Michal Walicki

Algebraic approaches to nondeterminism—an overview

Mathematics never saw much of a reason to deal with something called nondeterminism. It works with values, functions, sets, and relations. In computing science, however, nondeterminism has been an issue from the very beginning, if only in the form of nondeterministic Turing machines or nondeterministic finite state machines. Early references to nondeterminism in computer science go back to the 1960s [Floyd 1967; McCarthy 1963]. A great variety of theories and formalisms dealing with it have been developed during the last two decades. There are the denotational models based on power domains, the predicate transformers for the choice construct, and modifications of the l-calculus [de Liguoro and Piperno 1992; Astesiano and Costa 1979; Hennessy 1980]. Nondeterminism arises in a natural way when discussing concurrency, and models of concurrency typically also model nondeterminism. There are numerous variants of process languages and algebra, event structures, state transition systems [Manna and Pnueli 1992; Lehmann and Shelah 1983], and Petri nets [Petri 1977; Reisig 1985]. In terms of modeling, nondeterminism may be considered a purely operational notion. However, one of the main reasons for considering nondeterminism in computer science is the need for abstraction, allowing one to disregard irrelevant aspects of actual computations. Typically, we prefer to work with models that do not include all the details of the physical environment of computations such as timing, temperature, representation on hardware, and the like. Since we do not want to model all these complex dependencies, we may instead represent them by nondeterministic choices. The nondeterminism of concurrent systems usually arises as an ab-

A complete calculus for the multialgebraic and functional semantics of nondeterminism

The current algebraic models for nondeterminism focus on the notion of possibility rather than necessity and consequently equate (nondeterministic) terms that one would intuitively not consider equal. Furthermore, existing models for nondeterminism depart radically from the standard models for (equational) specifications of deterministic operators. One would prefer that a specification language for nondeterministic operators be based on an extension of the standard model concepts, preferably in such a way that the reasoning system for (possibly nondeterministic) operators becomes the standard equational one whenever restricted to the deterministic operators—the objective should be to minimize the departure from the standard frameworks. In this article we define a specification language for nondeterministic operators and multialgebraic semantics. The first complete reasoning system for such specifications is introduced. We also define a transformation of specifications of nondeterministic operators into derived specifications of deterministic ones, obtaining a “computational” semantics of nondeterministic specification by adopting the standard semantics of the derived specification as the semantics of the original one. This semantics turns out to be a refinement of multialgebra semantics. The calculus is shown to be sound and complete also with respect to the new semantics.

Multialgebras, Power Algebras and Complete Calculi of Identities and Inclusions

After motivating the introduction of nondeterministic operators into algebraic specifications, a language L with two primitive predicates, identity and inclusion, for specifying nondeterministic operations is introduced. It is given a multialgebraic semantics which captures the singular (call-time-choice) strategy of passing nondeterministic parameters. A calculus NEQ, with restricted substitutivity rules, is introduced. NEQ is sound and complete wrt. the multialgebraic semantics.

Sequence partitioning for process mining with unlabeled event logs

Finding the case id in unlabeled event logs is arguably one of the hardest challenges in process mining research. While this problem has been addressed with greedy approaches, these usually converge to sub-optimal solutions. In this work, we describe an approach to perform complete search over the search space. We formulate the problem as a matter of finding the minimal set of patterns contained in a sequence, where patterns can be interleaved but do not have repeating symbols. This represents a new problem that has not been previously addressed in the literature, with NP-hard variants and conjectured NP-completeness. We solve it in a stepwise manner, by generating and verifying a list of candidate solutions. The techniques, introduced to address various subtasks, can be applied independently for solving more specific problems. The approach has been implemented and applied in a case study with real data from a business process supported in a software application.

Propositional discourse logic

A novel normal form for propositional theories underlies the logic pdl, which captures some essential features of natural discourse, independent from any particular subject matter and related only to its referential structure. In particular, pdlallows to distinguish vicious circularity from the innocent one, and to reason in the presence of inconsistency using a minimal number of extraneous assumptions, beyond the classical ones. Several, formally equivalent decision problems are identified as potential applications: non-paradoxical character of discourses, admissibility of arguments in argumentation networks, propositional satisfiability, and the existence of kernels of directed graphs. Directed graphs provide the basis for the semantics of pdl and the paper concludes by an overview of relevant graph-theoretical results and their applications in diagnosing paradoxical character of natural discourses.

