arXiv subjects

Clemens Grabmayer

Publications and source records attributed to Clemens Grabmayer.

At least 19 recordsLinked to original sources

arXiv2025

Bisimilarity and Simulatability of Processes Parameterized by Join Interactions

Departing from Larsen's concept of parameterized bisimilarity of processes with respect to interaction with environments, we start an exploration of its natural weakening: bisimilarity of unrestricted join interactions with environments. Parameterized bisimilarity relates processes p and q with respect to an environment e if p and q behave bi-similarly while joining -- respectively the same -- transitions from e. The weakened variant relates processes p and q with respect to environment e if the join-interaction processes p & e and q & e of p and q with e are bisimilar. (Hereby join interactions r & f facilitate a step with label a to r' & f' if and only if r and f permit a-steps to r' and f' , respectively.) Join-interaction parameterized (ji-parameterized) bisimilarity coincides with parameterized bisimilarity for deterministic environments, but that it is a coarser equivalence in general. We explain how Larsen's concept can be recovered from ji-parameterized bisimilarity by 'determinizing' interactions. We show that by adaptation to simulatability (simulation preorder) the same concept arises: parameterized simulatability coincides with ji-parameterized simulatability. For the discrimination preorder of (ji-)parameterized simulatability on environments we obtain the same result as Larsen did for parameterized bisimilarity. Also, we give a modal-logic characterization of (ji-)parameterized simulatability. Finally we gather open problems, and provide an outlook on our current related work.

cs.PL↗

arXiv2024

From Compactifying Lambda-Letrec Terms to Recognizing Regular-Expression Processes

As a supplement to my talk at the workshop, this extended abstract motivates and summarizes my work with co-authors on problems in two separate areas: first, in the lambda-calculus with letrec, a universal model of computation, and second, on Milner's process interpretation of regular expressions, a proper subclass of the finite-state processes. The aim of my talk was to motivate a transferal of ideas for workable concepts of structure-constrained graphs: from the problem of finding compact graph representations for terms in the lambda-calculus with letrec to the problem of recognizing finite process graphs that can be expressed by regular expressions. In both cases the construction of structure-constrained graphs was expedient in order to enable to go back and forth easily between, in the first case, lambda-terms and term graphs, and in the second case, regular expressions and process graphs. The main focus here is on providing pointers to my work with co-authors, in both areas separately. A secondary focus is on explaining directions of my present projects, and describing research questions of possibly general interest that have developed out of my work in these two areas.

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arXiv2023

The Image of the Process Interpretation of Regular Expressions is Not Closed under Bisimulation Collapse

Axiomatization and expressibility problems for Milner's process semantics (1984) of regular expressions modulo bisimilarity have turned out to be difficult for the full class of expressions with deadlock 0 and empty step~1. We report on a phenomenon that arises from the added presence of 1 when 0 is available, and that brings a crucial reason for this difficulty into focus. To wit, while interpretations of 1-free regular expressions are closed under bisimulation collapse, this is not the case for the interpretations of arbitrary regular expressions. Process graph interpretations of 1-free regular expressions satisfy the loop existence and elimination property LEE, which is preserved under bisimulation collapse. These features of LEE were applied for showing that an equational proof system for 1-free regular expressions modulo bisimilarity is complete, and that it is decidable in polynomial time whether a process graph is bisimilar to the interpretation of a 1-free regular expression. While interpretations of regular expressions do not satisfy the property LEE in general, we show that LEE can be recovered by refined interpretations as graphs with 1-transitions refined interpretations with 1-transitions (which are similar to silent steps for automata). This suggests that LEE can be expedient also for the general axiomatization and expressibility problems. But a new phenomenon emerges that needs to be addressed: the property of a process graph `to can be refined into a process graph with 1-transitions and with LEE' is not preserved under bisimulation collapse. We provide a 10-vertex graph with two 1-transitions that satisfies LEE, and in which a pair of bisimilar vertices cannot be collapsed on to each other while preserving the refinement property. This implies that the image of the process interpretation of regular expressions is not closed under bisimulation collapse.

