Greatest Fixpoint Research Articles

Safety of Nöcker's strictness analysis

This paper proves correctness of Nöcker's method of strictness analysis, implemented in the Clean compiler, which is an effective way for strictness analysis in lazy functional languages based on their operational semantics. We improve upon the work Clark, Hankin and Hunt did on the correctness of the abstract reduction rules in two aspects. Our correctness proof is based on a functional core language and a contextual semantics, thus proving a wider range of strictness-based optimizations as correct, and our method fully considers the cycle detection rules, which contribute to the strength of Nöcker's strictness analysis.Our algorithm SAL is a reformulation of Nöcker's strictness analysis algorithm in a functional core language LR. This is a higher order call-by-need lambda calculus with case, constructors, letrec, and seq, which is extended during strictness analysis by set constants like Top or Inf, denoting sets of expressions, which indicate different evaluation demands. It is also possible to define new set constants by recursive equations with a greatest fixpoint semantics. The operational semantics of LR is a small-step semantics. Equality of expressions is defined by a contextual semantics that observes termination of expressions. Basically, SAL is a nontermination checker. The proof of its correctness and hence of Nöcker's strictness analysis is based mainly on an exact analysis of the lengths of evaluations, i.e., normal-order reduction sequences to WHNF. The main measure being the number of “essential” reductions in evaluations.Our tools and results provide new insights into call-by-need lambda calculi, the role of sharing in functional programming languages, and into strictness analysis in general. The correctness result provides a foundation for Nöcker's strictness analysis in Clean, and also for its use in Haskell.

Efficient static analysis of XML paths and types

We present an algorithm to solve XPath decision problems under regular tree type constraints and show its use to statically type-check XPath queries. To this end, we prove the decidability of a logic with converse for finite ordered trees whose time complexity is a simple exponential of the size of a formula. The logic corresponds to the alternation free modal μ-calculus without greatest fixpoint, restricted to finite trees, and where formulas are cycle-free. Our proof method is based on two auxiliary results. First, XML regular tree types and XPath expressions have a linear translation to cycle-free formulas. Second, the least and greatest fixpoints are equivalent for finite trees, hence the logic is closed under negation. Building on these results, we describe a practical, effective system for solving the satisfiability of a formula. The system has been experimented with some decision problems such as XPath emptiness, containment, overlap, and coverage, with or without type constraints. The benefit of the approach is that our system can be effectively used in static analyzers for programming languages manipulating both XPath expressions and XML type annotations (as input and output types).

Deciding inclusion of set constants over infinite non-strict data structures

Various static analyses of functional programming languages that permit infinite data structures make use of set constants like Top, Inf, and Bot, denoting all terms, all lists not eventually ending in Nil, and all non-terminating programs, respectively. We use a set language that permits union, constructors and recursive definition of set constants with a greatest fixpoint semantics in the set of all, also infinite, computable trees, where all term constructors are non-strict. This paper proves decidability, in particular DEXPTIME-completeness, of inclusion of co-inductively defined sets by using algorithms and results from tree automata and set constraints. The test for set inclusion is required by certain strictness analysis algorithms in lazy functional programming languages and could also be the basis for further set-based analyses.

Infinite Computation in the Equivalent Transformation Model

There are many logic programs that do not terminate but perform useful computation in some sense. The usual theory of logic programming adopts least fixpoints to define the meaning of programs, which fails to capture the intended meaning of infinite computation. To give an appropriate sense to useful infinite computation, the theory of logic programming has adopted greatest fixpoints in place of least fixpoints. However, this solution developed in logic paradigm can not explain finite and infinite computation in a unified manner. This paper proposes a new approach to infinite computation based on the equivalent transformation paradigm, where infinite computation is regarded as repeated equivalent transformations and is given appropriate sense in the same way as the usual finite computation.

TCTL Inevitability Analysis of Dense-Time Systems: From Theory to Engineering

Inevitability properties in branching temporal logics are of the syntax foralldiamphi, where phi is an arbitrary (timed) CTL (computation tree logic) formula. Such inevitability properties in dense-time logics can be analyzed with the greatest fixpoint calculation. We present algorithms to model-check inevitability properties. We discuss a technique for early decision on greatest fixpoint calculation which has shown promising performance against several benchmarks. We have experimented with various issues which may affect the performance of TCTL inevitability analysis. Specifically, our algorithms come with a parameter for the measurement of time-progress. We report the performance of our implementation with regard to various parameter values and with or without the non-Zeno computation requirement in the evaluation of greatest fixpoints. We have also experimented with safe abstraction techniques for model-checking TCTL inevitability properties. The experiment results help us in deducing rules for setting the parameter for verification performance. Finally, we summarize suggestions for configurations of efficient TCTL inevitability evaluation procedure

