Monadic First-order Logic Research Articles

A Syntactic Proof of the Decidability of First-Order Monadic Logic

Decidability of monadic first-order classical logic was established by Löwenheim in 1915. The proof made use of a semantic argument and a purely syntactic proof has never been provided. In the present paper we introduce a syntactic proof of decidability of monadic first-order logic in innex normal form which exploits G3-style sequent calculi. In particular, we introduce a cut- and contraction-free calculus having a (complexity-optimal) terminating proof-search procedure. We also show that this logic can be faithfully embedded in the modal logic T.

INTERLEAVING LOGIC AND COUNTING

AbstractReasoning with quantifier expressions in natural language combines logical and arithmetical features, transcending strict divides between qualitative and quantitative. Our topic is this cooperation of styles as it occurs in common linguistic usage and its extension into the broader practice of natural language plus ‘grassroots mathematics’.We begin with a brief review of $\mathsf {FO}(\#)$ , first-order logic with counting operators and cardinality comparisons. This system is known to be of very high complexity, and drowns out finer aspects of the combination of logic and counting. We therefore move to a small fragment that can represent numerical syllogisms and basic reasoning about comparative size: monadic first-order logic with counting, $\mathsf {MFO}(\#)$ . We provide normal forms that allow for axiomatization, determine which arithmetical notions can be defined on finite and on infinite models, and conversely, we discuss which logical notions can be defined out of purely arithmetical ones, and what sort of (non-)classical logics can be induced.Next, we investigate a series of strengthenings of $\mathsf {MFO}(\#)$ , again using normal form methods. The monadic second-order version is close, in a precise sense, to additive Presburger Arithmetic, while versions with the natural device of tuple counting take us to Diophantine equations, making the logic undecidable. We also define a system $\mathsf {ML}(\#)$ that combines basic modal logic over binary accessibility relations with counting, needed to formulate ubiquitous reasoning patterns such as the Pigeonhole Principle. We prove decidability of $\mathsf {ML}(\#)$ , and provide a new kind of bisimulation matching the expressive power of the language.As a complement to the fragment approach pursued here, we also discuss two other ways of lowering the complexity of $\mathsf {FO}(\#)$ by changing the semantics of counting in natural ways. A first approach replaces cardinalities by abstract but well-motivated values of ‘mass’ or other mereological aggregating notions. A second approach keeps the cardinalities but generalizes the meaning of counting to work in models that allow dependencies between variables.Finally, we return to our starting point in natural language, confronting the architecture of our formal systems with linguistic quantifier vocabulary and syntax, as well as with natural reasoning modules such as the monotonicity calculus. In addition to these encounters with formal semantics, we discuss the role of counting in semantic evaluation procedures for quantifier expressions and determine, for instance, which binary quantifiers are computable by finite ‘semantic automata’. We conclude with some general thoughts on yet further entanglements of logic and counting in formal systems, on rethinking the qualitative/quantitative divide, and on connecting our analysis to empirical findings in cognitive science.

Model theory of monadic predicate logic with the infinity quantifier

This paper establishes model-theoretic properties of texttt {M} texttt {E} ^{infty }, a variation of monadic first-order logic that features the generalised quantifier exists ^infty (‘there are infinitely many’). We will also prove analogous versions of these results in the simpler setting of monadic first-order logic with and without equality (texttt {M} texttt {E} and texttt {M} , respectively). For each logic texttt {L} in { texttt {M} , texttt {M} texttt {E} , texttt {M} texttt {E} ^{infty }} we will show the following. We provide syntactically defined fragments of texttt {L} characterising four different semantic properties of texttt {L} -sentences: (1) being monotone and (2) (Scott) continuous in a given set of monadic predicates; (3) having truth preserved under taking submodels or (4) being truth invariant under taking quotients. In each case, we produce an effectively defined map that translates an arbitrary sentence varphi to a sentence varphi ^mathsf{p} belonging to the corresponding syntactic fragment, with the property that varphi is equivalent to varphi ^mathsf{p} precisely when it has the associated semantic property. As a corollary of our developments, we obtain that the four semantic properties above are decidable for texttt {L} -sentences.

First-order rewritability of ontology-mediated queries in linear temporal logic

We investigate ontology-based data access to temporal data. We consider temporal ontologies given in linear temporal logic LTL interpreted over discrete time (Z,<). Queries are given in LTL or MFO(<), monadic first-order logic with a built-in linear order. Our concern is first-order rewritability of ontology-mediated queries (OMQs) consisting of a temporal ontology and a query. By taking account of the temporal operators used in the ontology and distinguishing between ontologies given in full LTL and its core, Krom and Horn fragments, we identify a hierarchy of OMQs with atomic queries by proving rewritability into either FO(<), first-order logic with the built-in linear order, or FO(<,≡), which extends FO(<) with the standard arithmetic predicates x≡0(modn), for any fixed n>1, or FO(RPR), which extends FO(<) with relational primitive recursion. In terms of circuit complexity, FO(<,≡)- and FO(RPR)-rewritability guarantee OMQ answering in uniform ▪ and, respectively, ▪.We obtain similar hierarchies for more expressive types of queries: positive LTL-formulas, monotone MFO(<)- and arbitrary MFO(<)-formulas. Our results are directly applicable if the temporal data to be accessed is one-dimensional; moreover, they lay foundations for investigating ontology-based access using combinations of temporal and description logics over two-dimensional temporal data.