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arXiv2023

A Coinductive Reformulation of Milner's Proof System for Regular Expressions Modulo Bisimilarity

Milner (1984) defined an operational semantics for regular expressions as finite-state processes. In order to axiomatize bisimilarity of regular expressions under this process semantics, he adapted Salomaa's proof system that is complete for equality of regular expressions under the language semantics. Apart from most equational axioms, Milner's system Mil inherits from Salomaa's system a non-algebraic rule for solving single fixed-point equations. Recognizing distinctive properties of the process semantics that render Salomaa's proof strategy inapplicable, Milner posed completeness of the system Mil as an open question. As a proof-theoretic approach to this problem we characterize the derivational power that the fixed-point rule adds to the purely equational part Mil$^-$ of Mil. We do so by means of a coinductive rule that permits cyclic derivations that consist of a finite process graph with empty steps that satisfies the layered loop existence and elimination property LLEE, and two of its Mil$^{-}$-provable solutions. With this rule as replacement for the fixed-point rule in Mil, we define the coinductive reformulation cMil as an extension of Mil$^{-}$. In order to show that cMil and Mil are theorem equivalent we develop effective proof transformations from Mil to cMil, and vice versa. Since it is located half-way in between bisimulations and proofs in Milner's system Mil, cMil may become a beachhead for a completeness proof of Mil. This article extends our contribution to the CALCO 2022 proceedings. Here we refine the proof transformations by framing them as eliminations of derivable and admissible rules, and we link coinductive proofs to a coalgebraic formulation of solutions of process graphs.

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arXiv2023

Proceedings Twelfth International Workshop on Computing with Terms and Graphs

The workshop TERMGRAPH 2022 took place at Technion in Haifa, Israel, on August 1, 2022, in the Pre-FLoC workshop block (July 31-August 1) of FLoC 2022 (Federated Logic Conference 2022, July 31-August 12). As such, TERMGRAPH 2022 was a one-day satellite event of the conference FSCD 2022 (Formal Structures of Computation and Deduction 2020, August 2-5).

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arXiv2022

Milner's Proof System for Regular Expressions Modulo Bisimilarity is Complete (Crystallization: Near-Collapsing Process Graph Interpretations of Regular Expressions)

Milner (1984) defined a process semantics for regular expressions. He formulated a sound proof system for bisimilarity of process interpretations of regular expressions, and asked whether this system is complete. We report conceptually on a proof that shows that Milner's system is complete, by motivating, illustrating, and describing all of its main steps. We substantially refine the completeness proof by Grabmayer and Fokkink (2020) for the restriction of Milner's system to `1-free' regular expressions. As a crucial complication we recognize that process graphs with empty-step transitions that satisfy the layered loop-existence/elimination property LLEE are not closed under bisimulation collapse (unlike process graphs with LLEE that only have proper-step transitions). We circumnavigate this obstacle by defining a LLEE-preserving `crystallization procedure' for such process graphs. By that we obtain `near-collapsed' process graphs with LLEE whose strongly connected components are either collapsed or of `twin-crystal' shape. Such near-collapsed process graphs guarantee provable solutions for bisimulation collapses of process interpretations of regular expressions.

cs.LO↗

arXiv2021

A Coinductive Version of Milner's Proof System for Regular Expressions Modulo Bisimilarity

By adapting Salomaa's complete proof system for equality of regular expressions under the language semantics, Milner (1984) formulated a sound proof system for bisimilarity of regular expressions under the process interpretation he introduced. He asked whether this system is complete. Proof-theoretic arguments attempting to show completeness of this equational system are complicated by the presence of a non-algebraic rule for solving fixed-point equations by using star iteration. We characterize the derivational power that the fixed-point rule adds to the purely equational part $\text{Mil$^{\boldsymbol{-}}$}$ of Milner's system $\text{$\text{Mil}$}$: it corresponds to the power of coinductive proofs over $\text{Mil$^{\boldsymbol{-}}$}$ that have the form of finite process graphs with the loop existence and elimination property $\text{LEE}$. We define a variant system $\text{cMil}$ by replacing the fixed-point rule in $\text{Mil}$ with a rule that permits $\text{LEE}$-shaped circular derivations in $\text{Mil$^{\boldsymbol{-}}$}$ from previously derived equations as a premise. With this rule alone we also define the variant system $\text{CLC}$ for merely combining $\text{LEE}$-shaped coinductive proofs over $\text{Mil$^{\boldsymbol{-}}$}$. We show that both $\text{cMil}$ and $\text{CLC}$ have proof interpretations in $\text{Mil}$, and vice versa. As this correspondence links, in both directions, derivability in $\text{Mil}$ with derivation trees of process graphs, it widens the space for graph-based approaches to finding a completeness proof of Milner's system. This report is the extended version of a paper with the same title presented at CALCO 2021.