On the verification of finite failure

In Gori [An abstract interpretation framework to reason on finite failure and other properties of finite and infinite computations, Theoret. Comput. Sci. 290(1) (2003) 863–936] a new fixpoint semantics which correctly models finite failure has been defined. This semantics is And-compositional, compositional w.r.t. instantiation and is based on a co-continuous operator. Based on this fixpoint semantics a new inductive method able to verify a program w.r.t. the property of finite failure can be defined. In this paper we show how Ferrand's approach, using both a least fixpoint and greatest fixpoint semantics, can be adapted to finite failure. The verification method is not effective. Therefore, we consider an approximation from above and an approximation from below of our semantics, which give two different finite approximations. These approximations are used for effective program verification.

Memoryless Strategies for Stochastic Games via Domain Theory

In Formal Methods we use mathematical structures to model real systems we want to build, or to reason about; in this paper we are concerned principally with game-based models.In an earlier work [Morgan, C. and A. McIver, Cost analysis of games using program logic, in: Proc. of the 8th Asia-Pacific Software Engineering Conference (APSEC 2001), 2001, abstract only: full text available at [13, key MDP01].] we presented an approach to stochastic minimising games based on our logic for probabilistic programs [Morgan, C., A. McIver and K. Seidel, Probabilistic predicate transformers, ACM Transactions on Programming Languages and Systems 18 (1996) 325–53, doi.acm.org/10.1145/229542.229547, McIver, A. and C. Morgan, Abstraction, Refinement and Proof for Probabilistic Systems, Technical Monographs in Computer Science, Springer Verlag, New York, 2004]. The contribution here is to extend our logical approach to maximising games. That maximising and minimising are not (simply) duals is due to the use of least fixed-points in both cases, as is normal for iterating programs. (For duality, we would have to use greatest fixed-points in the maximising case.)The significance of that extension is that it makes a further link between the program-logical approach and the more general methods for finding control strategies that “solve” games of this kind, in the sense of giving the player a “recipe” which, if followed, will deliver the theoretically optimal result.

Game Over: The Foci Approach to LTL Satisfiability and Model Checking

Focus games have been shown to yield game-theoretical characterisations for the satisfiability and the model checking problem for various temporal logics. One of the players is given a tool – the focus – that enables him to show the regeneration of temporal operators characterised as least or greatest fixpoints. His strategy usually is build upon a priority list of formulas and, thus, is not positional. This paper defines foci games for satisfiability of LTL formulas. Strategies in these games are trivially positional since they parallelise all of the focus player's choices, thus resulting in a 1-player game in effect. The games are shown to be correct and to yield smaller (counter-)models than the focus games. Finally, foci games for model checking LTL are defined as well.

On Modal μ-Calculus and Non-Well-Founded Set Theory

A finitary characterization for non-well-founded sets with finite transitive closure is established in terms of a greatest fixpoint formula of the modal μ-calculus. This generalizes the standard result in the literature where a finitary modal characterization is provided only for wellfounded sets with finite transitive closure. The proof relies on the concept of automaton, leading then to new interlinks between automata theory and non-well-founded sets.

Timing analysis with crosstalk is a fixpoint on a complete lattice

Increasing delay variation due to capacitive and inductive crosstalk has a dramatic impact on deep submicron technologies. It is now impossible to exclude crosstalk from timing analysis. However, timing analysis with crosstalk is a mutual dependence problem since the crosstalk effect in turn depends on the timing behavior of a circuit. In this paper, we establish a theoretical foundation for timing analysis with crosstalk. We show that solutions to the problem are fixpoints on a complete lattice. Based on that, we prove in general the convergence of any iterative approach. We also show that, starting from different initial solutions, an iterative approach will reach different fixpoints. The current prevailing practice, which starts from the worst case solution, will always reach the greatest fixpoint, which is the loosest solution. In order to reach the least fixpoint, we need to start from the best case solution. The convergence rates for both discrete and continuous models are discussed. Based on chaotic iteration and heterogeneous structures of coupled circuits, techniques to speed up iterations are also provided.

