On the other hand, pl is decidable there does exist an algorithm for deciding if a pl formula f is valid, e. However, you cannot prove that a given formula is not a tautol. Introduction our goal in the next two lectures is to identify decidable folt. 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. Firstorder logic uses quantified variables over nonlogical objects and allows the use of sentences that contain variables, so that rather than propositions such as socrates is a man. Function symbols of arity 0 are known as constant symbols.
Decidable reasoning in a logic of limited belief with function symbols. Further, via the foml translation above, we can show that the monodic restriction of tml based on the guarded fragment of first order logic and monadic first order logic are decidable 18. A survey of decidable firstorder fragments and description logics u. Before turning to the technical definitions, we first introduce the two notions of belief informally. The undecidability of first order logic a first order logic is given by a set of function symbols and a set of predicate symbols. In summary, the decidable firstorder logic presented here forms an important first step toward building decidable seman ticallymotivated kr systems. There is a single most general unifier mgu that is unique up. Now terms are recursively defined by variables are terms, and. Find, read and cite all the research you need on researchgate. 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.
The monadic monodic fragment with flexible functions can be decided with expspacecomplete complexity. To find the solution of this problem, we can easily. Background and definitions investigation of properties of firstorder intuitionistic logic with decidable propositional atoms will be based on using sequent calculus as a. These logics have the expressive power of standard first order logic along with an inference algorithm that will always terminate, both important considerations for knowledge representation. Decidable fragments of firstorder logic and of firstorder. Besides standard first order time structures, we consider also those that have only finite first order domains, and extend the results mentioned above. Decidable fragments of firstorder logic modulo linear rational arithmetic marco voigt july 0916, 2019. Decidable fragments of firstorder logic and of first. 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. The logic obl the key aspects of this logic are the semantics and properties of belief and onlybelieving, which ulti mately gives us the specification of decidable, intro spective reasoning in first order kbs. A firstorder language of the real numbers is the set of all wellformed sentences of firstorder logic that involve universal and existential quantifiers and logical combinations of equalities and inequalities of expressions over real variables. Decidable and undecidable fragments of firstorder branching temporal logics.
However, the monadic monodic fragment with rigid functions, where no two distinct. Decidability of firstorder theories of the real numbers. On a decidable generalized quantifier logic corresponding to. Author links open overlay panel ian hodkinson a frank wolter b michael zakharyaschev c. But that means todays subject matter is firstorder logic, which is extending propositional logic so that we can talk about things. But that means todays subject matter is firstorder logic, which is extending propositional logic. If there is gas in the tank and the fuel line is okay, then there is gas in the engine. The schemas motivate the main result of this paper because the decidability of a class of logic.
The corresponding first order theory is the set of sentences that are actually true of the real. Journal of logic, language and information link to publication citation for published version apa. Firstorder logic fol 2 2 firstorder logic fol also called predicate logic or predicate calculus fol syntax variables x,y,z. The logic obl the key aspects of this logic are the semantics and properties of belief and onlybelieving, which ulti mately gives us the specification of decidable, intro spective reasoning in firstorder kbs. A single rigid function is sufficient to make the logic not recursively enumerable. Decidable fragments of firstorder and fixedpoint logic. Tableaubased reasoning for decidable fragments of first. 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.
Validity undecidable in first order logic mathematics. Introduction model theory basics decidable bslra fragments application. Pdf this article presents first order logic decidable. Many researchers have contributed to this endeavor and till today numerous decidable and undecidable fragments of rstorder logic have been identi ed. Practice questions on propositional and firstorder logic 1. Decidable and undecidable fragments of first order branching temporal logics. Undecidability of monadic firstorder lineartime temporal. Propositional and first order logic background knowledge. The steps of the transformations are also mechanically checked, ensuring the soundness of. On a decidable generalized quantifier logic corresponding to a decidable fragment of first order logic alechina, n. Our methodology involves modeling protocols using general uninterpreted firstorder logic, and then systematically transforming the model to obtain a model and an inductive invariant that are decidable to check.
If there is power to the plugs and the plugs are clean, a good spark is. In this paper, we examine the computational complexity of various natural onevariable fragments of firstorder modal logics with the addition of arbitrary counting quantifiers. 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. Journal of logic, language and information, 43, 177189.
A decidable fragment of second order logic with applications to synthesis. The fact that firstorder logic with some nontriviality constraints is undecidable means that no algorithm can decide correctly whether a given firstorder formula is true or not. Suppose we are asked to compute all the prime numbers in the range of to 2000. We consider the decision problem for cases of firstorder temporal logic with function symbols and without equality. On a decidable generalized quantifier logic corresponding to a decidable fragment of firstorder logic. Besides standard firstorder time structures, we consider also those that have only finite firstorder domains, and extend the results mentioned above. Decidable firstorder logics with reasonable modeltheoretic semantics have several benefits for knowledge representation. If b is arrived at, then a implies b in every interpretation. Decidable firstorder modal logics with counting quantifiers. The corresponding firstorder theory is the set of sentences that are actually true of the real numbers. Firstorder logic formalizes fundamental mathematical concepts expressive turingcomplete not too expressive not axiomatizable. Firstorder logic propositional logic only deals with facts, statements that may or may not be true of the world, e. A valid first order statement is always provably valid.