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arXiv2021

Structure-Constrained Process Graphs for the Process Semantics of Regular Expressions

Milner (1984) introduced a process semantics for regular expressions as process graphs. Unlike for the language semantics, where every regular (that is, DFA-accepted) language is the interpretation of some regular expression, there are finite process graphs that are not bisimilar to the process interpretation of any regular expression. For reasoning about graphs that are expressible by regular expressions modulo bisimilarity it is desirable to have structural representations of process graphs in the image of the interpretation. For '1-free' regular expressions, their process interpretations satisfy the structural property LEE (loop existence and elimination). But this is not in general the case for all regular expressions, as we show by examples. Yet as a remedy, we describe the possibility to recover the property LEE for a close variant of the process interpretation. For this purpose we refine the process semantics of regular expressions to yield process graphs with 1-transitions, similar to silent moves for finite-state automata.

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arXiv2020

Structure-Constrained Process Graphs for the Process Semantics of Regular Expressions

Milner (1984) introduced a process semantics for regular expressions as process graphs. Unlike for the language semantics, where every regular (that is, DFA-accepted) language is the interpretation of some regular expression, there are finite process graphs that are not bisimilar to the process interpretation of any regular expression. For reasoning about graphs that are expressible by regular expressions modulo bisimilarity it is desirable to have structural representations of process graphs in the image of the interpretation. For `1-free' regular expressions, their process interpretations satisfy the structural property LEE (loop existence and elimination). But this is not in general the case for all regular expressions, as we show by examples. Yet as a remedy, we describe the possibility to recover the property LEE for a close variant of the process interpretation. For this purpose we refine the process semantics of regular expressions to yield process graphs with 1-transitions, similar to silent moves for finite-state automata. This report accompanies the paper with the same title in the post-proceedings of the workshop TERMGRAPH 2020. Here we give the proofs of not only one but of both of the two central theorems.

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arXiv2020

A Complete Proof System for 1-Free Regular Expressions Modulo Bisimilarity

Robin Milner (1984) gave a sound proof system for bisimilarity of regular expressions interpreted as processes: Basic Process Algebra with unary Kleene star iteration, deadlock 0, successful termination 1, and a fixed-point rule. He asked whether this system is complete. Despite intensive research over the last 35 years, the problem is still open. This paper gives a partial positive answer to Milner's problem. We prove that the adaptation of Milner's system over the subclass of regular expressions that arises by dropping the constant 1, and by changing to binary Kleene star iteration is complete. The crucial tool we use is a graph structure property that guarantees expressibility of a process graph by a regular expression, and is preserved by going over from a process graph to its bisimulation collapse.

cs.LO↗

arXiv2019

Linear Depth Increase of Lambda Terms along Leftmost-Outermost Beta-Reduction

Performing $n$ steps of $β$-reduction to a given term in the $λ$-calculus can lead to an increase in the size of the resulting term that is exponential in $n$. The same is true for the possible depth increase of terms along a $β$-reduction sequence. We explain that the situation is different for the leftmost-outermost strategy for $β$-reduction: while exponential size increase is still possible, depth increase is bounded linearly in the number of steps. For every $λ$-term $M$ with depth $d$, in every step of a leftmost-outermost $β$-reduction rewrite sequence starting from $M$ the term depth increases by at most $d$. Hence the depth of the $n$-th reduct of $M$ in such a rewrite sequence is bounded by $d\cdot (n+1)$. We prove the lifting of this result to $λ$-term representations as orthogonal first-order term rewriting systems, which can be obtained by the lambda-lifting transformation. For the transfer to lambda-calculus, we rely on correspondence statements via lambda-lifting. We argue that the linear-depth-increase property can be a stepping stone for an alternative proof of, and so can shed new light on, a result by Accattoli and Dal Lago (2015) that states: leftmost-outermost $β$-reduction rewrite sequences of length $n$ in the lambda-calculus can be implemented on a reasonable machine with an overhead that is polynomial in $n$ and the size of the initial term.

