Minimal Proof Search for Modal Logic K Model Checking

Most modal logics such as S5, LTL, or ATL are extensions of Modal Logic K. While the model checking problems for LTL and to a lesser extent ATL have been very active research areas for the past decades, the model checking problem for the more basic Multi-agent Modal Logic K (MMLK) has important applications as a formal framework for perfect information multi-player games on its own. We present Minimal Proof Search (MPS), an effort number based algorithm solving the model checking problem for MMLK. We prove two important properties for MPS beyond its correctness. The (dis)proof exhibited by MPS is of minimal cost for a general definition of cost, and MPS is an optimal algorithm for finding (dis)proofs of minimal cost. Optimality means that any comparable algorithm either needs to explore a bigger or equal state space than MPS, or is not guaranteed to find a (dis)proof of minimal cost on every input. As such, our work relates to A* and AO* in heuristic search, to Proof Number Search and DFPN+ in two-player games, and to counterexample minimization in software model checking.Comment: Extended version of the JELIA 2012 paper with the same titl

Algoritmi određivanja Nashove ravnoteže u teoriji igara

U radu je objašnjeno kako pronaći Nashovu ravnotežu i subgame-perfect ravnotežu u igrama s dva igrača. Opisan je zapis igara u normalnoj i proširenoj formi te pronalazak dominantnih strategija u svakoj od formi. U četvrtom poglavlju su objašnjeni neki algoritmi pomoću kojih pronalazimo Nashovu ravnotežu u igrama s dva igrača. Pomoću tih algoritama računalo donosi odluku koju strategiju će odabrati. Poseban fokus smo stavili na algoritam $\alpha-\beta$ podrezivanje. Taj algoritam je korišten za razvoj softvera u igri križić-kružić koji je priložen uz rad. Radi ograničenih računalnih resursa, efikasnost računala u pronalasku svoje najbolje strategije u nekoj igri ovisi o veličini stabla pretraživanja. Stablo pretraživanja je veće ukoliko je broj mogućih poteza u igri veći. Na primjer, igra križić kružić ima manji broj poteza od igre šah, pa je njeno stablo pretraživanja mnogo manje nego stablo pretraživanja u šahu. Što je stablo pretraživanja manje to će računalo točnije odrediti svoju najbolju strategiju i samim tim čovjeku će biti teže pobijediti računalo u nekoj igri.In this thesis, the finding of Nash equilibrium and subgame-perfect equilibrium was explained for two-player games. We considered normal-form and extended-form game representation (and their respective dominant strategies). Algorithms for finding Nash equilibrium in two-player games, as explained in the fourth chapter, are the base for optimal computer strategy decision making. The focus of this dissertation is the $\alpha-\beta$ pruning algorithm used for the making of the tic-tac-toe game attached to the thesis. The efficiency of finding optimal game strategy is greatly affected by limited computer resources. Therefore it is also closely affiliated with the size of the game search tree. The greater the game search tree, so grows the number of possible outcomes. Therefore, the smaller the search tree, the computer accuracy of finding the optimal strategy grows, and so does the difficulty of defeating the computer

Algoritmi određivanja Nashove ravnoteže u teoriji igara

U radu je objašnjeno kako pronaći Nashovu ravnotežu i subgame-perfect ravnotežu u igrama s dva igrača. Opisan je zapis igara u normalnoj i proširenoj formi te pronalazak dominantnih strategija u svakoj od formi. U četvrtom poglavlju su objašnjeni neki algoritmi pomoću kojih pronalazimo Nashovu ravnotežu u igrama s dva igrača. Pomoću tih algoritama računalo donosi odluku koju strategiju će odabrati. Poseban fokus smo stavili na algoritam $\alpha-\beta$ podrezivanje. Taj algoritam je korišten za razvoj softvera u igri križić-kružić koji je priložen uz rad. Radi ograničenih računalnih resursa, efikasnost računala u pronalasku svoje najbolje strategije u nekoj igri ovisi o veličini stabla pretraživanja. Stablo pretraživanja je veće ukoliko je broj mogućih poteza u igri veći. Na primjer, igra križić kružić ima manji broj poteza od igre šah, pa je njeno stablo pretraživanja mnogo manje nego stablo pretraživanja u šahu. Što je stablo pretraživanja manje to će računalo točnije odrediti svoju najbolju strategiju i samim tim čovjeku će biti teže pobijediti računalo u nekoj igri.In this thesis, the finding of Nash equilibrium and subgame-perfect equilibrium was explained for two-player games. We considered normal-form and extended-form game representation (and their respective dominant strategies). Algorithms for finding Nash equilibrium in two-player games, as explained in the fourth chapter, are the base for optimal computer strategy decision making. The focus of this dissertation is the $\alpha-\beta$ pruning algorithm used for the making of the tic-tac-toe game attached to the thesis. The efficiency of finding optimal game strategy is greatly affected by limited computer resources. Therefore it is also closely affiliated with the size of the game search tree. The greater the game search tree, so grows the number of possible outcomes. Therefore, the smaller the search tree, the computer accuracy of finding the optimal strategy grows, and so does the difficulty of defeating the computer

Algoritmi određivanja Nashove ravnoteže u teoriji igara

U radu je objašnjeno kako pronaći Nashovu ravnotežu i subgame-perfect ravnotežu u igrama s dva igrača. Opisan je zapis igara u normalnoj i proširenoj formi te pronalazak dominantnih strategija u svakoj od formi. U četvrtom poglavlju su objašnjeni neki algoritmi pomoću kojih pronalazimo Nashovu ravnotežu u igrama s dva igrača. Pomoću tih algoritama računalo donosi odluku koju strategiju će odabrati. Poseban fokus smo stavili na algoritam $\alpha-\beta$ podrezivanje. Taj algoritam je korišten za razvoj softvera u igri križić-kružić koji je priložen uz rad. Radi ograničenih računalnih resursa, efikasnost računala u pronalasku svoje najbolje strategije u nekoj igri ovisi o veličini stabla pretraživanja. Stablo pretraživanja je veće ukoliko je broj mogućih poteza u igri veći. Na primjer, igra križić kružić ima manji broj poteza od igre šah, pa je njeno stablo pretraživanja mnogo manje nego stablo pretraživanja u šahu. Što je stablo pretraživanja manje to će računalo točnije odrediti svoju najbolju strategiju i samim tim čovjeku će biti teže pobijediti računalo u nekoj igri.In this thesis, the finding of Nash equilibrium and subgame-perfect equilibrium was explained for two-player games. We considered normal-form and extended-form game representation (and their respective dominant strategies). Algorithms for finding Nash equilibrium in two-player games, as explained in the fourth chapter, are the base for optimal computer strategy decision making. The focus of this dissertation is the $\alpha-\beta$ pruning algorithm used for the making of the tic-tac-toe game attached to the thesis. The efficiency of finding optimal game strategy is greatly affected by limited computer resources. Therefore it is also closely affiliated with the size of the game search tree. The greater the game search tree, so grows the number of possible outcomes. Therefore, the smaller the search tree, the computer accuracy of finding the optimal strategy grows, and so does the difficulty of defeating the computer