A fixpoint theory for non-monotonic parallelism

This paper studies parallel recursion. The trace specification language used in this paper incorporates sequentially, nondeterminism, reactiveness (including infinite traces), three forms of parallelism (including conjunctive, fair-interleaving and synchronous parallelism) and general recursion. In order to use Tarski's theorem to determine the fixpoints of recursions, we need to identify a well-founded partial order. Several orders are considered, including a new order called the lexical order, which tends to simulate the execution of a recursion in a similar manner as the Egli-Milner order. A theorem of this paper shows that no appropriate order exists for the language. Tarski's theorem alone is not enough to determine the fixpoints of parallel recursions. Instead of using Tarski's theorem directly, we reason about the fixpoints of terminating and nonterminating behaviours separately. Such reasoning is supported by the laws of a new composition called partition. We propose a fixpoint technique called the partitioned fixpoint, which is the least fixpoint of the nonterminating behaviours after the terminating behaviours reach their greatest fixpoint. The surprising result is that although a recursion may not be lexical-order monotonic, it must have the partitioned fixpoint, which is equal to the least lexical-order fixpoint. Since the partitioned fixpoint is well defined in any complete lattice, the results are applicable to various semantic models. Existing fixpoint techniques simply become special cases of the partitioned fixpoint. For example, an Egli-Milner-monotonic recursion has its least Egli-Milner fixpoint, which can be shown to be the same as the partitioned fixpoint. The new technique is more general than the least Egli-Milner fixpoint in that the partitioned fixpoint can be determined even when a recursion is not Egli-Milner monotonic. Examples of non-monotonic recursions are studied. Their partitioned fixpoints are shown to be consistent with our intuition.

A fixpoint theory for non-monotonic parallelism

This paper studies parallel recursion. The trace specification language used in this paper incorporates sequentially, nondeterminism, reactiveness (including infinite traces), three forms of parallelism (including conjunctive, fair-interleaving and synchronous parallelism) and general recursion. In order to use Tarski's theorem to determine the fixpoints of recursions, we need to identify a well-founded partial order. Several orders are considered, including a new order called the lexical order, which tends to simulate the execution of a recursion in a similar manner as the Egli-Milner order. A theorem of this paper shows that no appropriate order exists for the language. Tarski's theorem alone is not enough to determine the fixpoints of parallel recursions. Instead of using Tarski's theorem directly, we reason about the fixpoints of terminating and nonterminating behaviours separately. Such reasoning is supported by the laws of a new composition called partition. We propose a fixpoint technique called the partitioned fixpoint, which is the least fixpoint of the nonterminating behaviours after the terminating behaviours reach their greatest fixpoint. The surprising result is that although a recursion may not be lexical-order monotonic, it must have the partitioned fixpoint, which is equal to the least lexical-order fixpoint. Since the partitioned fixpoint is well defined in any complete lattice, the results are applicable to various semantic models. Existing fixpoint techniques simply become special cases of the partitioned fixpoint. For example, an Egli-Milner-monotonic recursion has its least Egli-Milner fixpoint, which can be shown to be the same as the partitioned fixpoint. The new technique is more general than the least Egli-Milner fixpoint in that the partitioned fixpoint can be determined even when a recursion is not Egli-Milner monotonic. Examples of non-monotonic recursions are studied. Their partitioned fixpoints are shown to be consistent with our intuition.

Deciding low levels of tree-automata hierarchy

The paper concerns hierarchy of indices of finite automata over infinite objects. This hierarchy corresponds exactly to the hierarchy of alternations of least and greatest fixpoints in the mu-calculus. It is also connected to quantifier hierarchies in monadic second-order logic. The open question is to find a procedure that given a regular tree language decides its level in the index hierarchy. Here, decision procedures are presented for low levels of the hierarchy. It is shown that these procedures have optimal complexity.

On Greatest Fixpoint Semantics of Logic Programming

The study of fixpoints has long been at the heart of logic programming. However, whereas least fixpoint semantics works well for SLD‐refutations (i.e. is sound and complete), there is no satisfactory (i.e. complete) fixpoint semantics for infinite derivations. In this paper, we focus on this problem. Standard approaches in this area consist in concentrating on infinite derivations that can be seen as computing, in the limit, some infinite object. This is usually done by extending the domain of computation with infinite elements and then defining the meaning of programs in terms of greatest fixpoints. The main drawback of these approaches is that the semantics defined is not complete. Hence, since defining a greatest fixpoint semantics for logic programs amounts to consider a program as a set of co‐inductive definitions, we focus on this identification at a deeper level by considering infinite derivations as proof‐terms in a co‐inductive set. This reading leads into considering derivations as proofs rather than computations and allows one to show that for the subclass of infinite derivations over the domain of finite terms, a complete greatest fixpoint semantics can be obtained. Our main result is that the greatest fixpoint of the one‐step‐inference operator for the C‐semantics corresponds to atoms that have a non‐failing fair derivation with the additional property that complete information over a variable is obtained after finitely many steps.

