We propose a fragment of manysorted second order logic esmt and show that checking satisfiability of sentences in this fragment is decidable. Our goal in the next two lectures is to identify decidable folt fragments beyond bernaysschonfinkel with simple. Decidable fragments of firstorder logic decidable fragments of first. Introduction our goal in the next two lectures is to identify decidable folt. 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. 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. Decidable fragments of firstorder logic modulo linear rational. Decidable fragments of firstorder temporal logics spiral. If b is arrived at, then a implies b in every interpretation. Decidable fragments of firstorder modal logic are usually obtained by restricting the occurrences of variables particularly in the scope of cf. Introduction model theory basics decidable bslra fragments application. Monodic fragments of probabilistic firstorder logic.
Perhaps because they sit inside the twovariable fragment of first order logic. The design of decision procedures for rstorder theories and their combinations has been a very active research subject for thirty. 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 francois schwarzentruber1 pr eparation agr egation compl ement january 31, 20 1ens rennes, france 129. A survey of decidable firstorder fragments and description.
A survey of decidable firstorder fragments and description logics. Firstorder logic has a long tradition and is one of the most prominent and most important formalisms in computer science and mathematics. Decidable fragments of first order modal logic are usually obtained by restricting the occurrences of variables particularly in the scope of cf. Gf, the guarded fragment of fol is the least set of formulas such that. Transitivity and equivalence in decidable fragments of first. Each function or predicate symbol comes with an arity, which is natural number. 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. This reduction is then used to single out a number of decidable fragments of firstorder temporal logics and of twosorted first order. On the other hand, it is proved that by restricting applications of firstorder quantifiers to state i. Inria decidable fragments of firstorder logic and of.
Decidable fragments of firstorder logic and of firstorder linear arithmetic with uninterpreted predicates marco voigt to cite this version. Decidable fragments of first order logic and of first order linear arithmetic with uninterpreted predicates. Transitivity and equivalence in decidable fragments of. Combinations of theories for decidable fragments of rstorder logic. This reduction is then used to single out a number of decidable fragments of firstorder temporal logics. Function symbols of arity 0 are known as constant symbols.
Forwhichclassesxfoissatx decidable, whicharereductionclasses. Validity undecidable in first order logic mathematics. First order logic sebastian rudolph1 and mantas simkus 2 1 computational logic group, tu dresden, germany 2 institute of logic and computation, tu wien, austria abstract. Firstorder logic sebastian rudolph1 and mantas simkus 2 1 computational logic group, tu dresden, germany 2 institute of logic and computation, tu wien, austria abstract. 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. Author links open overlay panel ian hodkinson a frank wolter b michael zakharyaschev c. Mar 28, 2018 bundled fragments of firstorder modal logic. Firstorder logic formalizes fundamental mathematical concepts expressive turingcomplete not too expressive not axiomatizable. Further offspring of this amsterdambudapest collaboration in the.
Guarded fragments of other logics like dataloglite, and second order logics. Firstorder logic syntax, semantics, resolution ruzica piskac yale university ruzica. Decidable fragments of firstorder logic modulo linear. The criterion is then used to single out a number of new, in a sense optimal, decidable fragments of various predicate modal logics. 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. Various translation mappings including the relational translation, the functional. Dec 15, 2017 consequently, if the background theories admit individually a decidable satisfiability problem for the firstorder. The design of decision procedures for rst order theories and their combinations has been a very active research subject for thirty. Firstorder temporal logics are notorious for their bad computational behavior. Combinations of theories for decidable fragments of rst order logic.
Why doesnt completeness imply decidability for first order logic. The fragment is onedimensional in the sense that quantification is limited. Combinations of theories for decidable fragments of rstorder. 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. 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. 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. Decidable fragments of firstorder logic and of firstorder. Most description logics dls can be translated into wellknown decidable fragments of rst order logic fo, including the guarded fragment gf and the twovariable fragment fo2. Decidable and undecidable fragments of firstorder branching temporal logics. In section 3 we focus on four decidable fragments of firstorder logic, namely, the guarded fragment, the two variable fragment, maslovs class k, and fluted logic. However, you cannot prove that a given formula is not a tautol. Erichgradel decidablefragmentsoffirstorderandfixedpointlogic.
A sentence over l is satis able if it holds in some lstructure. Decidable fragments of firstorder logic and of firstorder linear arithmetic with uninterpreted predicates. Modal languages and bounded fragments of predicate logic. Decidable fragments of firstorder logic and of first. Decidable and undecidable fragments of first order branching temporal logics. Propositional and first order logic background knowledge. Because they correspond to a guarded fragment of first order logic.
