational basis of syntax can be derived, without losing descriptive accuracy, from the proposed treatment of predication

Physical (A)Causality: Determinism, Randomness and Uncaused Events

Physical indeterminism; Randomness in physics; Physical random number generators; Physical chaos; Self-reflexive knowledge; Acausality in physics; Irreducible randomnes

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum