Fixed-point Logic Research Articles

Connection between logical and algebraic approaches to concurrent systems

The logical and algebraic approaches are regarded as two of the dominant methodologies for the development of reactive and concurrent systems. It is well known that the logic approach is more abstract, but lacks compositionality; while the algebraic approach is inherently compositional, but lacks abstractness. However, connecting the two approaches is a major challenge in computer science, and many efforts have been directed to resolving the problem. Linking the algebraic approach to the logical approach has been satisfactorily resolved through the notion of characteristic formulae. But very limited success has been achieved so far in the other direction, as most of the established results have been developed only with respect to a simple semantics, which has usually been strong bisimulation. However, in practice, an observational semantics like weak bisimulation, which is much more complicated, is thought to be more useful. In this paper, we investigate how to connect the logical and algebraic approaches with respect to the observational preorder, which is a generalisation of weak bisimulation that takes divergence into account. We show the following results. First, we prove that the non-deterministic operator of process algebra can be defined in modal and temporal logics (such as the μ-calculus and the Fixpoint Logic with Chop) with respect to the observational preorder (in fact, the kernel of its precongruence). In this way, we can apply the logical approach to the design of a complex system in a compositional way. Second, we present two algorithms for constructing the characteristic formulae for a context-free process up to the preorder and its precongruence, respectively. The effect of this is that all the reductions for processes that are usually done in an algebraic setting can be handled in a logical setting.

Completeness for flat modal fixpoint logics

This paper exhibits a general and uniform method to prove axiomatic completeness for certain modal fixpoint logics. Given a set Γ of modal formulas of the form γ ( x , p 1 , … , p n ) , where x occurs only positively in γ , we obtain the flat modal fixpoint language L ♯ ( Γ ) by adding to the language of polymodal logic a connective ♯ γ for each γ ∈ Γ . The term ♯ γ ( φ 1 , … , φ n ) is meant to be interpreted as the least fixed point of the functional interpretation of the term γ ( x , φ 1 , … , φ n ) . We consider the following problem: given Γ , construct an axiom system which is sound and complete with respect to the concrete interpretation of the language L ♯ ( Γ ) on Kripke structures. We prove two results that solve this problem. First, let K ♯ ( Γ ) be the logic obtained from the basic polymodal K by adding a Kozen–Park style fixpoint axiom and a least fixpoint rule, for each fixpoint connective ♯ γ . Provided that each indexing formula γ satisfies a certain syntactic criterion, we prove this axiom system to be complete. Second, addressing the general case, we prove the soundness and completeness of an extension K ♯ + ( Γ ) of K ♯ ( Γ ) . This extension is obtained via an effective procedure that, given an indexing formula γ as input, returns a finite set of axioms and derivation rules for ♯ γ , of size bounded by the length of γ . Thus the axiom system K ♯ + ( Γ ) is finite whenever Γ is finite.

Generic Infinite Traces and Path-Based Coalgebraic Temporal Logics

This paper gives a general coalgebraic account of the notions of possibly infinite trace and possibly infinite execution in state-based, dynamical systems, by extending the generic theory of finite traces and executions developed by Hasuo and coauthors [I. Hasuo, B. Jacobs, and A. Sokolova. Generic trace semantics via coinduction. Logical Methods in Computer Science, 3:1–36, 2007]. The systems we consider are modelled as coalgebras of endofunctors obtained as the composition of a computational type (e.g. nondeterministic or stochastic) with a general transition type. This generalises existing work by Jacobs [B. Jacobs. Trace semantics for coalgebras. In Proc. CMCS 2004, volume 106 of ENTCS, 2004] that only accounts for a nondeterministic computational type. We subsequently introduce path-based temporal (including fixpoint) logics for coalgebras of such endofunctors, whose semantics is based upon the notion of possibly infinite execution. Our approach instantiates to both nondeterministic and stochastic computations, yielding, in particular, path-based fixpoint logics in the style of CTL* for nondeterministic systems, as well as generalisations of the logic PCTL for probabilistic systems.

A Solver for Modal Fixpoint Logics

We present MLSolver, a tool for solving the satisfiability and validity problems for modal fixpoint logics. The underlying technique is based on characterisations of satisfiability through infinite (cyclic) tableaux in which branches have an inner thread structure mirroring the regeneration of least and greatest fixpoint constructs in the infinite. Well-foundedness for unfoldings of least fixpoints is checked using deterministic parity automata. This reduces the satisfiability and validity problems to the problem of solving a parity game. MLSolver then uses a parity game solver in order to decide satisfiability and derives example models from the winning strategies in the parity game. Currently supported logics are the modal and linear-time μ-calculi, CTL*, and PDL (and therefore also CTL and LTL). MLSolver is designed to allow easy extensions in the form of further modal fixpoint logics.