Free μ-lattices

A μ-lattice is a lattice with the property that every unary polynomial has both a least and a greatest fix-point. In this paper we define the quasivariety of μ-lattices and, for a given partially ordered set P, we construct a μ-lattice J P whose elements are equivalence classes of games in a preordered class J(P) . We prove that the μ-lattice J P is free over the ordered set P and that the order relation of J P is decidable if the order relation of P is decidable. By means of this characterization of free μ-lattices we infer that the class of complete lattices generates the quasivariety of μ-lattices.

Mu-depth 3 is more than 2: a game-theoretic proof

We consider the propositional modal mu-calculus, a logic proposed by Kozen in 1983. In this logic two operators μ and v are present, which express the least and greatest fixpoints of monotone operators on sets. Bradfield in 1998 proved for any n the existence of a mu-calculus formula that requires n alternations of μ and v. In this paper we consider the particular case n = 3 and we exhibit a new formula requiring 3 alternations. Our proof is independent of the technique of Bradfield, and is based on a new kind of game on infinite trees.

Free mu-lattices

A mu-lattice is a lattice with the property that every unary <br />polynomial has both a least and a greatest fix-point. In this paper<br />we define the quasivariety of mu-lattices and, for a given partially<br />ordered set P, we construct a mu-lattice JP whose elements are<br />equivalence classes of games in a preordered class J (P). We prove<br />that the mu-lattice JP is free over the ordered set P and that the<br />order relation of JP is decidable if the order relation of P is <br />decidable. By means of this characterization of free mu-lattices we<br />infer that the class of complete lattices generates the quasivariety<br />of mu-lattices.<br />Keywords: mu-lattices, free mu-lattices, free lattices, bicompletion<br />of categories, models of computation, least and greatest fix-points,<br />mu-calculus, Rabin chain games.

Recursion and Corecursion Have the Same Equational Logic

This paper is concerned with the equational logic of corecursion, that is of definitions involving final coalgebra maps. The framework for our study is iteration theories (cf. e.g. [1,2]), recently reintroduced as models of the FLR0 fragment of the Formal Language of Recursion [5–7]. We present a new class of iteration theories derived from final coalgebras. This allows us to reason with a number of types of fixed-point equations which heretofore seemed to require to metric or order-theoretic ideas. All of the work can be done using finality properties and equational reasoning.Having a semantics, we obtain the following completeness result: the equations involving fixed-point terms which are valid for final coalgebra interpretations are exactly those valid in a number of contexts pertaining to recursion. For example, they coincide with the equations valid for least-fixed point recursion on cpo's.

Applying the mu-calculus in planning and reasoning about action

Planning algorithms have traditionally been geared towards achievement goals in single-agent environments. Such algorithms essentially produce plans to reach one of a specified set of states. More general approaches for planning based on temporal logic (TL) are emerging. Current approaches tend to use linear TL, and can handle sets of sequences of states. However, they assume deterministic actions with all changes effected solely by one agent. By contrast, we use a branching model of time that can express concurrent actions by multiple agents and the environment, leading to nondeterministic effects of an agent's actions. For this reason, we view plans not as sequences of actions, but as decision graphs describing the agent's actions in different situations. Thus, although we consider single-agent decision graphs, our approach is better suited to multiagent systems. We also consider an expressive formalism, which allows a wider variety of goals, including achievement and maintenance goals. Achievement corresponds to traditional planning, but maintenance is more powerful than traditional maintenance goals, and may require nonterminating plans. To formalize decision graphs requires a means to ‘alternate’ the agent's and the environment's choices. From logics of program, we introduce the prepositional mu-calculus;, which has operators for least and greatest fixpoints. We give a semantics, a fixpoint characterization, and an algorithm to compute decision graphs.

Extracting schema from semistructured data

Semistructured data is characterized by the lack of any fixed and rigid schema, although typically the data has some implicit structure. While the lack of fixed schema makes extracting semistructured data fairly easy and an attractive goal, presenting and querying such data is greatly impaired. Thus, a critical problem is the discovery of the structure implicit in semistructured data and, subsequently, the recasting of the raw data in terms of this structure. In this paper, we consider a very general form of semistructured data based on labeled, directed graphs. We show that such data can be typed using the greatest fixpoint semantics of monadic datalog programs. We present an algorithm for approximate typing of semistructured data. We establish that the general problem of finding an optimal such typing is NP-hard, but present some heuristics and techniques based on clustering that allow efficient and near-optimal treatment of the problem. We also present some preliminary experimental results.