Revisiting separation: Algorithms and complexity

Abstract Linear temporal logic (LTL) with Since and Until modalities is expressively equivalent, over the class of complete linear orders, to a fragment of first-order logic known as FOMLO (first-order monadic logic of order). It turns out that LTL, under some basic assumptions, is expressively complete if and only if it has the property, called separation, that every formula is equivalent to a Boolean combination of formulas that each refer only to the past, present or future. Herein we present simple algorithms and their implementations to perform separation of the LTL with Since and Until, over discrete and complete linear orders, and translation from FOMLO formulas into equivalent temporal logic formulas. We additionally show that the separation of a certain fragment of LTL results in at most a double exponential size growth.

Reasoning about Social Choice and Games in Monadic Fixed-Point Logic

Whether it be in normal form games, or in fair allocations, or in voter preferences in voting systems, a certain pattern of reasoning is common. From a particular profile, an agent or a group of agents may have an incentive to shift to a new one. This induces a natural graph structure that we call the improvement graph on the strategy space of these systems. We suggest that the monadic fixed-point logic with counting, an extension of monadic first-order logic on graphs with fixed-point and counting quantifiers, is a natural specification language on improvement graphs, and thus for a class of properties that can be interpreted across these domains. The logic has an efficient model checking algorithm (in the size of the improvement graph).

Problems of Synthesis of Σ-Automata Specified in Languages LP and LF of First Order Logic

For two fragments LP and LF of monadic first-order logic with bounded quantifiers, the corresponding versions of the proposed specification theorem are formulated and proved that allow one to reduce the procedure for synthesizing Σ-automata specified by formulas of these logics to an equivalent transformation of formulas.

Some Subsets of Monadic First-Order Logic (MFO) Used for the Specification and Synthesis of Σ-Automata

Two fragments, LP and LF, of first-order logic with bounded quantifiers are considered. These fragments are used for the specification of transducers. The fragment LP allows the characterization of the current behavior of a system on the basis of its past behavior, and LF allows doing this on the basis of its future behavior. Two kinds of semantics are defined for these logics, and properties of automata specified in these logics are investigated.

No Future without (a hint of) Past: A Finite Basis for ‘Almost Future’ Temporal Logic

Kamp's theorem established the expressive completeness of the temporal modalities Until and Since for the First-Order Monadic Logic of Order (FOMLO) over real and natural time flows. Over natural time, a single future modality (Until) is sufficient to express all future FOMLO formulas. These are formulas whose truth value at any moment is determined by what happens from that moment on. Yet this fails to extend to real time domains: here no finite basis of future modalities can express all future FOMLO formulas. In this paper we show that finiteness can be recovered if we slightly soften the requirement that future formulas must be totally past-independent: we allow formulas to depend just on the arbitrarily recent past, and maintain the requirement that they be independent of the rest – actually – of most of the past. We call them ‘almost future’ formulas, and show that there is a finite basis of almost future modalities which is expressively complete (over all Dedekind complete time flows) for the almost future fragment of FOMLO.

The systematic design of visual languages applied to logical reasoning

Visual languages are distinguished by a number of graphical objects and their relations, usually arranged in the two-dimensional plane. While objects and relations are syntactical containers which are used to represent some information, the question arises how to systematically treat all possible syntactical containers given the richness and complexity of the underlying geometry. This paper adopts the intersection paradigm applied in the context of spatial reasoning, which ensures the systematic identification of all conceivable well-formed diagrams. This allows the exhaustive analysis of a visual language. As an example, it is shown how this method enables a thorough understanding of the relations of the graphical elements of linear diagrams which represent monadic first-order logic. The consideration of indeterminate sets even demonstrates the effectiveness of this approach for a representation that includes a total of 512 relations.

The diamond of contraries

A number of approaches for logical reasoning with diagrams have been proposed. This paper considers the question, how the expressiveness of such systems can be raised without losing the visual power of less expressive diagrams. The antagonism between expressiveness and diagrammatic simplicity is coped with by a set of jointly exhaustive and contrary relations, modelling definite knowledge within a new diagrammatic representation. The restriction on actual knowledge reduces the expressiveness of these diagrams, but strengthens their visual power by avoiding ambiguities and by providing a close correspondence between diagrammatic syntax and set-theoretic semantics. The extension towards compound diagrams enables the representation of uncertain knowledge, but have a negative impact on the clarity of these diagrams. It is shown how formulae of monadic first-order logic are treated within this approach.

PaL diagrams: A linear diagram-based visual language

Linear diagrams have recently been shown to be more effective than Euler diagrams when used for set-based reasoning. However, unlike the growing corpus of knowledge about formal aspects of Euler and Venn diagrams, there has been no formalisation of linear diagrams. To fill this knowledge gap, we present and formalise Point and Line (PaL) diagrams, an extension of simple linear diagrams containing points, thus providing a formal foundation for an effective visual language. We prove that PaL diagrams are exactly as expressive as monadic first-order logic with equality, gaining, as a corollary, an equivalence with the Euler diagram extension called spider diagrams. The method of proof provides translations between PaL diagrams and sentences of monadic first-order logic.

