[Paper Review] The algebra of non-deterministic programs: demonic operators, orders and axioms

Robin Hirsch, Szabolcs Mikulás+1 more|arXiv (Cornell University)|Sep 25, 2020

Computability, Logic, AI Algorithms9 references4 citations

TL;DR

This paper establishes the finite axiomatizability of algebras of binary relations under demonic composition and refinement, extending Zarecki’s work on angelic operations. It proves that the class of such algebras is finitely axiomatized as ordered semigroups, while mixed signatures involving both demonic and angelic operations are not finitely axiomatizable, highlighting a key distinction in algebraic structure complexity.

ABSTRACT

Demonic composition, demonic refinement and demonic union are alternatives to the usual "angelic" composition, angelic refinement (inclusion) and angelic (usual) union defined on binary relations. We first motivate both the angelic and demonic via an analysis of the behaviour of non-deterministic programs, with the angelic associated with partial correctness and demonic with total correctness, both cases emerging from a richer algebraic model of non-deterministic programs incorporating both aspects. Zareckii has shown that the isomorphism class of algebras of binary relations under angelic composition and inclusion is finitely axiomatised as the class of ordered semigroups. The proof can be used to establish that the same axiomatisation applies to binary relations under demonic composition and refinement, and a further modification of the proof can be used to incorporate a zero element representing the empty relation in the angelic case and the full relation in the demonic case. For the signature of angelic composition and union, it is known that no finite axiomatisation exists, and we show the analogous result for demonic composition and demonic union by showing that the same axiomatisation holds for both. We show that the isomorphism class of algebras of binary relations with the "mixed" signature of demonic composition and angelic inclusion has no finite axiomatisation. As a contrast, we show that the isomorphism class of partial algebras of binary relations with the partial operation of constellation product and inclusion (also a "mixed" signature) is finitely axiomatisable.

Motivation & Objective

• To extend Zarecki’s finite axiomatization of ordered semigroups under angelic operations to the demonic setting.
• To investigate whether the isomorphism class of algebras of binary relations under demonic composition and refinement admits a finite axiomatization.
• To analyze the expressive power and axiomatizability of mixed signatures combining demonic and angelic operations, such as demonic composition with angelic inclusion.
• To explore the finite axiomatizability of partial algebras with constellation product and inclusion, contrasting them with full relational algebras.
• To determine the status of various relational algebras involving zero elements (empty or full relations) under demonic and angelic operations.

Proposed method

• Adapts Zarecki’s Cayley-type representation for ordered semigroups to the demonic setting, using partial algebras and domain conditions.
• Defines demonic composition as $ s*t = (s;t) \cap \{ (x,y) : s(x) \subseteq \text{dom}(t) \} $, ensuring total correctness semantics.
• Introduces demonic refinement $ s \sqsubseteq t $ as $ \text{dom}(t) \subseteq \text{dom}(s) \land s|_{\text{dom}(t)} \subseteq t $, forming a partial order.
• Introduces the constellation product $ s \cdot t $, a partial operation defined only when $ \text{ran}(s) \subseteq \text{dom}(t) $, generalizing demonic composition.
• Uses representation theorems to show that every ordered pre-constellation embeds into a relational algebra over a set $ X $, with $ 0 $ mapped to the empty relation.
• Applies model-theoretic techniques to prove non-finite axiomatizability for mixed signatures like $ (*, \subseteq) $, using infinite families of quasi-equations.

Experimental results

Research questions

• RQ1 Is the class of algebras of binary relations under demonic composition and refinement finitely axiomatizable?
• RQ2 Does the mixed signature of demonic composition and angelic inclusion ($ *, \subseteq $) admit a finite axiomatization?
• RQ3 Is the class of partial algebras of binary relations under constellation product and inclusion ($ \cdot, \subseteq $) finitely axiomatizable?
• RQ4 What is the relationship between the varieties generated by $ R(*, \sqcup\kern-6.0pt\sqcup, \emptyset) $ and $ T(\mathbin{;}, \cup, \nabla) $, and are they finitely axiomatizable?
• RQ5 Can the class $ VL_0(\mathbin{;}, \cup, \mathbf{0}) $ be finitely axiomatized, and how does it compare to $ VR(\mathbin{;}, \cup, \mathbf{1}') $?

Key findings

• The isomorphism class of algebras of binary relations under demonic composition and refinement is finitely axiomatized as the class of ordered semigroups.
• The same finite axiomatization applies to algebras with a zero element representing the empty relation in the demonic case.
• The mixed signature $ (*, \subseteq) $ — combining demonic composition and angelic inclusion — is not finitely axiomatizable, as shown by an infinite family of quasi-equations.
• The class of partial algebras of binary relations under constellation product and inclusion ($ \cdot, \subseteq $) is finitely axiomatizable as the class of ordered pre-constellations.
• The class $ R(*, \sqcup\kern-6.0pt\sqcup, \emptyset) $ is contained in $ T(\mathbin{;}, \cup, \nabla) $, but equality and finite axiomatizability of either class remain open.
• The class $ R(\mathbin{;}, \cup, \mathbf{0}) $ is not finitely axiomatizable, and the same holds for $ R(*, \subseteq) $, confirming a structural asymmetry between angelic and demonic operations in mixed signatures.