Similarly, the set of logical consequences of any recursively enumerable set of firstorder axioms is semidecidable. 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. With functions, entailment is not semidecidable and satisfiability is not semidecidable. Classical firstorder logic introduction universidade do minho. In order to do this, we introduced the notion of a computably enumerable set c. Inference in firstorder logic school of informatics. References anderson and belnap, 1975 anderson, alan r. Automatic theorem proving university of rhode island. On the other hand, pl is decidable there does exist an algorithm for deciding if a pl formula f is valid, e. Cdclbased abstract state transition system for coherent logic. That is, why can i verify the validity of a theorem in a finite number of steps, but if the.
If there is gas in the engine and a good spark, the engine runs. 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. Show that the following statement is a logical consequence of the statements above. If the answer is yes, that is, if there is a proof, then the theorem prover will eventually halt and say so. Now terms are recursively defined by variables are terms, and. Incompleteness applies to mathematical systems you need numbers because you need a. Problem solving planning example cornell university. I fol is semi decidable there is a procedure that always halts and says \yes if f is valid, but may not halt if f is invalid. I know the practical consequences of semidecidability of first order logic. How is first order logic complete but not decidable. A propositionalization of a theory in firstorder logic fol is a set of propositional sentences that is satis. We usually say which logical system we are interested in, and the main logic of interest is rst order logic. Translating into logic is a skill that takes some practice to get used to, but once you get the hang of it, its actually not too bad and honestly it can be.
To find the solution of this problem, we can easily. For firstorder logic, any sentence that is entailed by another set of sentences can be proved from that set godel did not suggest a proof procedure in 1965 robinson published his resolution algorithm entailment in firstorder logic is semidecidable, that is, we can show that. Thus, resolution in firstorder logic will always tell you if a sentence is entailed however, it might not be able to tell you for what values it is satisfiable similar to the semidecidable nature of fo logic, resolution is complete if entailment can be found in a finite number of inferences or resolves resolution in fo logic. The rst argument of a function symbol is allowed to. On the other hand, i pl is decidable there does exist an algorithm for deciding if a pl formula f is valid, e. Then we can build an algorithm that enumerates all these statements.
We found that resolution for propositional logic was sound and complete. Firstorder logic fol is a richer language than propositional logic. If you enter a modal formula, you will see a choice of how the accessibility relation should be constrained. A problem is semidecidable if there exists a procedure that. Dont query using firstorder logic fol unless you absolutely have to. Sound and complete for firstorder definite clauses proof similar to propositional proof. The weaker form of the theorem can be proved from the undecidability of the halting problem as follows. Thus, separating intensions from extensions makes it possible to have an equational theory over predicate and function names intensions that is separate from the extensional equality of relations and functions. Semidecidable very slow in practice generalized modus ponens inference with definite clauses replace instantiation step with unification unifyp,q. Npcomplete semidecidable, may run forever when no entailment fol csl302 artificial intelligence 17. Function symbols of arity 0 are known as constant symbols. Herbrand logic firstorder syntax and herbrand semantics. Npcomplete semidecidable, may run forever when no entailment fol csl452 artificial intelligence 17.
On a positive note, we show that validity is semidecidable or recursively. First order linear temporal logic over finite temporal structures is not semidecidable article pdf available april 1999 with 25 reads how we measure reads. In case of fol it means that there is an algorithm to prove that a given formula is a tautology e. Ona,c onc,table, ond,b, onb,table, cleara, cleard plus rules for moving things around prove. Decidability implies semidecidability, so the language of. Pdf first order linear temporal logic over finite temporal.
Semidecidability of first order logic mathematics stack. A decidable firstorder logic for knowledge representation. We cannot translate a rbitrary fol theories to propositional logic because fol is only semidecidable. The validity problem of predicate logic is only semidecidable. Logic and proof hilary 2016 undecidability of validity and satis ability james worrell in this lecture we consider the computational problems of determining validity and satis ability of rst order formulas. Suppose we are asked to compute all the prime numbers in the range of to 2000. For first order logic, any sentence that is entailed by another set of sentences can be proved from that set godel. First order logic fol 2 2 first order logic fol also called predicate logic or predicate calculus fol syntax variables x,y,z. In last fridays lecture, we talked about how to translate statements from english into frst order logic. Thus, resolution in firstorder logic will always tell you if a sentence is entailed however, it might not be able to tell you for what values it is satisfiable similar to the semidecidable nature of fo logic, resolution is complete if entailment can be found in a finite number of. I know the practical consequences of semi decidability of first order logic.
We present a new, cdclbased approach for automated theorem proving in coherent logic an expressive semi decidable fragment of first order logic that provides potential for obtaining human. If there is gas in the tank and the fuel line is okay, then there is gas in the engine. A problem is said to be decidable if we can always construct a corresponding algorithm that can answer the problem correctly. Examples qall men are mortal qsocrates is a man qsocrates is mortal qall purple mushrooms are poisonous. 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.
We present a new, cdclbased approach for automated theorem proving in coherent logic an expressive semidecidable fragment of firstorder logic that provides potential for obtaining human. Firstorder logic propositional logic only deals with facts, statements that may or may not be true of the world, e. If there is power to the plugs and the plugs are clean, a good spark is. However, if we work only with clauses is it correct to say that, due to the soundness and completeness of resolution principle, we achieved decidability. Outline n reducing firstorder inference to propositional inference n generalized modus ponens n forward chaining n.