cs.LO↗

arXiv2019

Modeling Terms by Graphs with Structure Constraints (Two Illustrations)

In the talk at the workshop my aim was to demonstrate the usefulness of graph techniques for tackling problems that have been studied predominantly as problems on the term level: increasing sharing in functional programs, and addressing questions about Milner's process semantics for regular expressions. For both situations an approach that is based on modeling terms by graphs with structure constraints has turned out to be fruitful. In this extended abstract I describe the underlying problems, give references, provide examples, indicate the chosen approaches, and compare the initial situations as well as the results that have been obtained, and some results that are being developed at present.

cs.LO↗

arXiv2015

The Degree of Squares is an Atom (Extended Version)

We answer an open question in the theory of degrees of infinite sequences with respect to transducibility by finite-state transducers. An initial study of this partial order of degrees was carried out in (Endrullis, Hendriks, Klop, 2011), but many basic questions remain unanswered. One of the central questions concerns the existence of atom degrees, other than the degree of the `identity sequence' 1 0^0 1 0^1 1 0^2 1 0^3 .... A degree is called an `atom' if below it there is only the bottom degree 0, which consists of the ultimately periodic sequences. We show that also the degree of the `squares sequence' 1 0^0 1 0^1 1 0^4 1 0^9 1 0^{16} ... is an atom. As the main tool for this result we characterise the transducts of `spiralling' sequences and their degrees. We use this to show that every transduct of a `polynomial sequence' either is in 0 or can be transduced back to a polynomial sequence for a polynomial of the same order.

cs.FL↗

arXiv2015

Nested Term Graphs (Work In Progress)

We report on work in progress on 'nested term graphs' for formalizing higher-order terms (e.g. finite or infinite lambda-terms), including those expressing recursion (e.g. terms in the lambda-calculus with letrec). The idea is to represent the nested scope structure of a higher-order term by a nested structure of term graphs. Based on a signature that is partitioned into atomic and nested function symbols, we define nested term graphs both in a functional representation, as tree-like recursive graph specifications that associate nested symbols with usual term graphs, and in a structural representation, as enriched term graph structures. These definitions induce corresponding notions of bisimulation between nested term graphs. Our main result states that nested term graphs can be implemented faithfully by first-order term graphs. keywords: higher-order term graphs, context-free grammars, cyclic lambda-terms, higher-order rewrite systems

cs.LO↗

arXiv2015

Regularity Preserving but not Reflecting Encodings

Encodings, that is, injective functions from words to words, have been studied extensively in several settings. In computability theory the notion of encoding is crucial for defining computability on arbitrary domains, as well as for comparing the power of models of computation. In language theory much attention has been devoted to regularity preserving functions. A natural question arising in these contexts is: Is there a bijective encoding such that its image function preserves regularity of languages, but its pre-image function does not? Our main result answers this question in the affirmative: For every countable class C of languages there exists a bijective encoding f such that for every language L in C its image f[L] is regular. Our construction of such encodings has several noteworthy consequences. Firstly, anomalies arise when models of computation are compared with respect to a known concept of implementation that is based on encodings which are not required to be computable: Every countable decision model can be implemented, in this sense, by finite-state automata, even via bijective encodings. Hence deterministic finite-state automata would be equally powerful as Turing machine deciders. A second consequence concerns the recognizability of sets of natural numbers via number representations and finite automata. A set of numbers is said to be recognizable with respect to a representation if an automaton accepts the language of representations. Our result entails that there is one number representation with respect to which every recursive set is recognizable.

