28CHUNCK 3. SJÁLFVIRKAR YTINGARVELAR OG TURING VÉLAR

b"n0a"b'2n)n0.APD Aneedstodetermine, af terprocessingaseriesof'a's, ifitsh deterministicP D Acandiver geintotworoutes : onetryingtomatcha"b" andtheothere deterministicrouteswillsucceed.T hisideaisdeeplyexaminedinacademicliteratur el 1608.02640], whichclari fiestheconstraintso f DP DAsandtherequirementofnon— determinism f orthecompleteseto fC F Ls.T hepracticalconsequenceo fthisnon— determinismisthat, althoughallC F Lscanbepar sed, noteveryonecanbepar sede f fic down(e.g., LL(k) )orbottom—up(e.g., LR(k))parsingmethodswithoutalterationsor free, ortheapplicationo f moreintricateparsingmethods f orambiguousornon— deterministiccontext— freelanguages.Ontheotherhand, aTuringM achine(T M )e purposecomputation.Itisof ficiallycharacterizedasa7—tuple, MTM = (Q,,,,q0, B, F).Inthis formulation, Qrepresentsafinitesetof statessimilartothel 0 represents the starting condition. BT > signifies the distinct blank symbol that occupies all endless sections of the tape not used the input, establishing a definitive limit for the input string and an unlimited workspace. F represents the collection of final states, where entering signifies the acceptance of the input string. The primary feature of a TM lies in its memory structure: an unbounded, theoretically endless tape that allows random access, facilitating operations for reading, writing, and two-way movement of the tape head. In contrast to the stack’s LIFO limitation, the tape head of the TM can traverse left or right across any segment of the tape, enabling information to be stored, fetched, and altered at any location. The ability for random access, combined with the tape’s limitless capacity, is the core reason for the TM's remarkable strength, allowing it to emulate any other computational model and execute any imaginable algorithm. A TM’s configuration is typically expressed as (q, 1 X 2), which clearly indicates the current state q, the complete tape content 1X 2 (where 1 represents the non-blank content to the left of the head, X is the symbol beneath the head, and 2 signifies the non-blank content to the right), while the head is situated directly above the symbol X. The underscore underneath X distinctly marks the location of the head. The transition function connects QxI to QxIxL,R, indicating that

29

a computational step depends on the current state and the symbol currently beneath the tape head. This action leads to a change to a different state, the inscription of a new symbol on the tape at the current head location, and a directional shift of the tape head (Left or Right). For example, if (q 1 ,’a’)=(q

2 ;b',R), then in state q 1 with ’a’ under the head, the TM transitions to state q 2 , writes 'b' over ’a’, and shifts its head one position to the right. The yield relation for a TM, explaining a single transition, can be expressed as follows: (q, 1X 2) TM(q , 1Y 2 )if (q,X)=(q ,Y,R)and 2 is 2 with the first symbol shifted to 1 (q, 1X 2) TM(q „1 Y 2)if (q,X)=(q ,Y,L) and 1 is 1 with the last symbol shifted to 2 The reflexive and transitive closure of this relation, TM x , facilitates the description of a complete computation, from the starting configuration to a halting (accepting or rejecting) configuration. Turing Machines’ computational power clearly exceeds that of Pushdown Automata to a significant extent. TMs can identify recursively enumerable languages (RELs), referred to as Turingrecognizable languages. This broad category of languages includes all context-free languages and also covers a wide range of others, particularly context-sensitive languages (CSLs) and even issues that are undecidable more straightforward models, like the Halting Problem. The connection among these different language classes is marked a definitive hierarchy, commonly known as the Chomsky Hierarchy: L Regular L CFL L CSL L REL. This means that all regular languages are also context-free, all context-free languages are context-sensitive, and all context-sensitive languages are recursively enumerable, with each inclusion being proper. For instance, L Regular comprises languages such as a x b * , L CFL contains an bn, L CSL consists of a nbncn, and L REL encompasses the collection of all valid.

Descriptions of Turing Machines. Moreover, the esteemed ChurchTuring Thesis asserts that any problem suitable for an algorithmic solution can, in theory, be addressed a Turing Machine, thus position-

32CHUNCK 4. KYNNING A NIDURDREPANDI UPPTEKNUM BEVER

1. The field of theoretical computer science is significantly influenced the examination of automata, which are abstract computational models offering a solid foundation for grasping the boundaries and functionalities of computation. From the most basic Finite State Automata (FSA) to the universally capable Turing Machines (TMs), every model presents unique memory mechanisms that determine its computational strength and, importantly, the decidability of its characteristics. This chuck investigates the upper limits of halting times for Pushdown Automata (PDAs), a distinctive class of machines in this hierarchy, presenting and examining two new functions: (PDA) and pudau(n). These functions seek to measure the longest potential computation paths for PDAs, creating an intriguing parallel with and a significant contrast to the established Busy Beaver function for Turing Machines. 1.1 Background in Computability Theory and Automata Computability theory, a fundamental aspect of theoretical computer science and mathematical logic, investigates the essential inquiries regarding what is computable and what is not. At its core is the idea of an algorithm and the structured models that represent it. Automata theory, as a subfield, offers a structured categorization of these computational models according to their memory functions and the types of formal languages they can identify.

of a sequence of characters onto the stack, which can consist of the empty string, thus effectively executing a pure pop. The formal representation of the yield relation for a PDA, illustrating one computational step, is as follows: (q,aw,X ) PDA (q ,w, )if(q ,) (q,a,X). In this case, (q,aw,X ) indicates the configuration before the transition, where q signifies the current state, a is the input symbol (or ) being read, X is the top symbol on the stack, and w and represent the remaining input and stack, respectively. The derived configuration (q ,w, ) reflects the updated state q , the remaining input w after a has been handled, and the stack content , where denotes the string that was added to the stack after X was removed. The reflexive

33

and transitive closure of this relation, PDA *, represents a series of zero or more computational actions, allowing for the examination of the entire processing of an input string. PDAs are widely recognized as the computational framework for context-free languages (CFLs), a category of languages crucial for the accurate representation of programming language syntax, encompassing nested constructs such as if-else statements and function invocations. It’s crucial to understand that non-determinism in PDAs is not just a supplementary trait but a vital attribute that greatly enhances their expressive capability; the category of languages accepted non-deterministic PDAs (NPDAs) is clearly and strictly a superset of those accepted deterministic PDAs (DPDAs). This essential difference is clearly illustrated languages like palindromes (e.g., L=wa,b * w=w R ), the identification of which fundamentally requires non-deterministic decisions for effective handling, since the reversal point in the string cannot be deterministically determined without infinite lookahead. Such an automaton could nondeterministically choose to move from a

At the bottom level of this computational hierarchy are Finite State Automata (FSAs). These devices have a limited number of states and no additional memory. Their processing capability is confined to identifying regular languages, which are defined straightforward, repetitive patterns. The actions of an FSA are completely dictated its present state and the input symbol it processes. Moving up the hierarchy, Pushdown Automata (PDAs) signify an important advancement in computational ability. In automata theory, PDAs are considered one of the essential structures, defined finite-state controllers enhanced with a unique memory component known as a stack. This stack functions based on a First-In, Last-Out (FILO/LIFO) principle, indicating that the most recently added item is the first to be taken out. This unlimited stack memory enables PDAs to identify context-free languages (CFLs), a wider category of languages than regular languages, which includes structures needing memory beyond finite states, such as recur-