How much effort and time is spent on evaluating a query is controlled through a parameter that specifies how many case splits the reasoner may investigate. Subramani1 1lane department of computer science and electrical engineering west virginia university 6 february, february 20 subramani first order logic. This pa per presents a variant of firstorder relevance logic that has a decidable algorithm for determining tautological en tailment. Dec, 2005 so im a bit confused about these metatheorems about first order logic, partly because i havent read any of the real proofs, but i just want to know the results for right now. Decidability of firstorder theories of the real numbers. Function symbols of arity 0 are known as constant symbols. Our main result is a decidable fragment of manysorted firstorder logic that captures many.
First order logic first order semantic structures, formal language, variables and quantifiers, satisfaction, entailment. We usually say which logical system we are interested in, and the main logic of interest is rst order logic. If there is gas in the engine and a good spark, the engine runs. First order intuitionistic logic with decidable propositional atoms has a balanced mix of classical and intuitionistic characteristics. So, if there were an effective means of deciding in first order logic in general whether gamma superset d, we would have a way to solve the halting problem. Citeseerx decidability of firstorder logic queries over. Decidability and undecidability in toc geeksforgeeks. First order logic huixing fang school of information engineering yangzhou university. I dont have time to fix this right now, but i marked the two places where the second definition is intended, while the lede gives only the first definition. Starting with the basics of set theory, induction and computability, it covers. The fact that first order logic with some nontriviality constraints is undecidable means that no algorithm can decide correctly whether a given first order formula is true or not. A first order logic is given by a set of function symbols and a set of predicate symbols. The undecidability of first order logic stanford university.
This is a systematic and wellpaced introduction to mathematical logic. The schemas motivate the main result of this paper because the decidability of a class of logic. Views currently occupy a central place in database research due to their role in applications such as information integration and data warehousing. Due to the fact that it is algorithmically impossible to determine whether a given sentence of rst order logic is valid or not, various restrictions of rst order logic have been devised to obtain languages that are less expressive but can be reasoned about by computers more. If b is arrived at, then a implies b in every interpretation. Citeseerx document details isaac councill, lee giles, pradeep teregowda. It has a gentle learning curve, with lots of exercises, and a.
The history of this decision problem goes all the way back to gottfried leibniz 1646 1716 and his dream of a universal computer. In this paper we show the surprising result that first order logic ceases to be decidable at order 3 and above. Reducing liveness to safety in first order logic source files. We study the problem of deciding satisfiability of first order logic queries over views, our aim being to delimit the boundary between the decidable and the undecidable fragments of this language. Decidablefragmentsoffirstorderandfixedpointlogic frompre. We prove the undecidability of the monadic fragment of pnuelis firstorder lineartime temporal logic fltl without function symbols nor equality. On the other hand, the theory of a boolean algebra with a distinguished subalgebra is undecidable. On the other hand, we have a thorough understanding of the robust decidability of propositional modal logics. Both the decidability results and undecidablity results extend in various ways to boolean algebras in extensions of first order logic. Fol is often called semidecidable since although there are. Logical systems extending first order logic, such as second order logic and type theory, are also undecidable. 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. Propositional and first order logic background knowledge. Finite model reasoning in expressive fragments of firstorder logic.
Proof in alonzo churchs and alan turings mathematical logic. Undecidability of monadic firstorder lineartime temporal logic. It is wellknown that the satis ability problem for full rst order logic is not solvable algorithmically we say that rst order logic is undecidable. First order logic clauses decidability mathematics stack. What i dont understand are the profound reasons for which the first order logic is semidecidable. It coincides with classical logic in its propositional part, and it coincides with intuitionistic logic on the set of formulas either not containing propositional. Because they correspond to a guarded fragment of first order logic. Prenex normal form wikipedia first order resolution. For our base languages the problems are decidable thanks to the. The firstorder theory of sets with cardinality constraints. Decidability and undecidability in toc identifying languages or problems as decidable, undecidable or partially decidable is a very common question in gate. The logic book by merrie bergmann, et al, used to be used to teach propositional logic and firstorder predicate logic to philosophy undergraduates at university college london ucl and at the university of oxford. Although this logic is considerably weaker than standard firstorder logic, it can be used.