cs.FL↗

arXiv2014

Maximal Sharing in the Lambda Calculus with letrec

Increasing sharing in programs is desirable to compactify the code, and to avoid duplication of reduction work at run-time, thereby speeding up execution. We show how a maximal degree of sharing can be obtained for programs expressed as terms in the lambda calculus with letrec. We introduce a notion of `maximal compactness' for lambda-letrec-terms among all terms with the same infinite unfolding. Instead of defined purely syntactically, this notion is based on a graph semantics. lambda-letrec-terms are interpreted as first-order term graphs so that unfolding equivalence between terms is preserved and reflected through bisimilarity of the term graph interpretations. Compactness of the term graphs can then be compared via functional bisimulation. We describe practical and efficient methods for the following two problems: transforming a lambda-letrec-term into a maximally compact form; and deciding whether two lambda-letrec-terms are unfolding-equivalent. The transformation of a lambda-letrec-term $L$ into maximally compact form $L_0$ proceeds in three steps: (i) translate L into its term graph $G = [[ L ]]$; (ii) compute the maximally shared form of $G$ as its bisimulation collapse $G_0$; (iii) read back a lambda-letrec-term $L_0$ from the term graph $G_0$ with the property $[[ L_0 ]] = G_0$. This guarantees that $L_0$ and $L$ have the same unfolding, and that $L_0$ exhibits maximal sharing. The procedure for deciding whether two given lambda-letrec-terms $L_1$ and $L_2$ are unfolding-equivalent computes their term graph interpretations $[[ L_1 ]]$ and $[[ L_2 ]]$, and checks whether these term graphs are bisimilar. For illustration, we also provide a readily usable implementation.

cs.PL↗

arXiv2014

Infinitary Term Rewriting for Weakly Orthogonal Systems: Properties and Counterexamples

We present some contributions to the theory of infinitary rewriting for weakly orthogonal term rewrite systems, in which critical pairs may occur provided they are trivial. We show that the infinitary unique normal form property fails by an example of a weakly orthogonal TRS with two collapsing rules. By translating this example, we show that this property also fails for the infinitary lambda-beta-eta-calculus. As positive results we obtain the following: Infinitary confluence, and hence the infinitary unique normal forms property, holds for weakly orthogonal TRSs that do not contain collapsing rules. To this end we refine the compression lemma. Furthermore, we establish the triangle and diamond properties for infinitary multi-steps (complete developments) in weakly orthogonal TRSs, by refining an earlier cluster-analysis for the finite case.

cs.LO↗

arXiv2013

Term Graph Representations for Cyclic Lambda-Terms

We study various representations for cyclic lambda-terms as higher-order or as first-order term graphs. We focus on the relation between `lambda-higher-order term graphs' (lambda-ho-term-graphs), which are first-order term graphs endowed with a well-behaved scope function, and their representations as `lambda-term-graphs', which are plain first-order term graphs with scope-delimiter vertices that meet certain scoping requirements. Specifically we tackle the question: Which class of first-order term graphs admits a faithful embedding of lambda-ho-term-graphs in the sense that: (i) the homomorphism-based sharing-order on lambda-ho-term-graphs is preserved and reflected, and (ii) the image of the embedding corresponds closely to a natural class (of lambda-term-graphs) that is closed under homomorphism? We systematically examine whether a number of classes of lambda-term-graphs have this property, and we find a particular class of lambda-term-graphs that satisfies this criterion. Term graphs of this class are built from application, abstraction, variable, and scope-delimiter vertices, and have the characteristic feature that the latter two kinds of vertices have back-links to the corresponding abstraction. This result puts a handle on the concept of subterm sharing for higher-order term graphs, both theoretically and algorithmically: We obtain an easily implementable method for obtaining the maximally shared form of lambda-ho-term-graphs. Also, we open up the possibility to pull back properties from first-order term graphs to lambda-ho-term-graphs. In fact we prove this for the property of the sharing-order successors of a given term graph to be a complete lattice with respect to the sharing order. This report extends the paper with the same title (http://arxiv.org/abs/1302.6338v1) in the proceedings of the workshop TERMGRAPH 2013.

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