There is a single most general unifier mgu that is unique up. We can intuitively understand decidable problems by considering a simple example. Description logic is essentially a decidable fragment of firstorder logic, reformulated in a manner that is good for talking about classes of entity and. Sentence validity according to first order logic semantics is semidecidable. On a positive note, we show that validity is semi decidable or recursively. Firstorder logic commits to the existence of objects and relations between objects an agent can use firstorder logic for reasoning about world states and for deciding on actions firstorder sentences are made of terms, predicates, functions, quantifiers and connectives firstorder logic is. Informatics 2d efficiency of forward chaining i incremental forward chaining. What i dont understand are the profound reasons for which the first order logic is semi decidable. Inference in firstorder logic colorado state university. P will accept and return a proof for some sentence s if s is valid. Firstorder logic introduction universidade do minho. We rst showed that if is semi decidable, then cn is semi decidable. The satis ability problem is not semi decidable, but some classes of clause sets are identi ed for which the proposed procedure is complete andor terminating.
I 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. Similarly for satis ability 2 22 semantic argument proof. I studied the concepts of decidability, semi decidability and undecidability. First order logic predicate calculus cps 270 ronald parr first order logic propositional logic is very restrictive cant make global statements about objects in the world tends to have very large kbs first order logic is more expressive relations, quantification, functions more expensive first order syntax sentences. The decision problem of validity in firstorder logic is semidecidable. Using first order logic the rest of chapter 8 is boring, so we will skip though good practice for logic representation we will go ahead into ch. Decidable and undecidable problems in theory of computation. First order logic commits to the existence of objects and relations between objects an agent can use first order logic for reasoning about world states and for deciding on actions first order sentences are made of terms, predicates, functions, quantifiers and connectives first order logic is complete but semi decidable. But that means todays subject matter is firstorder logic, which is extending propositional logic so that we can talk about things. Corollary 1 suppose is a semi decidable set of sentences of an e ectively presented rst order language. Decidability implies semi decidability, so the language of. Decidable fragments of firstorder logic and of firstorder linear. However, you cannot prove that a given formula is not a tautol. Entailment in firstorder logic is only semidecidable.
Why doesnt completeness imply decidability for first order logic. If b is arrived at, then a implies b in every interpretation. For example, the set of logical validities v of firstorder logic is semidecidable, but not decidable. Let us call a formal language semi decidable if there is a computable algorithm that, given a formula of the language as input, will output \yes if the formula is valid and will not say \yes and perhaps will not even halt otherwise. Completeness and decidability results for firstorder clauses with indices. In computability theory and computational complexity theory, an undecidable problem is a decision problem for which it is proved to be impossible to construct an algorithm that always leads to a correct yesorno answer. I studied the concepts of decidability, semidecidability and undecidability. Undecidability of validity and satisfiability 1 semidecidability 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. Herbrand universe theherbrand universe tfof a closed formula f in skolem form. Situation calculus in firstorder logic, once a statement is shown to be true, it remains true forever.
In this case, it is because there is no effective method for determining for an arbitrary formula a whether a is not in v. With functions, entailment is not semidecidable and satis. Expanding statespace search to reason about uncertainty. Logic and proof hilary 2016 undecidability of validity and satis ability james worrell in this lecture we consider the computational problems of determining validity and satis ability of rstorder formulas. Resolution is semidecidable puts burder on user to be decidable and tractable thus far.
Abdelkader kersani and nicolas peltier university of grenoble lig, cnrs abstract. Logic and reasoning indian institute of technology kharagpur. When we speak of the decision problem in this module, we mean this particular problem. Let us call a formal language semidecidable if there is a computable algorithm that, given a formula of the language as input, will output \yes if the formula is valid and will not say \yes and perhaps will not even halt otherwise.
Represent the following statements in first order logic. Each function or predicate symbol comes with an arity, which is natural number. In summary, the decidable first order logic presented here forms an important first step toward building decidable seman ticallymotivated kr systems. Firstorder logic godels completeness theorem showed that a proof procedure exists but none was demonstrated until robinsons 1965 resolution algorithm. For first order logic, any sentence that is entailed by another set of sentences can be proved from that set godel did not suggest a proof procedure in 1965 robinson published his resolution algorithm entailment in first order logic is semi decidable, that is, we can show that. Inference in firstorder logic chapter 9 1 2 outline reducing firstorder inference to propositional inference unification generalized modus ponens. For modal predicate logic, constant domains and rigid terms are assumed. First order logic clauses decidability mathematics stack. Undecidability of firstorder logic computer science new. Compact propositional encoding of firstorder theories. Completeness and decidability results for firstorder clauses. Besides classical propositional logic and firstorder predicate logic with functions, but without identity, a few normal modal logics are supported.
If something is intelligent, it has common sense deep blue does not have common sense. The expressive power of the considered logic is strictly greater than the one considered in our previous work. We show that both validity and satis ability are undecidable. For firstorder logic, any sentence that is entailed by another set of sentences can be proved from that set godel. Corollary 1 suppose is a semidecidable set of sentences of an e ectively presented rstorder language.
Apr, 2020 using first order logic the rest of chapter 8 is boring, so we will skip though good practice for logic representation we will go ahead into ch. We rst showed that if is semidecidable, then cn is semidecidable. Completeness and decidability results for firstorder. Inference in firstorder logic russell and norvig chapter 9.
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