Do the peano axioms form a consistent theory in first order predicate logic here consistency means that one can not prove everything or. Small substructures and decidability issues for first order logic with two variables. Artificial intelligence practice questions on propositional and firstorder logic 1. The emergence of firstorder logic stanford encyclopedia. The mathematics of boolean algebra stanford encyclopedia of. The latter result is proved using a general result of a. We prove that the twovariable fragment of firstorder intuitionistic logic is undecidable, even without constants and equality. We study the problem of deciding the satisfiability of first order logic queries over views, with our aim to delimit the boundary between the decidable and the undecidable fragments of this language. Most modifications to logic suggested for kr are either exten sions to firstorder logic e. Combinations of theories for decidable fragments of firstorder logic. Broadbent department of computer science, oxford university wolfson building, parks road, oxford, ox1 3qd, uk. Views currently occupy a central place in database research, due to their role in applications such as information integration and data warehousing. With correct knowledge and ample experience, this question becomes very easy to solve. Im looking at the theorem concerning the model of arithmetic.
A logical systems propositional or first order logic, for example is called decidable if the set of logical validities is computable. 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. Even the theory of boolean algebras with a distinguished ideal is decidable. We show that, allowing quanti cation of exible variables, the set of tautologies in this logic is not recursively. This new logic affords us much greater expressive power. The order 2 cpda graphs already have undecidable mso theories but it was only recently shown by kartzow kartzow 2010 that first order logic is decidable at the second level. Timm lampert, decidability of firstorder logic philpapers. It has a gentle learning curve, with lots of exercises, and a companion volume of selected answers. The limits of decidability for first order logic on cpda graphs. The limits of decidability for first order logic on cpda graphs christopher h. Artificial intelligence 20192020 semi decidability of first order logic 2 decidability and automation of l fo. Undecidability of monadic firstorder lineartime temporal logic abstract. Lecture 10 software engineering 2 first order logic.
However, if we work only with clauses is it correct to say that, due to the soundness and completeness of resolution principle, we achieved decidabi. Undecidability of firstorder intuitionistic and modal logics with two variables roman kontchakov, agi kurucz, and michael zakharyaschev abstract. We found that resolution for propositional logic was sound and complete. Check out the master branch for a more stable version. When we speak of the decision problem in this module, we mean this particular problem. The problems numbered 1, 2, and 10 which concern mathematical logic and which gave birth to what is called the entscheidungsproblem or the decision problem were eventually solved though in the negative by alonzo church and alan turing in their famous churchturing thesis. Log ical systems exte nding first orde r logic, suc h as second order logic and type theory, are also undecidable. Ive also seen that satisfiability in some first order formulas are decidable. For firstorder logic both problems are undecidable 56, 53, 54 and recursively inseparable 55. A decidable firstorder logic for knowledge representation. Say a set of sentences in first order logic has the finite countermodel property if any sentence in the set that is falsifiable is falsifiable on a finite domain.
Incompleteness and undecidability of predicate logic. Would you like to see this cheatsheet in your native language. That is, why can i verify the validity of a theorem in a finite number of steps, but if the formula is not a theorem. How is first order logic complete but not decidable.
Modularity for decidability of deductive verification with applications to distributed systems marcelo taube, giuliano losa, kenneth mcmillan, oded padon, mooly sagiv, sharon shoham, james r. If there is gas in the tank and the fuel line is okay, then there is gas in the engine. Decidability of cylindric set algebras of dimension two and first order logic with two variables. I struggle with understanding why the satisfiability in the first order logic is undecidable. Decidability and undecidability of theories with a predicate. Summary modal logics have decent algorithmic properties, useful for speci. For anybody schooled in modern logic, firstorder logic can seem an entirely natural object of study, and its discovery inevitable. The emergence of firstorder logic stanford encyclopedia of. Firstorder logic is undecidable by guram bezhanishvili nd lawrence s. First order logic, even restricted to only horn clauses, is semidecidable.
