Teek predikaatarvutuse väljendamisülesannete lahenduste kontrollimiseks
Date
2017
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Abstract
Käesoleva bakalaureusetöö eesmärk on implementeerida teek predikaatarvutuse väljendamisülesannete lahenduste kontrollimiseks. Teeki kuuluvad meetodid, mis võimaldavad kontrollida, kas sisend, s.t aritmeetika signatuuris σ = <0, 1; +, *; => kujutatud predikaatarvutuse valem, on korrektsel kujul ja kas valem väljendab etteantud predikaati. Valemi süntaktilise korrektsuse hindamiseks kontrollitakse muuhulgas järgnevat: valem sisaldab vaid etteantud signatuuri kuuluvaid konstant-, funktsionaal- ja predikaatsümboleid, valemis esinevad sulud on tasakaalus, lõppvalemis kasutatavad abipredikaadid on eelnevalt defineeritud, abipredikaadi defineerimisel ei kasutata predikaadi tähises vabu muutujaid, mida abipredikaadi valemis ei esine ja vastupidi – abipredikaadi valemis ei esine vabu muutujaid, mida tähises ei ole välja toodud. Selleks, et kontrollida, kas valem väljendab etteantud predikaati, kasutatakse käesolevas töös kahte meetodit. Tõesuspuu meetodiga üritatakse kindlaks teha, kas sisestatud valem ja õige valem on samaväärsed, teisisõnu, kas nende kahe valemi ekvivalents on samaselt tõene. Kui tõesuspuu meetod ei anna määratud aja jooksul kindlat tulemust, võetakse kasutusele meetod, mis kontrollib, kas kahe valemi tõeväärtused langevad kokku suurel lõplikul hulgal. Kuigi valemite samaväärsuse kontroll on algoritmiliselt mittelahenduv, võimaldavad arendatud teegi meetodid anda kindla vastuse laia alamhulga ülesannete korral, tõhustades seega praegust predikaatarvutuse väljendamisülesannete lahenduste kontrollimise töökorraldust Tartu Ülikooli matemaatilise loogika kursustel.
The goal of this bachelor thesis is to implement a library for verifying the solutions to the provided sentences of the predicate calculus. The library is intended to be used in automatizing the examination of students' answers in the predicate calculus courses in Tartu Ülikool. At present, the lab instructors check the answers manually. The implemented library contains methods that verify the user input from two aspects – the input is syntactically correct and it expresses the given sentence. Input is deemed syntactically correct if it contains the constant, functional and predicate symbols that are permitted with the arithmetic signature of σ = <0, 1; +, *; =>, a balanced amount of brackets and the helper predicates defined beforehand. For helper predicates, the program also checks the correct usage of free variables, defined as follows – all free variables used in the predicate are provided as arguments to the predicate and all the predicate's arguments are free variables in the predicate. To check if the user provided formula is equivalent to a correct formula, the program deploys two methods. First, it applies a method called the truth tree, which consists in analysing the logical structure of the equivalence formula, made up of the user input and the correct answer. The second method consists of evaluating the two formulas with a limited set of natural numbers and checking if the formulas' truth values coincide.Verifying the equivalence of two formulas is a problem that is known to be algorithmically unsolvable. The two selected methods complement each other – the truth tree method allows determining the equivalence of two formulas, while the brute-force evaluation allows determining their non-equivalence.
The goal of this bachelor thesis is to implement a library for verifying the solutions to the provided sentences of the predicate calculus. The library is intended to be used in automatizing the examination of students' answers in the predicate calculus courses in Tartu Ülikool. At present, the lab instructors check the answers manually. The implemented library contains methods that verify the user input from two aspects – the input is syntactically correct and it expresses the given sentence. Input is deemed syntactically correct if it contains the constant, functional and predicate symbols that are permitted with the arithmetic signature of σ = <0, 1; +, *; =>, a balanced amount of brackets and the helper predicates defined beforehand. For helper predicates, the program also checks the correct usage of free variables, defined as follows – all free variables used in the predicate are provided as arguments to the predicate and all the predicate's arguments are free variables in the predicate. To check if the user provided formula is equivalent to a correct formula, the program deploys two methods. First, it applies a method called the truth tree, which consists in analysing the logical structure of the equivalence formula, made up of the user input and the correct answer. The second method consists of evaluating the two formulas with a limited set of natural numbers and checking if the formulas' truth values coincide.Verifying the equivalence of two formulas is a problem that is known to be algorithmically unsolvable. The two selected methods complement each other – the truth tree method allows determining the equivalence of two formulas, while the brute-force evaluation allows determining their non-equivalence.