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. Author links open overlay panel ian hodkinson a frank wolter b michael zakharyaschev c. Guarded fragments of other logics like dataloglite, and second order logics. Introduction model theory basics decidable bslra fragments application. Decidable fragments of firstorder logic decidable fragments of first. In section 4, we investigate ways of generalizing decidable fragments from. 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. 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. Combinations of theories for decidable fragments of rstorder logic. Inria decidable fragments of firstorder logic and of.
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 first order modal logic are usually obtained by restricting the occurrences of variables particularly in the scope of cf. Decidable fragments of firstorder logic and of firstorder linear arithmetic with uninterpreted predicates marco voigt to cite this version. Combinations of theories for decidable fragments of first. Each function or predicate symbol comes with an arity, which is natural number. Summary modal logics have decent algorithmic properties, useful for speci. In this paper we consider some of the bestknown decidable fragments of firstorder logic with equality, including the lowenheim class monadic. A survey of decidable firstorder fragments and description. Introduction our goal in the next two lectures is to identify decidable folt.
Propositional and first order logic background knowledge. The triguarded fragment of firstorder logic 16 pages published. A survey of decidable firstorder fragments and description logics. What is your definition of decidable in firstorder logic. Decidable fragments of firstorder logic and of firstorder linear arithmetic with uninterpreted predicates. Combinations of theories for decidable fragments of firstorder logic. The undecidability of first order logic a first order logic is given by a set of function symbols and a set of predicate symbols. Transitivity and equivalence in decidable fragments of first. Decidable fragments of first order logic francois schwarzentruber1 pr eparation agr egation compl ement january 31, 20 1ens rennes, france 129. The design of decision procedures for rst order theories and their combinations has been a very active research subject for thirty.
Decidable fragments of firstorder logic modulo linear. Gilles barthe, geoff sutcliffe and margus veanes editors. Perhaps because they sit inside the twovariable fragment of first order logic. 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.
Decidable fragments of firstorder temporal logics spiral. 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. Why doesnt completeness imply decidability for first order logic. Satis ability in the triguarded fragment of firstorder logic. 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. First order logic isnt undecidable exactly, but rather often referred to as semidecidable. Tableaubased reasoning for decidable fragments of first. 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. If b is arrived at, then a implies b in every interpretation. Undecidable propositional bimodal logics and onevariable. On the other hand, it is proved that by restricting applications of firstorder quantifiers to state i. On the other hand, we have a thorough understanding of the robust decidability of propositional modal logics.
Modal languages and bounded fragments of predicate logic 219 this paper is the. Function symbols of arity 0 are known as constant symbols. 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. By classical results of abadi and halpern, validity for probabilistic. How is first order logic complete but not decidable. A decidable fragment of second order logic with applications. Some undecidable fragments of first order modal logic. This reduction is then used to single out a number of decidable fragments of firstorder temporal logics. 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. Inria decidable fragments of firstorder logic and of first. On the one hand, the decidable fragments of first order logic have been well mapped out during the last few decades.
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. As shown by turing and church, fol is undecidable, because there is no algorithm for deciding if a fol formula is valid. However, you cannot prove that a given formula is not a tautol. 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. 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. Decidable and undecidable fragments of first order branching temporal logics. Further offspring of this amsterdambudapest collaboration in the. Modal logic decidable fragments of first order logic francois schwarzentruber1 pr eparation agr egation compl ement january 31, 20 1ens rennes, france 129. Gf, the guarded fragment of fol is the least set of formulas such that. The present thesis sheds more light on the decidability boundary. Dec 15, 2017 consequently, if the background theories admit individually a decidable satisfiability problem for the firstorder. Decidable fragments of firstorder and fixedpoint logic.