The logic book by merrie bergmann, et al, used to be used to teach propositional logic and first order predicate logic to philosophy undergraduates at university college london ucl and at the university of oxford. A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. Excellent as a course text, the book presupposes only elementary background and can be used also for selfstudy by more ambitious students. On the one hand, the decidable fragments of first order logic have been well mapped out during the last few decades. Most of the below text refers to the old version in particular, the web demos are still running a version from 2017. Propositional and first order logic propositional logic first order logic decidability property propositional logic is decidable. Artificial intelligence 20192020 semi decidability of first order logic 1 artificial intelligence. The limits of decidability for first order logic on cpda. The present thesis sheds more light on the decidability boundary. The satisfiability problem for firstorder logic is undecidable. Decidability of first order logic queries over views. This was shown by alonzo church and alan turing in 1936 independently of each other. Decidability of firstorder logic queries over views.
Henkins proof of completeness 2 then well do some subset of the following topics, or others proposed by the class. The order 2 cpda graphs already have undecidable mso theories but it was only recently shown by kartzow 9 that first order logic is decidable at the second level. Perhaps because they sit inside the twovariable fragment of first order logic. Why doesnt completeness imply decidability for first order logic. Firstorder intuitionistic logic with decidable propositional. 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. Decidable fragments of firstorder and fixedpoint logic. Decidable fragments of firstorder logic and of firstorder. A first order language of the real numbers is the set of all wellformed sentences of first order logic that involve universal and existential quantifiers and logical combinations of equalities and inequalities of expressions over real variables.
The central observation is that it is possible to express the behaviour of an arbitrary turing machine in firstorder logic. Semenov on decidability of monadic theories, and a proof of semenovs result is presented. For anybody schooled in modern logic, first order logic can seem an entirely natural object of study, and its discovery inevitable. First order logic has a long tradition and is one of the most prominent and most important formalisms in computer science and mathematics. Views currently occupy a central place in database research, due to their role in applications such as information. Proof in alonzo churchs and alan turings mathematical. Undecidability of firstorder logic computer science new. In addition to studying decidability, we also show that uf 1 is incomparable in expressive power with both foc2 and gnfo. Mar 12, 2014 in the other direction, it is shown, from the same hypothesis, that the monadic secondorder theory of s, p. Satfo222nexptime theoremlewis80,furer84 satfo2isnexptimehard.
So im a bit confused about these metatheorems about first order logic, partly because i havent read any of the real proofs, but i just want to know the results for right now. I know the practical consequences of semidecidability of first order logic. Soundness, completeness, decidability in first order logic is one of those pick any two scenarios. Syntax we shall now introduce a generalisation of propositional logic called. First order logic is a widely used relational language of reasoning. Ever since the undecidability of firstorder classical.
Professor ildiko sain, senior research fellow at alfr ed r enyi institute of mathematics. Semidecidability of first order logic mathematics stack. Validity is decidable for several classes of formulas defined by syntactic restrictions on their form sect. In 18, vardi initiated an intriguing research e ort that aims to understand. Firstorder modal logic, decidability, bundled fragments. Each function or predicate symbol comes with an arity, which is natural number. The chapter surveys several important theoretical results in first order logic. Decidable fragments of firstorder logic and of first. In case of fol it means that there is an algorithm to prove that a given formula is a tautology e. The later turing and gumanskis attempts are criticized as inadequate or doubtful. The corresponding first order theory is the set of sentences that are actually true of the real. Soundness means that any derivation from the axioms and inference rules is still valid. Random graphs and the zeroone law one presentation.