We propose a fragment of manysorted second order logic esmt and show that checking satisfiability of sentences in this fragment is decidable. Some undecidable fragments of first order modal logic. Reducing liveness to safety in firstorder logic source files oded padon, jochen hoenicke, giuliano losa, andreas podelski, mooly sagiv, sharon shoham. First order logic isnt undecidable exactly, but rather often referred to as semidecidable. The triguarded fragment of firstorder logic 16 pages published. On the one hand, the decidable fragments of first order logic have been well mapped out during the last few decades. The design of decision procedures for rst order theories and their combinations has been a very active research subject for thirty. Several decidable fragments of rst order logic have been identi ed in the past as a generalisation of the standard translation of modal logic. In this paper we consider some of the bestknown decidable fragments of firstorder logic with equality, including the lowenheim class monadic fol with equality, but without functions, bernaysschonfinkelramsey theories finite sets of formulas of the form. If b is arrived at, then a implies b in every interpretation. As shown by turing and church, fol is undecidable, because there is no algorithm for deciding if a fol formula is valid. Erichgradel decidablefragmentsoffirstorderandfixedpointlogic. Why doesnt completeness imply decidability for first order logic. Various translation mappings including the relational translation, the functional.
The algebraic approach for fragments of rstorder logic over nite words has been very fruitful. Inria decidable fragments of firstorder logic and of first. The design of decision procedures for rstorder theories and their combinations has been a. Firstorder logic formalizes fundamental mathematical concepts expressive turingcomplete not too expressive not axiomatizable. Decidable fragments of firstorder logic modulo linear. Decidable fragments of firstorder logic and of first. Deductive veri cation in decidable fragments with ivy. Firstorder logic syntax, semantics, resolution ruzica piskac yale university ruzica. Function symbols of arity 0 are known as constant symbols. Automated deduction procedures for modal logics, and related decidable fragments of firstorder logic, are used in many realworld applications. Several decidable fragments of rstorder logic have been identi ed in the past as a generalisation of the standard translation of modal logic. Some undecidable fragments of firstorder modal logic.
Say a set of sentences in firstorder logic has the finite countermodel property if any sentence in the set that is falsifiable is falsifiable on a finite domain. The reader is referred to 3 for proofs, more examples of formalizing interesting properties using decidable logic, extensions for transitive closure, and a report on our. A decidable fragment of second order logic with applications to synthesis. A valid first order statement is always provably valid. We provide a short survey of some of these fragments and motivate why they are interest. A popular way of obtaining decision procedures for these logics is to base them on semantic tableau calculi. Decidability frontier for fragments of rstorder logic with transitivity. Decidable and undecidable fragments of first order branching temporal logics.
Introduction model theory basics decidable bslra fragments application. However, you cannot prove that a given formula is not a tautol. Decidability frontier for fragments of rstorder logic with. Combinations of theories for decidable fragments of rst order logic. First order logic is complete, which means i think given a set of sentences a and a sentence b, then either b or b can be arrived at through the rules of inference being applied to a. Decidable fragments of first order logic and of first. Logical systems extending firstorder logic, such as secondorder logic and type theory, are also undecidable. Examples of such background theories include presburger arithmetic, the theory of realclosed fields, and the theory of linear real arithmetic. Abstract past research into decidable fragments of firstorder logic fo has produced two very prominent fragments. Transitivity and equivalence in decidable fragments of.
We focus on calculi that use unification, instead of the more widely employed approach of generating ground instantiations. Decidable fragments of firstorder logic modulo linear rational. Dec 01, 2000 we show that the satisfiability problem for monodic formulas in various linear time structures can be reduced to the satisfiability problem for a certain fragment of classical first order logic. Many researchers have contributed to this endeavor and till today numerous decidable and undecidable fragments of rst order logic have been identi ed. Author links open overlay panel ian hodkinson a frank wolter b. Decidability frontier for fragments of rst order logic with transitivity. Gf, the guarded fragment of fol is the least set of formulas such that. We introduce a novel decidable fragment of firstorder logic. This fact highlights a fundamental limitation of computing devices in general and. Undecidable propositional bimodal logics and onevariable. Quantified modal logic provides a natural logical language for reasoning about modal attitudes even while retaining the richness of quantification for referring to predicates over domains. Fol decidable classes and combinations cardinalities and decidable fragments decidable classes bernaysschon.
Decidable fragments of firstorder logic and of firstorder linear arithmetic with uninterpreted predicates marco voigt to cite this version. Deductive veri cation in decidable fragments with ivy 5 3 expressiveness of decidable fragments of first order logic much of the veri cation using ivy is done in a manysorted extension of the epr fragment of rst order logic that allows only allows strati ed or acyclic quanti er alternations and function symbols. Tableaubased reasoning for decidable fragments of first. An important problem in computational logic is to determine fragments of wellknown logics such as firstorder logic that are as expressive as possible yet are decidable or more strongly have low computational complexity. Many description logics are decidable fragments of first order logic, however, some description logics have more features than fol, and many spatial, temporal, and fuzzy description logics are undecidable as well.
Further offspring of this amsterdambudapest collaboration in the. This reduction is then used to single out a number of decidable fragments of firstorder temporal logics and of twosorted firstorder logics in which one. For all valid statements, there is a decidable, sound and complete proof calculus. The fragment is onedimensional in the sense that quantification is limited. Tableaubased reasoning for decidable fragments of first order logic hilverd reker a thesis submitted to the university of manchester for the degree of doctor of philosophy, 2012 automated deduction procedures for modal logics, and related decidable fragments of rst order logic, are used in many realworld applications. Firstorder temporal logics are notorious for their bad computational behavior. Logical systems extending first order logic, such as second order logic and type theory, are also undecidable. Decidable fragments on structures where two variable logic is undecidable. Combinations of theories for decidable fragments of first. Summary modal logics have decent algorithmic properties, useful for speci. The satisfiability problem for firstorder logic on finite structures is undecidable. Many description logics are decidable fragments of firstorder logic, however, some description logics have more features than fol, and many spatial, temporal, and fuzzy description logics are undecidable as well.