Decidable fragments on structures where two variable logic is undecidable. Gilles barthe, geoff sutcliffe and margus veanes editors. Our main result is a decidable fragment of manysorted firstorder logic that captures many real world specifications. On the one hand, the decidable fragments of first order logic have been well mapped out during the last few decades. A valid first order statement is always provably valid. Decidable fragments of first order logic and of first order linear arithmetic with uninterpreted predicates marco voigt to cite this version.
How is first order logic complete but not decidable. The triguarded fragment of firstorder logic 16 pages published. Inria decidable fragments of firstorder logic and of first. Logical systems extending first order logic, such as second order logic and type theory, are also undecidable. We provide a short survey of some of these fragments and motivate why they are interest. Modal logic decidable fragments of first order logic francois schwarzentruber1 pr eparation agr egation compl ement january 31, 20 1ens rennes, france 129. As shown by turing and church, fol is undecidable, because there is no algorithm for deciding if a fol formula is valid. Tableaubased reasoning for decidable fragments of first.
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. Modal languages and bounded fragments of predicate logic 219 this paper is the. This fact highlights a fundamental limitation of computing devices in general and of automated. Main results the main results in this paper are decidable fragments of manysorted.
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. In case of fol it means that there is an algorithm to prove that a given formula is a tautology e. In this paper we consider some of the bestknown decidable fragments of firstorder logic with equality, including the lowenheim class monadic. It is wellknown that the satisfiability problem for full firstorder logic is not solvable algorithmically we say that firstorder logic is undecidable. Pdf decidable and undecidable fragments of firstorder. Decidable fragments of firstorder logic modulo linear rational arithmetic marco voigt july 0916, 2019. The satisfiability problem for firstorder logic on finite structures is undecidable. What is your definition of decidable in firstorder logic. Combinations of theories for decidable fragments of rst.
Some undecidable fragments of firstorder modal logic. Undecidable propositional bimodal logics and onevariable. Automated deduction procedures for modal logics, and related decidable fragments of firstorder logic, are used in many realworld applications. We focus on calculi that use unification, instead of the more widely employed approach of generating ground instantiations over the course of. Abstract past research into decidable fragments of firstorder logic fo has produced two very prominent fragments. Some undecidable fragments of first order modal logic. For all valid statements, there is a decidable, sound and complete proof calculus. Satis ability in the triguarded fragment of firstorder logic. Logical systems extending firstorder logic, such as secondorder logic and type theory, are also undecidable. But then most fragments of the logic are undecidable, over many model classes. In section 4, we investigate ways of generalizing decidable fragments from. 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. Forwhichclassesxfoissatxdecidable, whicharereductionclasses. Most description logics dls can be translated into wellknown decidable fragments of rstorder logic fo, including the guarded fragment gf and the twovariable fragment fo2.
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. The undecidability of first order logic a first order logic is given by a set of function symbols and a set of predicate symbols. 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. Deductive veri cation in decidable fragments with ivy. Decidable fragments of firstorder and fixedpoint logic. Combinations of theories for decidable fragments of first. Completeness and decidability of general firstorder logic. 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. Over the years, only a few fragments such as the monodic have been shown to be decidable.
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. Monodic fragments of probabilistic firstorder logic jean christoph jung 1, carsten lutz, sergey goncharov2, and lutz schroder. 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. Examples of such background theories include presburger arithmetic, the theory of realclosed fields, and the theory of linear real arithmetic. On the other hand, we have a thorough understanding of the robust decidability of propositional modal logics. By classical results of abadi and halpern, validity for probabilistic. In this paper we consider some of the bestknown decidable fragments of first order logic with equality, including the lowenheim class monadic. 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. A set of sentences has a decidable satisfiability problem. A decidable fragment of second order logic with applications to synthesis. Combinations of theories for decidable fragments of firstorder logic. A decidable fragment of second order logic with applications.
Propositional and first order logic propositional logic first order logic basic concepts propositional logic is the simplest logic illustrates basic ideas usingpropositions p 1, snow is whyte p 2, otday it is raining p 3, this automated reasoning course is boring p i is an atom or atomic formula each p i can be either true or false but never both. 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. This reduction is then used to single out a number of decidable fragments of first order temporal logics and of twosorted first order logics in which. A popular way of obtaining decision procedures for these logics is to base them on semantic tableau calculi.
Decidable fragments of first order logic modulo linear rational arithmetic marco voigt july 0916, 2019. We introduce a novel decidable fragment of firstorder logic. Summary modal logics have decent algorithmic properties, useful for speci. First order logic isnt undecidable exactly, but rather often referred to as semidecidable. The present thesis sheds more light on the decidability boundary.