This reduction is then used to single out a number of decidable fragments of firstorder temporal logics and of twosorted first order. Transitivity and equivalence in decidable fragments of. We propose a fragment of manysorted second order logic esmt and show that checking satisfiability of sentences in this fragment is decidable. 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. Firstorder logic syntax, semantics, resolution ruzica piskac yale university ruzica. Erichgradel decidablefragmentsoffirstorderandfixedpointlogic. Automated deduction procedures for modal logics, and related decidable fragments of firstorder logic, are used in many realworld applications. 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. Firstorder temporal logics are notorious for their bad computational behavior. 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. Decidable fragments of first order logic modulo linear rational arithmetic marco voigt july 0916, 2019. Mar 28, 2018 bundled fragments of firstorder modal logic.
In case of fol it means that there is an algorithm to prove that a given formula is a tautology e. Various translation mappings including the relational translation, the functional. 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. 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. Decidable fragments on structures where two variable logic is undecidable. Deductive veri cation in decidable fragments with ivy. Monodic fragments of probabilistic firstorder logic jean christoph jung 1, carsten lutz, sergey goncharov2, and lutz schroder. Pdf decidable and undecidable fragments of firstorder.
Decidable fragments of first order logic and of first order linear arithmetic with uninterpreted predicates marco voigt to cite this version. Combinations of theories for decidable fragments of rstorder. Decidable fragments of firstorder logic modulo linear rational. We introduce a novel decidable fragment of firstorder logic. Decidable fragments of first order logic and of first order linear arithmetic with uninterpreted predicates. Monodic fragments of probabilistic firstorder logic. 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. 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. Main results the main results in this paper are decidable fragments of manysorted.
Forwhichclassesxfoissatxdecidable, whicharereductionclasses. The fragment is onedimensional in the sense that quantification is limited. 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. A decidable fragment of second order logic with applications to synthesis. Firstorder logic has a long tradition and is one of the most prominent and most important formalisms in computer science and mathematics.
Firstorder logic formalizes fundamental mathematical concepts expressive turingcomplete not too expressive not axiomatizable. Modal languages and bounded fragments of predicate logic. Decidable fragments of firstorder logic and of firstorder. This reduction is then used to single out a number of decidable fragments of. A sentence over l is satis able if it holds in some lstructure.
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. Decidable fragments of firstorder modal logic are usually obtained by restricting the occurrences of variables particularly in the scope of cf. For all valid statements, there is a decidable, sound and complete proof calculus. We focus on calculi that use unification, instead of the more widely employed approach of generating ground instantiations over the course of. The design of decision procedures for rstorder theories and their combinations has been a very active research subject for thirty. The criterion is then used to single out a number of new, in a sense optimal, decidable fragments of various predicate modal logics. 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. 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. Logical systems extending firstorder logic, such as secondorder logic and type theory, are also undecidable. This fact highlights a fundamental limitation of computing devices in general and of automated. Over the years, only a few fragments such as the monodic have been shown to be decidable. Abstract past research into decidable fragments of firstorder logic fo has produced two very prominent fragments. The satisfiability problem for firstorder logic on finite structures is undecidable.
Decidable fragments of firstorder logic modulo linear rational arithmetic marco voigt july 0916, 2019. Our main result is a decidable fragment of manysorted firstorder logic that captures many real world specifications. Because they correspond to a guarded fragment of first order logic. Decidable fragments of firstorder logic and of first. Our goal in the next two lectures is to identify decidable folt fragments beyond bernaysschonfinkel with simple. Logical systems extending first order logic, such as second order 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. A valid first order statement is always provably valid. A set of sentences has a decidable satisfiability problem. But then most fragments of the logic are undecidable, over many model classes. Combinations of theories for decidable fragments of rst. We provide a short survey of some of these fragments and motivate why they are interest.
537 9 361 898 1200 235 521 500 1427 441 589 785 687 916 606 1179 1013 1038 109 1477 1212 151 557 1586 805 1049 269 1445 263 1301 441 1538 117 1213 1587 321 66 32 298 159 953 1464 435 826 988 36 1432 1347