Combinations of theories for decidable fragments of rstorder logic. A set of sentences has a decidable satisfiability problem. Monodic fragments of probabilistic firstorder logic jean christoph jung 1, carsten lutz, sergey goncharov2, and lutz schroder. It is wellknown that the satisfiability problem for full firstorder logic is not solvable algorithmically we say that firstorder logic is undecidable. Over the years, only a few fragments such as the monodic have been shown to be decidable. Further results are language theoretic and very often decidable algebraic characterizations of. Finite model reasoning in expressive fragments of firstorder logic. Decidable fragments of first order modal logic are usually obtained by restricting the occurrences of variables particularly in the scope of cf. In this paper we consider some of the bestknown decidable fragments of first order logic with equality, including the lowenheim class monadic. Combinations of theories for decidable fragments of rst. Decidable fragments of firstorder modal logic are usually obtained by restricting the occurrences of variables particularly in the scope of cf. A sentence over l is satis able if it holds in some lstructure. Finite model reasoning in expressive fragments of firstorder. Decidable fragments of firstorder logic modulo linear rational arithmetic marco voigt july 0916, 2019.
The criterion is then used to single out a number of new, in a sense optimal, decidable fragments of various predicate modal logics. Combinations of theories for decidable fragments of firstorder logic. Modal languages and bounded fragments of predicate logic 219 this paper is the. Forwhichclassesxfoissatxdecidable, whicharereductionclasses.
Validity undecidable in first order logic mathematics. An important problem in computational logic is to determine fragments of wellknown logics such as first order logic that are as expressive as possible yet are decidable or more strongly have low computational complexity. This reduction is then used to single out a number of decidable fragments of. But then most fragments of the logic are undecidable, over many model classes. Decidability frontier for fragments of rstorder logic. Decidable reasoning about distributed protocols oded padon, giuliano losa, mooly sagiv, sharon shoham. Decidable fragments of rst order logic for the moment, let l be a vocabulary of rst order logic that for every arity contains countably many relations symbols of this arity. It is known that even the twovariable monadic fragment is highly undecidable over various linear timelines, and over branching time even onevariable fragments might be undecidable.
Deductive veri cation in decidable fragments with ivy 5 3 expressiveness of decidable fragments of firstorder logic much of the veri cation using ivy is done in a manysorted extension of the epr fragment of rst order logic that allows only allows strati ed or acyclic quanti er alternations and function symbols. Decidable fragments of firstorder logic and of firstorder. Monodic fragments of probabilistic firstorder logic. Introduction our goal in the next two lectures is to identify decidable folt. Decidable fragments of first order logic modulo linear rational arithmetic marco voigt july 0916, 2019. Transitivity and equivalence in decidable fragments of first. Dec 15, 2017 consequently, if the background theories admit individually a decidable satisfiability problem for the firstorder. Our goal in the next two lectures is to identify decidable folt fragments beyond bernaysschonfinkel with simple. A decidable fragment of second order logic with applications. Gilles barthe, geoff sutcliffe and margus veanes editors. Decidable fragments of firstorder and fixedpoint logic. In section 4, we investigate ways of generalizing decidable fragments from. Perhaps because they sit inside the twovariable fragment of first order logic.
Decidable fragments of rstorder logic for the moment, let l be a vocabulary of rstorder logic that for every arity contains countably many relations symbols of this arity. Inria decidable fragments of firstorder logic and of. The undecidability of first order logic a first order logic is given by a set of function symbols and a set of predicate symbols. How is first order logic complete but not decidable. Modal languages and bounded fragments of predicate logic. This fact highlights a fundamental limitation of computing devices in general and of automated. Decidable fragments of firstorder logic decidable fragments of first. Combinations of theories for decidable fragments of rstorder. Main results the main results in this paper are decidable fragments of manysorted. Each function or predicate symbol comes with an arity, which is natural number. On the other hand, it is proved that by restricting applications of firstorder quantifiers to state i. Because they correspond to a guarded fragment of first order logic. Our main result is a decidable fragment of manysorted firstorder logic that captures many real world specifications.
Forwhichclassesxfoissatx decidable, whicharereductionclasses. Firstorder logic has a long tradition and is one of the most prominent and most important formalisms in computer science and mathematics. In case of fol it means that there is an algorithm to prove that a given formula is a tautology e. Pdf decidable and undecidable fragments of firstorder. Decidable and undecidable fragments of firstorder branching temporal logics. Mar 28, 2018 bundled fragments of firstorder modal logic. A survey of decidable firstorder fragments and description. Tableaubased reasoning for decidable fragments of firstorder logic hilverd reker a thesis submitted to the university of manchester for the degree of doctor of philosophy, 2012 automated deduction procedures for modal logics, and related decidable fragments of rstorder logic, are used in many realworld applications. Guarded fragments of other logics like dataloglite, and second order logics. On the other hand, we have a thorough understanding of the robust decidability of propositional modal logics. Among those technical hurdles, finding useful decidable fragments of f o m l has been a major one preventing the use of f o m l in computational applications. The present thesis sheds more light on the decidability boundary. Decidable fragments of firstorder logic and of firstorder linear arithmetic with uninterpreted predicates. What is your definition of decidable in firstorder logic.