Game Solution, Epistemic Dynamics and Fixed-Point Logics

Current methods for solving games embody a form of procedural rationality that invites logical analysis in its own right. This paper is a brief case study of Backward Induction for extensive games, replacing earlier static logical definitions by stepwise dynamic ones. We consider a number of analysis from recent years that look different conceptually, and find that they are all mathematically equivalent. This shows how an abstract logical perspective can bring out basic invariant structure in games. We then generalize this to an exploration of fixed-point logics on finite trees that best fit game-theoretic equilibria. We end with some open questions that suggest a broader program for merging current computational logics with notions and results from game theory. This paper is largely a program for opening up an area: an extended version of the technical results will be found in the forthcoming dissertation [26].

Multi-Valued Modal Fixed Point Logics for Model Checking

In this paper, I will show how multi-valued logics are used for model checking. Model checking is an automatic technique to analyze correctness of hardware and software systems. A model checker is based on a temporal logic or a modal fixed point logic. That is to say, a system to be checked is formalized as a Kripke model, a property to be satisfied by the system is formalized as a temporal formula or a modal formula, and the model checker checks that the Kripke model satisfies the formula. Although most existing model checkers are based on 2-valued logics, recently new attempts have been made to extend the underlying logics of model checkers to multi-valued logics. I will summarize these new results.

Natural deduction for the finite least fixed point logic with an infinitary rule

Natural deduction for the finite least fixed point logic with an infinitary rule

Fixpoint Logics over Hierarchical Structures

Hierarchical graph definitions allow a modular description of graphs using modules for the specification of repeated substructures. Beside this modularity, hierarchical graph definitions also allow to specify graphs of exponential size using polynomial size descriptions. In many cases, this succinctness increases the computational complexity of decision problems. In this paper, the model-checking problem for the modal μ-calculus and (monadic) least fixpoint logic on hierarchically defined input graphs is investigated. In order to analyze the modal μ-calculus, parity games on hierarchically defined input graphs are investigated. Precise upper and lower complexity bounds are derived. A restriction on hierarchical graph definitions that leads to more efficient model-checking algorithms is presented.

Recursive Definitions and Fixed-Points

An expression such as @?x(P(x)@[email protected](P)), where P occurs in @f(P), does not always define P. When such expression implicitly defines P, in the sense of Beth [Beth, E. W., On Padoa's method in the theory of definitions., Indagationes Mathematicae 15 (1953), pp. 330-339] and Padoa [Padoa, A., Essai d'une theorie algebrique des nombres entiers, precede d'une introduction logique a une theorie deductive quelconque, in: Bibliotheque Du Congres International de Philosophie, 1900, pp. 118-123], we call it a recursive definition. In the Least Fixed-Point Logic (LFP), we have theories where interesting relations can be recursively defined [Ebbinghaus, H.-D. and J. Flum, ''Finite Model Theory,'' Springer-Verlag, 1995; Libkin, L., ''Elements of Finite Model Theory,'' Springer, 2004]. We will show that for some sorts of recursive definitions there are explicit definitions on sufficiently strong theories of LFP. It is known that LFP, restricted to finite models, does not have Beth's Definability Theorem [Gurevich, Y. and S. Shelah, On finite rigid structures, Journal of Symbolic Logic 61 (1996), pp. 549-562; Hodkinson, I.M., Finite variable logics, Bulletin of the EATCS 51 (1993), pp. 111-140; Dawar, A., L. Hella and P.G. Kolaitis, Implicit definability and infinitary logic in finite model theory, in: ICALP '95: Proceedings of the 22nd International Colloquium on Automata, Languages and Programming (1995), pp. 624-635]. Beth's Definability Theorem states that, if a relation is implicitly defined, then there is an explicit definition for it. We will also give a proof that Beth's Definability Theorem fails for LFP without this finite model restriction. We intend to investigate fragments of LFP for which Beth's Definability Theorem holds.

Type Inference for Correspondence Types

We present a correspondence type/effect system for authenticity in a @p-calculus with polarized channels, dependent pair types and effect terms and show how one may, given a process P and an a priori type environment E, generate constraints that are formulae in the Alternating Least Fixed-Point (ALFP) logic. We then show how a reasonable model of the generated constraints yields a type/effect assignment such that P becomes well-typed with respect to E if and only if this is possible. The formulae generated satisfy a finite model property; a system of constraints is satisfiable if and only if it has a finite model. As a consequence, we obtain the result that type/effect inference in our system is polynomial-time decidable.