On the expressiveness of second-order spider diagrams

Existing diagrammatic notations based on Euler diagrams are mostly limited in expressiveness to monadic first-order logic with an order predicate. The most expressive monadic diagrammatic notation is known as spider diagrams of order. A primary contribution of this paper is to develop and formalise a second-order diagrammatic logic, called second-order spider diagrams, extending spider diagrams of order. A motivation for this lies in the limited expressiveness of first-order logics. They are incapable of defining a variety of common properties, like ‘is even’, which are second-order definable. We show that second-order spider diagrams are at least as expressive as monadic second-order logic. This result is proved by giving a method for constructing a second-order spider diagram for any regular expression. Since monadic second-order logic sentences and regular expressions are equivalent in expressive power, this shows second-order spider diagrams can express any sentence of monadic second-order logic.

On the expressiveness of spider diagrams and commutative star-free regular languages

Spider diagrams provide a visual logic to express relations between sets and their elements, extending the expressiveness of Venn diagrams. Sound and complete inference systems for spider diagrams have been developed and it is known that they are equivalent in expressive power to monadic first-order logic with equality, MFOL[=]. In this paper, we further characterize their expressiveness by articulating a link between them and formal languages. First, we establish that spider diagrams define precisely the languages that are finite unions of languages of the form K▪Γ⁎, where K is a finite commutative language and Γ is a finite set of letters. We note that it was previously established that spider diagrams define commutative star-free languages. As a corollary, all languages of the form K▪Γ⁎ are commutative star-free languages. We further demonstrate that every commutative star-free language is also such a finite union. In summary, we establish that spider diagrams define precisely: (a) languages definable in MFOL[=], (b) the commutative star-free regular languages, and (c) finite unions of the form K▪Γ⁎, as just described.

Spider Diagrams

AbstractThe use of diagrams in mathematics has traditionally been restricted to guiding intuition and communication. With rare exceptions such as Peirce's alpha and beta systems, purely diagrammatic formal reasoning has not been in the mathematician's or logician's toolkit. This paper develops a purely diagrammatic reasoning system of “spider diagrams” that builds on Euler, Venn and Peirce diagrams. The system is known to be expressively equivalent to first-order monadic logic with equality. Two levels of diagrammatic syntax have been developed: an ‘abstract’ syntax that captures the structure of diagrams, and a ‘concrete’ syntax that captures topological properties of drawn diagrams. A number of simple diagrammatic transformation rules are given, and the resulting reasoning system is shown to be sound and complete.

Future temporal logic needs infinitely many modalities

Kamp’s theorem states that there is a temporal logic with two modalities (“until” and “since”) which is expressively complete for the first-order monadic logic of order over the real line. In this paper we show that there is no temporal logic with finitely many modalities which is expressively complete for the future fragment of first-order monadic logic of order over the real line (a future formula over the real time line is a formula whose truth value at a point is independent of what happened in the past).

Star free expressions over the reals

An interpretation of star free expressions over the reals is provided. The expressive power of star free expressions is compared to the expressive power of monadic first-order logic of order over the reals. It is proved that these formalisms have the same expressive power. This result provides a generalization of the classical McNaughton–Papert theorem (1971) from the finite orders to the order of the reals.

Expressive Completeness of Duration Calculus

This paper compares the expressive power of first-order monadic logic of order, a fundamental formalism in mathematical logic and the theory of computation, with that of the propositional version of duration calculus (PDC), a formalism for the specification of real-time systems. Our results show that the propositional duration calculus is expressively complete for first-order monadic logic of order. Our semantics for PDC conservatively extends the standard semantics to all positive (including infinite) length intervals. Hence, in view of the expressive completeness, liveness properties can be specified in PDC. This observation refutes a widely believed misconception that the duration calculus cannot specify liveness properties.

Succinctness Gap between Monadic Logic and Duration Calculus

In [8, 11] the expressive completeness of the Propositional fragment of Duration Calculus relative to monadic first-order logic of order was established. In this paper we show that there is at least an exponential blow-up in every meaning preserving translation from monadic logic to PDC. Hence, there exists an exponential gap between the succinctness of monadic logic and that of duration calculus.

Resolution-based approach to compatibility analysis of interacting automata

The problem of compatibility analysis of two interacting automata arises in the development of algorithms for reactive systems if partial nondeterministic automata serve as models for both the reactive system and its environment. Intuitively, two interacting automata are compatible if a signal from one automaton always induces a defined transition in the other. We present the method for solving this problem in the automated design of reactive system algorithms specified by formulas of a first-order monadic logic. Compatibility analysis is automatically performed both for initial specifications of the algorithm and its environment and for any specification of the same kind obtained as a result of changes made by the designer. So, there is a need in the development of methods for solving this problem at the level of logical specification. The methods based on the resolution principle proved to be very helpful for this purpose.