Complete axiomatizations of finite syntactic epistemic states

An agent who bases his actions upon explicit logical formulae has at any given point in time a finite set of formulae he has computed. Closure or consistency conditions on this set cannot in general be assumed – reasoning takes time and real agents frequently have contradictory beliefs. This paper discusses a formal model of knowledge as explicitly computed sets of formulae. It is assumed that agents represent their knowledge syntactically, and that they can only know finitely many formulae at a given time. In order to express interesting properties of such finite syntactic epistemic states, we extend the standard epistemic language with an operator expressing that an agent knows at most a particular finite set of formulae, and investigate axiomatization of the resulting logic. This syntactic operator has also been studied elsewhere without the assumption about finite epistemic states. A strongly complete logic is impossible, and the main results are non-trivial characterizations of the theories for which we can get completeness. The paper presents a part of a general abstract theory of resource bounded agents. Interesting results, e.g., complex algebraic conditions for completeness, are obtained from very simple assumptions, i.e., epistemic states as arbitrary finite sets and operators for knowing at least and at most.

Expressive Power of Digraph Solvability

A kernel of a directed graph is a set of vertices without edges between them, such that every other vertex has a directed edge to a vertex in the kernel. A digraph possessing a kernel is called solvable. We show that solvability of digraphs is equivalent to satisfiability of theories of propositional logic. Based on a new normal form for such theories, this equivalence relates finitely branching digraphs to propositional logic, and arbitrary digraphs to infinitary propositional logic. Furthermore, we show that the existence of a kernel for a digraph is equivalent to the existence of a kernel for its lifting to an infinitely-branching dag. While the computational complexity of solvability differs between finite dags (trivial, since always solvable) and finite digraphs (NP-complete), this difference disappears in the infinite case: we prove that solvability for recursive dags and digraphs is Σ11-complete. This implies that satisfiability for recursive theories of infinitary propositional logic is also Σ11 -complete. Finally, using solvability of dags we formulate a new equivalent of the axiom of choice.

Finding kernels or solving SAT

We begin by offering a new, direct proof of the equivalence between the problem of the existence of kernels in digraphs, KER, and satisfiability of propositional theories, SAT, giving linear reductions in both directions. Having introduced some linear reductions of the input graph, we present new algorithms for KER, with variations utilizing solvers of boolean equations. In the worst case, the algorithms try all assignments to either a feedback vertex set, F, or a set of nodes E touching only all even cycles. Hence KER is fixed parameter tractable not only in the size of F, as observed earlier, but also in the size of E. A slight modification of these algorithms leads to a branch and bound algorithm for KER which is virtually identical to the DPLL algorithm for SAT. This suggests deeper analogies between the two problems and the probable scenario of KER research facing the challenges known from the work on SAT. The algorithm gives also the upper bound O^@?(1.427^|^G^|) on the time complexity of general KER and O^@?(1.286^|^G^|) of KER for oriented graphs, where |G| is the number of vertices.

Strongly complete axiomatizations of “knowing at most” in syntactic structures

Syntactic structures based on standard syntactic assignments model knowledge directly rather than as truth in all possible worlds as in modal epistemic logic, by assigning arbitrary truth values to atomic epistemic formulae. This approach to epistemic logic is very general and is used in several logical frameworks modeling multi-agent systems, but has no interesting logical properties — partly because the standard logical language is too weak to express properties of such structures. In this paper we extend the logical language with a new operator used to represent the proposition that an agent “knows at most” a given finite set of formulae and study the problem of strongly complete axiomatization of syntactic structures in this language. Since the logic is not semantically compact, a strongly complete finitary axiomatization is impossible. Instead we present, first, a strongly complete infinitary system, and, second, a strongly complete finitary system for a slightly weaker variant of the language.

Mining sequences for patterns with non-repeating symbols

Finding the case id in unlabeled event logs is arguably one of the hardest challenges in process mining research. While this problem can be addressed with greedy approaches, these usually converge to sub-optimal solutions. In this paper, we describe an approach to perform complete search over the search space. We formulate the problem as a matter of finding the minimal set of patterns contained in a sequence, where patterns can be interleaved but do not have repeating symbols. We show that for practical purposes it is possible to reduce the search space to maximal disjoint occurrences of these patterns. Experimental results suggest that, whenever this approach finds a solution, it usually finds a minimal one.