Affine systems of equations and counting infinitary logic

We study the definability of constraint satisfaction problems (CSPs) in various fixed-point and infinitary logics. We show that testing the solvability of systems of equations over a finite Abelian group, a tractable CSP that was previously known not to be definable in Datalog , is not definable in the infinitary logic with finitely many variables and counting. This implies that it is not definable in least fixed-point logic or its extension with counting. We relate definability of CSPs to their classification obtained from tame congruence theory of the varieties generated by the algebra of polymorphisms of the template structure. In particular, we show that if this variety admits either the unary or affine type, the corresponding CSP is not definable in the infinitary logic with counting.

Descriptive Complexity Theories

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Open answer set programming with guarded programs

Open answer set programming (OASP) is an extension of answer set programming where one may ground a program with an arbitrary superset of the program's constants. We define a fixed-point logic (FPL) extension of Clark's completion such that open answer sets correspond to models of FPL formulas and identify a syntactic subclass of programs, called ( loosely ) guarded programs . Whereas reasoning with general programs in OASP is undecidable, the FPL translation of (loosely) guarded programs falls in the decidable (loosely) guarded fixed-point logic (μ(L)GF). Moreover, we reduce normal closed ASP to loosely guarded OASP, enabling, for the first time, a characterization of an answer set semantics by μLGF formulas. We further extend the open answer set semantics for programs with generalized literals. Such generalized programs (gPs) have interesting properties, for example, the ability to express infinity axioms. We restrict the syntax of gPs such that both rules and generalized literals are guarded . Via a translation to guarded fixed-point logic, we deduce 2-EXPTIME-completeness of satisfiability checking in such guarded gPs (GgPs). Bound GgPs are restricted GgPs with EXPTIME-complete satisfiability checking, but still sufficiently expressive to optimally simulate computation tree logic (CTL). We translate Datalog lite programs to GgPs, establishing equivalence of GgPs under an open answer set semantics, alternation-free μGF, and Datalog LITE.

SUPERVALUATION FIXED-POINT LOGICS OF TRUTH

Michael Kremer defines fixed-point logics of truth based on Saul Kripke’s fixed point semantics for languages expressing their own truth concepts. Kremer axiomatizes the strong Kleene fixed-point logic of truth and the weak Kleene fixed-point logic of truth, but leaves the axiomatizability question open for the supervaluation fixed-point logic of truth and its variants. We show that the principal supervaluation fixed point logic of truth, when thought of as consequence relation, is highly complex: it is not even analytic. We also consider variants, engendered by a stronger notion of ‘fixed point’, and by variant supervaluation schemes. A ‘logic’ is often thought of, not as a consequence relation, but as a set of sentences – the sentences true on each interpretation. We axiomatize the supervaluation fixed-point logics so conceived.

Modular Games for Coalgebraic Fixed Point Logics

We build on existing work on finitary modular coalgebraic logics [C. Cîrstea. A compositional approach to defining logics for coalgebras. Theoretical Computer Science, 327:45–69, 2004; C. Cîrstea and D. Pattinson. Modular construction of complete coalgebraic logics. Theoretical Computer Science, 388:83–108, 2007], which we extend with general fixed points, including CTL- and PDL-like fixed points, and modular evaluation games. These results are generalisations of their correspondents in the modal μ-calculus, as described e.g. in [C. Stirling. Modal and Temporal Properties of Processes. Springer, 2001]. Inspired by recent work of Venema [Y. Venema. Automata and fixed point logic: a coalgebraic perspective. Information and Computation, 204:637–678, 2006], we provide our logics with evaluation games that come equipped with a modular way of building the game boards. We also study a specific class of modular coalgebraic logics that allow for the introduction of an implicit negation operator.