Most modifications to logic suggested for kr are either exten sions to firstorder logic e. A decidable firstorder logic for knowledge representation. In this paper, we examine the computational complexity of various natural onevariable fragments of first order modal logics with the addition of arbitrary counting quantifiers. Soundness means that any derivation from the axioms and inference rules is still valid. A careful examination eals rev that prop ositional mo dal logic can in fact b e ed view as a t fragmen of ariable 2v rstorder. A survey of decidable firstorder fragments and description. If the answer is yes, that is, if there is a proof, then the theorem prover will eventually halt and say so. Logical systems extending first order logic, such as seco nd order logic and type theory, are also undecidable. 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.
In mer92, it is shown that the corresponding monadic fragment of fltl is also decidable. Even if rstorder logic were decidable, it would have had dramatic consequences. Combinations of theories for decidable fragments of first. Why doesnt completeness imply decidability for first order logic.
Some strong evidence, thanks to the work of erich gr. However, satisfiability is undecidable as a consequence of churchs. General rights it is not permitted to download or to forwarddistribute the text or part of it without the. 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. Firstorder logicalso known as predicate logic, quantificational logic, and firstorder predicate calculusis a collection of formal systems used in mathematics, philosophy, linguistics, and computer science.
This is proven, as in fol, by a reduction to the validity problem of propositional temporal logic by standard arguments. Undecidability of monadic firstorder lineartime temporal logic. The firstorder theory of sets with cardinality constraints. We will see that if the decision problem for second order logic had a positive solution, then it would in principle be possible to solve every mathematical problem in a purely mechanical way. On a decidable generalized quantifier logic corresponding to a decidable fragment of. The addition of counting quantifiers provides us a rich language with. Pdf decidable firstorder modal logics with counting. Decidable cases of firstorder temporal logic with functions. However, the monadic monodic fragment with rigid functions, where no two distinct terms have.
Even though logic has played an important role in knowl edge representation kr research, there has been little effort expended on devising decidable logics for kr. General rights it is not permitted to download or to forwarddistribute the text or part of it without the consent of the authors andor holders. 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 sort is intended for temporal reasoning. Each function or predicate symbol comes with an arity, which is natural number. References anderson and belnap, 1975 anderson, alan r. On a decidable generalized quantifier logic corresponding to a decidable fragment of first order logic. We propose a fragment of manysorted second order logic esmt and show that checking satisfiability of sentences in this fragment is decidable. If there is gas in the engine and a good spark, the engine runs.
In case of fol it means that there is an algorithm to prove that a given formula is a tautology e. Even if rst order logic were decidable, it would have had dramatic consequences. From the perspective of kurokawas research, firstorder logic with classical propositional atoms and intuitionistic predicate atoms is investigated here. Practice questions on propositional and first order logic 1. Decidable fragments of first order logic modulo linear rational arithmetic marco voigt july 0916, 2019. Because they correspond to a guarded fragment of first order logic. Validity undecidable in first order logic mathematics stack. We consider the decision problem for cases of first order temporal logic with function symbols and without equality. A problem is said to be decidable if we can always construct a corresponding algorithm that can answer the problem correctly. Firstorder intuitionistic logic with decidable propositional. Decidable fragments of firstorder logic modulo linear. Pdf decidable and undecidable fragments of firstorder.
These logics have the expressive power of standard first order logic along with an inference algorithm that will always terminate, both important considerations for. Logical systems extending first order logic, such as second order logic and type theory, are also undecidable. Decidable fragments of firstorder modal logics article pdf available in journal of symbolic logic 663 april 1999 with 69 reads how we measure reads. Decidable and undecidable problems in theory of computation. 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. Fol is semidecidable there is a procedure that always halts and says yes if f is valid, but may not halt if f is invalid. Decidable fragments of firstorder logic and of firstorder linear arithmetic with uninterpreted predicates. Decidable fragments of firstorder logic and of firstorder linear arithmetic with uninterpreted predicates marco voigt to cite this version. On a decidable generalized quantifier logic corresponding. We will see that if the decision problem for secondorder logic had a positive solution, then it would in principle be possible to solve every mathematical problem in a purely mechanical way. Decidable fragments on structures where two variable logic is undecidable. Decidable reasoning in a fragment of the epistemic situation calculus. We can intuitively understand decidable problems by considering a simple example.
First order logic isnt undecidable exactly, but rather often referred to as semidecidable. Decidable reasoning in a firstorder logic of limited conditional belief. How is first order logic complete but not decidable. Perhaps because they sit inside the twovariable fragment of first order logic. For all valid statements, there is a decidable, sound and complete proof calculus. Satfo222nexptime theoremlewis80,furer84 satfo2isnexptimehard. Decidable first order logics with reasonable modeltheoretic semantics have several benefits for knowledge representation. Dec 01, 2000 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 one sort is intended for temporal reasoning.
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