Abstract Recursion and Intrinsic Complexity

Boolean Monadic Recursive Schemes as a Logical Characterization of the Subsequential Functions

“…Boolean monadic recursive schemes are simple programs that operate over such string structures. They are a particular case of the recursive programs of Moschovakis [17]. We briefly review the syntax and semantics of such recursive programs in this particular signature, then impose (Definition 3) the pertinent syntactic restriction to obtain BMRSs.…”

“…In this paper, we define boolean monadic recursive schemes (BMRSs), a restriction on the general notion of a recursive program scheme in the sense of Moschovakis [17]. As indicated by the name, these schemes recursively define a series of unary functions that take as inputs indices from a string and return a boolean value.…”

A Behavioural Theory of Recursive Algorithms

“…The third condition is satisfied, because each local function symbol f of arity n is implicitly turned in a program instance into an (n + 1)-ary function symbol, namely by adding the additional agent as environment parameter for the evaluation of terms with f as leading function symbol. Therefore, each local function of the callee is different from each local function of the caller, and to execute the call rule means to create a new agent c, 30 which is CalledBy the agent self that executes the call, to equip c with the fresh program instance q c and Initialize its state by the values of t i , t ′ j . This makes the callee ready to run and puts the caller into Waiting mode, in the sense defined by Definition 2.11 (except the trivial case that q c is already Terminated when Called so that it will not be executed).…”

“…In [29] Moschovakis raised the question how recursive algorithms like the well-known mergesort are covered. He questions that algorithms can be adequately defined by machines (see also [30]). Although the perception that Gurevich used seq-ASMs as a definition for the general notion of algorithm-not only for its sequential instance-is a misinterpretation, unfortunately the response by Blass and Gurevich [6] to Moschovakis's criticism does not clarify the issue in a convincing way.…”

“…Instead of admitting that an extended behavioural theory for recursive algorithms still needs to be developed, distributed ASMs with a semantics defined through partial-order runs [22] are claimed to be sufficient to capture recursive algorithms. 7 As Börger and Bolognesi point out in their contribution to the debate [10], a much simpler extension of seq-ASMs suffices for the specification of algorithms of the mergesort kind, which define recursive functions (as do Moschovakis' systems of recursive equations called 'recursive programs' [30], see the discussion in Sect.6). Furthermore, the response by Blass and Gurevich blurs the subtle distinction between the axiomatic definition and the possibility to express any algorithm on an arbitrary level of abstraction by an abstract machine.…”