An Infinitary System for the Least Fixed-Point Logic restricted to Finite Models

The notion of the least fixed-point of an operator is widely applied in computer science as, for instance, in the context of query languages for relational databases. Some extensions of first-order classical logic (FOL) with fixed-point operators, as the least fixed-point logic (LFP), were proposed to deal with problems related to the expressivity of FOL. LFP captures the complexity class PTIME over the class of finite ordered structures. The descriptive characterization of computational classes is a central issue within finite model theory (FMT). Trakhtenbrot's theorem states that validity over finite models is not recursively enumerable, that is, completeness fails over finite models. This result is based on an underlying assumption that any deductive system is of finite nature. However, we can relax such assumption as done in the scope of proof theory for arithmetic. Motivated by Gödel incompleteness theorems, proof theory for arithmetic offer an example of a true mathematically meaningful principle non derivable in first-order arithmetic. One way of presenting this proof is based on a definition of a proof system with an infinitary rule, the ω-rule, that establishes the consistency of first-order arithmetic through a proof-theoretical perspective. Inspired by this proof, here we will propose an infinitary natural deduction system for FOL and LFP restricted to finite models, FOLfin and LFPfin, respectively, and we will prove soundness and completeness for them, and also a normalization theorem for fragments of these systems. With this infinitary deductive system for LFPfin, we aim to present a proof theory for a logic traditionally investigated within the scope of FMT. It opens up an alternative way of proving results already obtained within FMT and also new results through a proof theoretical perspective.

Choiceless polynomial time, counting and the Cai–Fürer–Immerman graphs

We consider Choiceless Polynomial Time ( C ̃ PT ), a language introduced by Blass, Gurevich and Shelah, and show that it can express a query originally constructed by Cai, Fürer and Immerman to separate fixed-point logic with counting ( IFP + C ) from P . This settles a question posed by Blass et al. The program we present uses sets of unbounded finite rank: we demonstrate that this is necessary by showing that the query cannot be computed by any program that has a constant bound on the rank of sets used, even in C ̃ PT(Card) , an extension of C ̃ PT with counting.

On Minimal Models

We investigate some logics which use the concept of minimal models in their definition. Minimal objects are widely used in Logic and Computer Science. They are applied in the context of Inductive Definitions, Logic Programming and Artificial Intelligence. An example of logic which uses this concept is the MIN(FO) logic due to van Benthem [20]. He shows that MIN(FO) is equivalent to the Least Fixed Point logic (LFP) in expressive power. In [6], we extended MIN(FO) to the MIN Logic and proved it is equivalent to second-order logic in expressive power. Here, we exhibit a fragment of MIN, the MIN Δ logic, which is more expressive than LFP, less expressive than MIN and closed under boolean connectives and first-order quantification. In order to do this, in the Section 2, we prove that the Downward Lowenheim-Skolem Theorem holds for arbitrary countable sets of LFP-formulas by showing that every infinite structure has a countable LFP-substructure. The method may be used to generalize this theorem to any set of LFP-formulas. We also analyse the expressive power of the Nested Abnormality Theories (NATs) of Lifschitz, another formalism based on minimal models used in Artificial Intelligence, and we demonstrate that for each second-order theory Γ there is a NAT which is a conservative extension of Γ. We give a translation from second-order sentences into such NATs which is linear in the size of the sentence in prenex normal form. Finally, we establish a hierarchy of expressiveness of these logics that deal with the concept of minimal models.

The Complexity of Model Checking Higher-Order Fixpoint Logic

Higher-Order Fixpoint Logic (HFL) is a hybrid of the simply typed \lambda-calculus and the modal \lambda-calculus. This makes it a highly expressive temporal logic that is capable of expressing various interesting correctness properties of programs that are not expressible in the modal \lambda-calculus. This paper provides complexity results for its model checking problem. In particular we consider those fragments of HFL built by using only types of bounded order k and arity m. We establish k-fold exponential time completeness for model checking each such fragment. For the upper bound we use fixpoint elimination to obtain reachability games that are singly-exponential in the size of the formula and k-fold exponential in the size of the underlying transition system. These games can be solved in deterministic linear time. As a simple consequence, we obtain an exponential time upper bound on the expression complexity of each such fragment. The lower bound is established by a reduction from the word problem for alternating (k-1)-fold exponential space bounded Turing Machines. Since there are fixed machines of that type whose word problems are already hard with respect to k-fold exponential time, we obtain, as a corollary, k-fold exponential time completeness for the data complexity of our fragments of HFL, provided m exceeds 3. This also yields a hierarchy result in expressive power.

Generalising automaticity to modal properties of finite structures

We introduce a complexity measure of modal properties of finite structures which generalises the automaticity of languages. It is based on graph-automata-like devices called labelling systems. We define a measure of the size of a structure that we call rank, and show that any modal property of structures can be approximated up to any fixed rank n by a labelling system. The function that takes n to the size of the smallest labelling system doing this is called the labelling index of the property. We demonstrate that this is a useful and fine-grained measure of complexity and show that it is especially well suited to characterise the expressive power of modal fixed-point logics. From this we derive several separation results of modal and non-modal fixed-point logics, some of which are already known whereas others are new.